TORONTO (IDN) - The Global Environment Facility (GEF) expects the twenty-first meeting of the Conference of the Parties (COP21) to the UN Framework Convention on Climate Change (UNFCCC) – from November 30 to December 11 in Paris – to be a "turning point".
It expects COP21 to send "a loud and clear signal to citizens, markets and the private sector that the transformation of the global economy is inevitable, beneficial, and already underway".
Governments from around the world are meeting in Paris to reach a new climate change agreement that aims to keep global average temperature rise below 2 degrees Celsius (3.6 degrees Fahrenheit) – the level beyond which there will be irreversible impacts.
The GEF refers to the Intergovernmental Panel on Climate Change (IPCC), which has made clear that "the longer we delay in tackling climate change, the higher the risks and costs". In view of this, the next couple of decades would be critical.
The Global Environment Facility was established on the eve of the 1992 Rio Earth Summit, to help tackle planet Earth's most pressing environmental problems. Since then, the GEF has provided over $14 billion in grants and mobilized in excess of $70 billion in additional financing for more than 4,000 projects. The GEF has become an international partnership of 183 countries, international institutions, civil society organizations, and private sector to address global environmental issues.
The GEF believes that halting the growth in global emissions and putting them on a downward path so as to prevent climate disruption is possible. But it necessitates a transformation of the global economy that not only addresses climate change, but also powers new growth.
"Shifting to a low-carbon and resilient trajectory will require coordinated, integrated solutions to catalyze transformation of three key economic systems: energy – how we power our homes, offices and industry, and move goods and people from one place to another; cities – how we live; and food production – how and where we produce food, and what we eat," Ishii continued.
Ishi said that, based on GEF's quarter century of experience and a wide network of partners, it is well-placed to support transformation.
The GEF's commitment to address climate change issues is unequivocal, said Ishi. In 2013-14, it committed a total $1.4 billion for adaptation and mitigation action. By the end of the current funding cycle in June 2018, it is estimated that the GEF will be making about $3 billion available to developing countries to help address climate change, with the potential of $25 billion to be leveraged from other sources.
As a financial mechanism of the UNFCCC, the GEF is supporting developing countries' shift towards a low-emission development path. Besides its ongoing support to countries in their UNFCCC obligations (such as national communications and biennial update reports), the GEF has also provided financial support to 46 countries as they prepared their Intended Nationally Determined Contributions, and stands ready to help make these "investment plans" operational.
In Paris, the GEF will also be actively supporting the 12 action tracks of the so-called Lima Paris Action Agenda to showcase coalitions, partnerships and integrated approaches for action on the ground in areas such as buildings, forests, transport and private finance.
Governments, the private sector and civil society are taking action on climate change because it is in their interest to do so, GEF said in a press release. Ahead of Paris, almost all countries, as well as states, cities, business and investors, have come forward with the most comprehensive set of pledges and plans ever seen to reduce emissions and bolster resilience, it added.
While Paris is a huge opportunity to demonstrate political ambition and action on climate change, according to GEF, what happens after is as, if not more, important.
The GEF was established in October 1991 as a $1 billion pilot program in the World Bank to assist in the protection of the global environment and to promote environmental sustainable development. The GEF would provide new and additional grants and concessional funding to cover the "incremental" or additional costs associated with transforming a project with national benefits into one with global environmental benefits.
The United Nations Development Programme, the United Nations Environment Program, and the World Bank were the three initial partners implementing GEF projects.
In 1994, the GEF was restructured and moved out of the World Bank system to become a permanent, separate institution. The decision to make the GEF an independent organization enhanced the involvement of developing countries in the decision-making process and in implementation of the projects. Since 1994, however, the World Bank has served as the Trustee of the GEF Trust Fund and provided administrative services.
As part of the restructuring, the GEF was entrusted to become the financial mechanism for both the UN Convention on Biological Diversity and the UN Framework Convention on Climate Change. In partnership with the Montreal Protocol of the Vienna Convention on Ozone Layer Depleting Substances, the GEF started funding projects that enable the Russian Federation and nations in Eastern Europe and Central Asia to phase out their use of ozone-destroying chemicals.


text:
Vanessa Riley's Regency Reflections
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Retro Read: The Country Gentleman by Fiona Hill
Ann Guilfoyleis a wealthy and independent young woman in Regency England, with her life planned before her. She opens her drawing room to what she considers the creme de la creme of thinking people and she intends to marry the exact right man. Then financial tragedy strikes and she finds herself 200 miles from London trying to settle herself into country life, a fate worse than death for a woman who considers herself sophisticated and intellectual. She thinks she can only mock thegentleman farmer Mr. Highet and his "gargantuan" mother. In short, Ann is a snob who thinks this country gentleman beneath her, yet when a different tragedy strikes, this one of the heart, she accepts his offer of help and her attitudes and heart begin to change.
This is not a story full of suspenseful, page-turning moments. The pace is almost as leisurely as the country life about which Hill writes. Yet the way in which Ann grows as a woman is so heart-warming, along with the love story, I have always listed this book among of my top fifty Regencies and worth the reread from time to time.
You will likely only find this sbook in llibraries or used bookstores. I've seen it for as little as $.01, which is a pity, and it is a quarter of a century old, definitely among the clean, sweet traditional Regencies of old, with an author who is a master character-driven storyteller.
Laurie Alice Shares A Secret
Laurie Alice here,
This month, I am writing about the first book I ever read by Jo Beverley, a book that got me hooked on Jo as an author.
The Stanforth Secrets.
Stanforth Secrets
Though the widowed Chloe Stanforth loves her house by the sea, a series of puzzling incidents has left her unsettled and anxious to find a new home. Her situation is complicated by the arrival of her husband's cousin, for whom she has long harbored a deep and guilty attraction.
Back from the war, Justin Delamere hopes he can finally woo Chloe, until he suspects her guilty of treason-and murder. Can he trust the woman he's always desired, and can Chloe surrender her most private secrets to the man who controls her freedom and now her heart?
As a long-time mystery lover, the notion of a Regency mystery thrilled me, and this book did not disappoint. I loved all the odd clues and details the author used, so Regency and so country-house mystery. She captures both genres well in this lovely little book.
A couple of months after I read this book, I met Jo Beverley. She had long since moved on from writing traditional Regencies. Indeed, Walker Publishing had stopped publishing Regencies by this time. I told Jo how much I enjoyed the book. After thanking me in her gracious way, she then told me she would never write such a book again. She said having to worry about everyone in the house was too much trouble. "You have to think about all the servants." Spoken in her lovely Lancastrian accent, "servants" came out almost three syllables with the barest hint of an r on the first emphasized syllable.
The author charmed me nearly as much as that first book and most of her books since.
Although The Stanforth Secrets was published in 1989, it is now available on Amazon. Highly recommended to those who want a clean, traditional Regency romance along with a fun mystery.
Originally posted 9/28/2014.
Marriage A Glimpse Of Heaven Or Hell
Vanessa here,
Glad you can join me here, today. Well, the porch at the Regency Reflections Blog now possesses new paint, a bit of a makeover. We've been posting here since 2012. We love being able to showcase different glimpses of Inspirational Regencies, talking about the stories and the motivations behind them. We've even given tastes of the Regency romances that hooked us long before the first traditionally printed Christian Regency was released.
But our fire had grown cold.
It was time find our love again.
The reason I write Regencies is because I found my voice in the 1800's. It sounds of a woman, with dreams of a happy-ever-after, challenged by the circumstances, the very skin she's born within. These stories, gifted by my first love, a passionate, merciful God must be told. It is my first love. And this blog will now share stories of authors and characters who possess the same fire.
Now some of my friend's stories may be secular authors. Before you throw holy water at me, I just have to say it. Not everyone is meant to entertain the pews. All types of stories are needed to edify, entertain, and to educate. All of my friends, regardless of what they write use their God-given talents to bring joy and hope into this world. This is something all should see.
The Fabulous Jude Knight
My first guest is the wonderful Jude Knight. Jude traveled all the way from New Zealand to have tea with me on my Atlanta porch. Jude writes strong determined heroines, heroes who can appreciate a clever capable woman, villains you'll love to loathe, and all with a leavening of humor.
"Jude, I know you must be tired, so have a seat. Let me fix you some tea. How would you like it?"
"Thank you, so much for your hospitality. I'd like a green tea with a slice of lemon, or black tea with a small dash of milk."
"Green it is. Here you go. It's quite hot. While it cools, tell the good readers what a happy ever after means to you."
"My view of 'happy ever after' is shaped by my life and my beliefs. Falling in love is not enough. A wedding is not enough. Good intentions are not enough. To believe that a romance has a happy ending, I need to believe that the couple's love will last for a lifetime; that they have what they need to work out the inevitable problems that will try to tear them apart."
"I don't mean to get you kicked out of any 'ABA' bad girl societies, but tell me what love and faith means to you."
"My beloved and I have known one another for 47 years, and been married for nearly 44. In that time, our love has been tested over and over, but each challenge we've surmounted has made our relationship stronger.
I joke that our marriage has survived because his parents and mine were both against it, and we were too stubborn to admit they were right. He always adds that it also survived because we lived at the end of a long country road and were very poor. Walking out on the marriage would have meant a literal walk — and it was a long way."
"I love that. Please continue."
"More to the point, though, we both believed that we had to work things out. Our Christian faith told us that marriage was a permanent commitment. We promised 'as long as we both shall live', and we meant it. And we both came from fractured marriages; we knew what disharmony did to children. We were determined to find our 'happy ever after', and we did."
"Tell me how your beliefs have shaped your writing."
"My husband is a Catholic, and I converted to Catholicism some eight or nine years after we first met. Catholic marriage theology holds that marriage is a sacrament — a visible sign of the presence of God in the world. Just as water is the sign of Baptism, and the bread and wine are the signs of the Eucharist (Holy Communion), so the man and the woman are the sign of Marriage. Water signifies (and becomes) the cleansing grace of God. Bread and wine signify (and become) the presence of Christ in the church community and each individual. The couple signify something very wonderful: Christ's union with the Church, God's union with His creation. A person could spend a lifetime thinking about the implications of this, and some people have."
"Wow. That's deep. I'm Baptist, and I get it. More so, I feel the same."
"For today, suffice it to say that building the kind of marriage that is a true signifier of this mystery is not a magic trick taking place in front of the altar on a couple's wedding day. It is the work of a lifetime together."
"Ok, tell the good folks about A Baron for Becky."
"A Baron for Becky is my Regency about marriage, which is why it is a book of two halves. In the first part of the book, my heroine — rescued from dreadful danger — becomes the mistress of a kindly libertine whose view of marriage is extremely jaundiced. Their relationship is founded on lust and convenience on his part, and gratitude on hers."
"Did I mention to you to tell the PG version for Regency Reflection readers? Just kidding. So in a A Baron for Becky, the heroine makes wrong choices, but that didn't disqualify her from finding true love. Now that is a message for today."
"Yes. The second half of the book is about just that between, Becky and Hugh. The libertine arranges their marriage, which takes place at the midpoint of the book. But Becky and Hugh build that marriage. I poured my heart into showing them falling in love; showing how their past experiences almost destroyed them; showing the slow painful process of rebuilding."
"He was sorry for hurting her, for not trusting her, for manipulating her into marriage, for being a representative of the men that had hurt her. He was sorry for it all, and he could never make it up to her. But he would live his life trying."
"Dear hubby and I just made 19 years. An accomplishment in this age, but tell us your 44-year secret."
"The trick of a happy marriage (and a happy life) is to go on loving one another between trials, and to consistently fall in love with the same person, over and over and over. Because love is not about being in love, pleasant though that state is. Love is an action, not a state, not a feeling. Love is making breakfast for the person you want to brain with the frypan. Love is listening to the same joke for the twentieth time and laughing yet again. Love is walking hand in hand for no better reason than that you are fond of one another. Love involves feelings: lust and affection, familiarity and friendship, mutual respect and regard. It grows on shared experiences, memories—both good and bad—of the things you've lived through together. But above all, love is what you do when your feelings prompt you against being loving.
How could it be otherwise when love is an echo of Love Himself, the One who loved His people even though they betrayed him, rejected him, and even killed Him? Love is far too important to depend on chemical soup. Love is an action."
Now that is a message to kick off the new Regency Reflections. Thank you Jude. While my friend takes a swig of her green tea, I just want to thank her for traveling to Atlanta and being my guest. Below is more about Jude and links for A Baron for Becky.
About Jude Knight
Jude Knight is the pen name of Judy Knighton. After a career in commercial writing, editing, and publishing, Jude is returning to her first love, fiction. Her novella, Candle's Christmas Chair, was released in December 2014, and is in the top ten on several Amazon bestseller lists in the US and UK. Her first novel Farewell to Kindness, was released on 1 April, and is first in a series: The Golden Redepennings.
Follow Jude on social media:
Visit Jude's Website http://judeknightauthor.com/
Like Jude on Facebook https://www.facebook.com/JudeKnightAuthor
Amazon http://amzn.to/1C3hFNl
Amazon UK http://amzn.to/1H3YmTk
Amazon Aus http://bit.ly/1HzUZ9R
Smashwords http://bit.ly/1HzUXPf
Barnes & Noble http://bit.ly/1GRTvkR
iBooks http://apple.co/1FVFNfU
Kobo http://bit.ly/1NzI2LK
Barbara Cartland's "Curse of the Clan" Set in 1822
Dame Barbara Cartland wrote over 723 books. Known for setting her novels in the Victorian era, she was an exceptionally popular novelist, peaking in the 1970s. (I remember my mother reading her novels.) After recently picking up two Cartland books at a used book sale, I was pleasantly surprised to find The Curse of the Clan (published in 1977) to be quite satisfying.
Imagine my further delight that the novel is set in 1822. Set in the late "Regency" to be sure, the tale follows an orphan who is elevated to the title of Scottish Duchess. Her fearsome, yet handsome husband marries her to gain revenge upon a neighboring clan who foisted an adulterous, now-dead, wife upon him.
The story boomed along with vivid action and upon reflection, would make an excellent movie, if historical films were popular. The scenes at the orphanage, a carriage accident (which affected the plot), a shooting attack, revelation of her true parentage, then the winning over of the husband…all would make for a delightful, picturesque movie.
I got a real kick out of finally trying a Cartland book, and wouldn't hesitate to read more –especially if I can ferret out which were set in the early 1800s.
Have you read any of Barbara Cartland's books? What do you think?
To Ruin a Lady is Quite Fun
Helloooo, thank you to the Regency authors for letting me join your lovely group. My first Regency is releasing from Love Inspired Historical in September and it was a hoot to write. So much so that I'm at work on another.
A great regency involves just a bit of ruination…at least for the heroine in my current manuscript. But how canI go about ruining her so that she's forced into a marriage of convenience?
As someone who has enjoyed regency romances since I was a teen, I still have so much to learn. I needed a ruination that was palatable for a Christian audience but still severe enough to force my heroine into the arms of my delectable yet decidedly anti-marriage hero. So the first place I looked was, of course, Google. My dear friend Google.
The search yielded many interesting titles but no specific reasons on why or how a lady could be ruined. I really wanted something concrete. Something unarguable. But that showed my misunderstanding of the Regency period. Thank goodness for writing friends! A dear author friend named who has been writing regencies for years informed me that all it could take is some gossip to rip my heroine's reputation into tatters.
And so I got to work in creating ruination. After all, my hero and heroine belong together, even if they don't know it yet!
Has gossip ever hurt you or ruined your reputation? How did you recover?
What in the Regency World is a Round Gown?
Susan Karsten here. I love historic costumes, but am by no means an expert, even though I took the subject in college. If you are at all like me (Regency fiction reader/fanatic), you've come across the archaic and forgotten term "round gown". Again, if you are like me, you will take a mental guess what that might be, and move on, flipping pages as fast as you can read them.
To the best of my research, the round gown appears to be a pre-Regency style that hung on, or was used for day-wear even as fashion moved to a different silhouette. Marie Antoinette is said to have inspired the round gown, then a dress and robe joined together and tied in the front Later came Josephine Bonaparte who ushered in the slim, high-waisted, gossamer thin chemise dress of the early 19th Century, that we think of first when we think of Regency dresses.
Back to the round gown, the Empire gown's precursor. The round gown had a soft, round skirt silhouette, with full gatherings at a slightly raised waist, a train, and straight, elbow-length sleeves. The round gown's train, which was common for a short time for day wear and lasted until 1805-06 for the evening, would be pinned up for the dance, as mentioned in Austen's Northanger Abbey. One shudders at the impracticality of these long white muslin dresses in England, a country renowned for wet weather and muddy roads.
So, when you encounter the term "round gown" in your favorite Regency fiction, think of probably a day dress, kept for wearing at home, and more modest than their evening counterparts. An earlier silhouette, and not in the first stare of fashion.
I so hope some of you will add to this description with more yummy details about the mysterious round gown.
Reminiscing about the traditional Regency novel
Hi, readers! Susan here with another blast from the past — 1987 to be exact. I expect many of you inspirational Regency-lovers are like me…you loved the older, clean Regencies that were so readily available a few decades ago, published by Signet (my faves), Zebra and the like.
In fact, my efforts toward a fiction-writing career began with a desire to try my hand at writing one of these thrilling, yet clean, romances…with a dash or more of the Christian faith included as a character developing element…sometimes even as a plot twist or a conflict-causing, stake-raising factor.
So today, I am bringing you a review of an old favorite, Mary Jo Putney's "The Diabolical Baron."
Don't let the title throw you, the book is a charming tale of true love, the twists and turns and the happily ever after. With two attractive suitors trying to lay claim to her heart and a father insisting she marry for a fortune, she has deep waters to navigate all the while trying to protect her beloved sister.
If you can find this title, I believe it might be one of your favorites too — though it is not a true inspirational romance. My hopes are that the Regency genre will grow in popularity again, with Christian writers bringing it to the fore.
Charming Quotes from Jane Austen
Hello, my Regency-loving friends. Interesting, isn't it, that the actual Regency last only from 1811-1820, but the periods before and after can also be considered part of the era? When trying to explain to the uninitiated, what the Regency is, I'll often bring up Jane Austen. I find that most, but not all, have heard of the book Pride and Prejudice and they can get a grasp on what kind of fiction genre I am writing.
So, to bring Jane Austen alive again, in our minds only, I bring you some of her delicious quotes.
What dreadful hot weather we have! It keeps me in a continual state of inelegance.
"A lady's imagination is very rapid; it jumps from admiration to love, from love to matrimony in a moment."
"There is nothing like staying at home for real comfort."
"The person, be it gentleman or lady, who has not pleasure in a good novel, must be intolerably stupid."
Based on these quotes alone, I do believe I would have enjoyed know Jane Austen.
Which author would you have liked to spend time with? Answer in the comments, please.
Blast from the Past: Marion Chesney's Regency novels
Hi all, Susan Karsten here!
…Back from an absence of about four months (that pesky tax job). Since I enjoy Camy's posts on older regency books so much, I am bringing you info about a book, and its author, and telling you about her extensive and delicious back-list of regency reading fun (over 90 titles). If the author Marion Chesney is not familiar to you — get thee to a bookstore — or library in this case — since she isn't (boo-hoo) writing regencies anymore.
No, she now only writes fabulously popular cozy mysteries now and you may know her as M.C. Beaton. However, her regencies are GREAT, and with some digging, are still available to the avid fan. She's got some of her backlist out as e-books lately, too.
Chesney's debut (writing under her own name) book, which I happen to own, is "The Poor Relation." Heroine and former debutante Amaryllis Duvane's fortunes have sunk low and she is reduced to the status of serving her wealthier relatives. Her past love, the Marquess of Merechester, shows up to court one of these wicked stepsister types, and the drama begins.
I'll happily admit to being a huge fan of Chesney, in all her genres. But the chance to read one of her first efforts makes me admire her career trajectory even more. As one familiarizes oneself with her work, it's clear that as she gained publishing popularity and confidence, more and more of Chesney's delicious humor comes out on the page. I can only hope to instigate half as many snickers for my own readers…someday…when I make my debut!
If you've ever enjoyed Chesney's regencies, please add a comment.
Susan Karsten
Poll: How to do you find new Regencies to read?
Camy here! I was having a discussion with a friend about how Regency romance lovers find new Regencies to read.
I usually do it by word of mouth—recommendations from readers on a Goodreads group forum board, or from blogs like this one.
I was curious how you find the new Regencies you read/buy/borrow?
You don't have to do this, but what I did was go look through my book catalogue database. I use Booxter, which is a Mac program that enables me to enter all the books I've read and/or own. I can organize it and search it as I like, which makes it very useful. I went to all the Regency romances I have and sorted it so that I could see the last 10 books I most recently obtained.
Four books were ebooks I bought from Regency authors I already know I enjoy. I get their newsletters and when they had a new Regency available on ebook, I bought it.
Two books were given to me as gifts from a friend who had extra copies of an author's books.
Two books were free ebooks that I saw advertised somewhere, either on Facebook or BookBub.
The last two books were actually two of three books that I got from Paperbackswap. They are out-of-print Traditional Regency Romances that were published by Signet in the 80s and 90s and are now only available as used paperback copies.
(On a side note, I really wish these old Regencies were available as ebooks! However, I know there's a lot of factors involved in putting an out-of-print book out in ebook—who owns the rights, if the right-holder has the resources or the time to format the book for e-publishing and get the cover, write the blurb and metadata, upload it to the websites, etc.)
So … how about you? You don't have to be as exact as I did, but how do you find new Regencies to read and/or buy?

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sec:intro}

Sequential (quickest) change-point detection is concerned with the design and analysis of statistical procedures for rapid ``on-the-go'' detection of possible spontaneous changes in the characteristics of a ``live'' monitored (random) process. Specifically, the process is assumed to be monitored continuously through sequentially made observations, and if there is an indication of a possible change in the process' characteristics, one is to detect it as soon as possible, subject to a tolerable level of the risk of false detection. This type of statistical process control is used, e.g., in industrial quality and process control (see, e.g.,~\citealp{Ryan:Book2011,Montgomery:Book2012,Kenett+Zacks:Book1998}), biostatistics, economics, seismology (see, e.g.,~\citealp[Section~11.1.2]{Basseville+Nikiforov:Book93}), forensics, navigation (see, e.g.,~\citealp[Section~11.1.1]{Basseville+Nikiforov:Book93}), cybersecurity (see, e.g.,~\citealp{Tartakovsky+etal:SM2006-discussion,Tartakovsky+etal:JSM2005,Tartakovsy+etal:IEEE-JSTSP2013,Polunchenko+etal:SA2012}), and communication systems. A sequential change-point detection procedure, a rule whereby one is to stop and declare that (apparently) a change is in effect, is defined as a stopping time, $\T$, that is adapted to the observed data, $\{X_n\}_{n\ge1}$.

It will be assumed in this work that the observations, $\{X_n\}_{n\ge1}$, are independent throughout the entire period of surveillance, and such that $X_1,\ldots,X_{\nu}$ are distributed according to a common density $f(x)$, while $X_{\nu+1},X_{\nu+2},\ldots$ each follow a common density $g(x)$; both $f(x)$ and $g(x)$ are fully specified, and $g(x)\not\equiv f(x)$. The serial number of the last $f(x)$-distributed observation, i.e., $\nu\ge0$, is the change-point, and we will regard it as unknown (but not random); the notation $\nu=0$ ($\nu=\infty$) is to be understood as the case when the density of $X_n$ is $g(x)$ ($f(x)$) for all $n\ge1$. This is schematically illustrated in Figure~\ref{fig:basic-iid-change-point}.
\begin{figure}[!htb]
    \centering
    \includegraphics[width=0.8\textwidth]{change_point_idea_iid_minimax}
    \caption{Observations model.}
    \label{fig:basic-iid-change-point}
\end{figure}

At this point, by appropriately choosing the detection delay loss function (which is to be minimized) the described observations model can be used to formulate the corresponding change-point detection problem under at least three different criteria: minimax, generalized (or quasi-) Bayesian and multi-cyclic. For a recent survey, see, e.g.,~\cite{Tartakovsky+Moustakides:SA10},~\cite{Polunchenko+Tartakovsky:MCAP2012} or~\cite{Polunchenko+etal:JSM2013}. However, the focus of this work is on the pre-change regime only. More specifically, the bulk of this paper is devoted to determining the pre-change performance of two related detection procedures: the original Shiryaev--Roberts (SR) procedure (due to the independent work of~\citealp{Shiryaev:SMD61,Shiryaev:TPA63} and that of~\citealp{Roberts:T66}) and its generalization -- the Shiryaev--Roberts--$r$ (SR--$r$) procedure introduced by~\cite{Moustakides+etal:SS11}. As the former procedure is a special case of the latter, we will refer to the SR--$r$ procedure as the Generalized SR (GSR) procedure, in analogy to the terminology used by~\cite{Tartakovsky+etal:TPA2012}.

It is a fact that there isn't much literature on numerical methodology to evaluate the GSR procedure's performance, even though the procedure is nearly minimax optimal in the sense of~\cite{Pollak+Siegmund:AS1975} and~\cite{Pollak:AS85}, and exactly optimal in Shiryaev's~\citeyearpar{Shiryaev:SMD61,Shiryaev:TPA63} multi-cyclic setup. See~\cite{Tartakovsky+etal:TPA2012,Pollak+Tartakovsky:SS09} and~\cite{Polunchenko+Tartakovsky:AS10} for the relevant results. A fair amount of research has been done on the Girschick--Rubin~\citeyearpar{Girschick+Rubin:AMS52} procedure, a ``precursor'' of the original SR procedure; see, e.g.,~\cite{Knoth:FSQC2006,Yashchin:ISR1993}. The original SR procedure was studied, e.g., by~\cite[Section~11.6]{Kenett+Zacks:Book1998}. Also, a comparative analysis of the original SR procedure against various ``mainstream'' charts was carried out, e.g., by~\cite{Mahmoud+etal:JAS2008,Mevorach+Pollak:AJMMS91,Moustakides+etal:CommStat09}. Further headway was recently made by~\cite{Moustakides+etal:CommStat09,Moustakides+etal:SS11,Tartakovsky+etal:IWSM2009}. They offered an efficient numerical method to compute a wide range of operating characteristics of not only the GSR procedure, but of a much larger family of procedures, spanning in particular the popular Cumulative Sum (CUSUM) chart due to~\cite{Page:B54} and the Exponentially Weighted Moving Average (EWMA) scheme proposed by~\cite{Roberts:T59}. However, the question of the method's accuracy was only partially answered, with no error bounds or convergence rates supplied. This agrees with the overall trend in the literature concerning the computational aspect of change-point detection: to deal with the accuracy question in an {\it ad hoc} manner, if at all. This work's aim is twofold: First, we build on to the previous work of~\cite{Moustakides+etal:CommStat09,Moustakides+etal:SS11,Tartakovsky+etal:IWSM2009} and develop a more accurate and robust numerical method to study the GSR procedure's Run-Length distribution. We target the pre-change regime only, and focus on the entire pre-change distribution of the corresponding stopping time and on its first moment (i.e., the usual Average Run Length to false alarm) and standard deviation. The method is based on the integral-equations approach and uses the collocation framework as the principal idea. The basis functions are selected to be linear ``hat'' functions so as to make use of a certain change-of-measure identity and also exploit a specific martingale property of the GSR procedure's detection statistic. Second, the design and analysis of the method are complete in that the question of the method's accuracy is properly addressed, providing a tight error bound and convergence rate. Due to the use of a change-of-measure identity in combination with a certain martingale property of the GSR procedure the method's accuracy is rather high even if the partition is rough, be it because the number of partition nodes is too low or because the partitioned interval is too wide; the latter is an issue when the procedure's detection threshold is high, i.e., in the asymptotic case. Furthermore, the method's accuracy is also robust with respect to the magnitude of the change (faint, moderate or contrast).

The rest of the paper is organized thus. We start with formally stating the problem and describing the GSR procedure in Section~\ref{sec:GSR+properties}. The numerical method and its accuracy analysis are presented in Section~\ref{sec:GSR-RL-evaluation}. In Section~\ref{sec:case-study} we put the numerical method to the test by considering a case study. The study is based on the problem of detecting a shift in the mean of a sequence of independent Gaussian random variables. For this scenario, the performance of the original SR procedure and its extensions has already been quantified and compared against that of the CUSUM chart; see, e.g.,~\cite{Moustakides+etal:CommStat09,Tartakovsky+etal:IWSM2009,Mahmoud+etal:JAS2008,Knoth:FSQC2006,Mevorach+Pollak:AJMMS91}. We tailor the case study to assessing the accuracy of the numerical method, and to computing the GSR procedure's actual Run-Length distribution and its characteristics. To the best of our knowledge, no such work has previously been done. In Section~\ref{sec:further-discussion} we comment on extending the proposed method to other performance measures as well as to other detection procedures. Finally, in Section~\ref{sec:conclusion} we draw conclusions.

\section{The Generalized Shiryaev--Roberts procedure}
\label{sec:GSR+properties}

We first set down some notation. Let $\T$ be a generic stopping time. Let $\Pr_\nu$ and $\EV_\nu$ be, respectively, the probability measure and the corresponding expectation given a known $0\le\nu\le\infty$. Particularly, $\Pr_\infty$ and $\EV_\infty$ denote, respectively, the probability measure and the corresponding expectation assuming the observations' distribution never changed (i.e., $\nu=\infty$). Likewise, $\Pr_0$ and $\EV_0$ are assuming the distribution had changed before the first observation was made (i.e., $\nu=0$).

One of the objects of interest in this work is the Average Run Length (ARL) to false alarm, defined as $\ARL(\T)\triangleq\EV_\infty[\T]$. This is the standard minimax measure of the false alarm risk introduced by Lorden's~\citeyearpar{Lorden:AMS71}. However, this is just the first moment of the $\Pr_\infty$-distribution of the corresponding stopping time. As such, it may not be an exhaustive measure of the risk of sounding a false alarm; see, e.g.,~\cite{Lai:IEEE-IT1998} and~\cite{Tartakovsky:IEEE-CDC05}. Therefore, we will also be interested in the entire $\Pr_\infty$-distribution of the corresponding stopping time, and in particular, in its standard deviation, which we will denote as $\sqrt{\Var_\infty[\T]}\triangleq\sqrt{\EV_\infty[\T^2]-(\EV_\infty[\T])^2}$.

To introduce the GSR procedure we first construct the corresponding likelihood ratio (LR). Let $\mathcal{H}_k\colon\nu=k$, where $0\le k<\infty$ and $\mathcal{H}_{\infty}\colon\nu=\infty$ be, respectively, the hypothesis that the change takes place at time moment $\nu=k$, where $0\le k<\infty$, and the alternative hypothesis that no change ever occurs. The joint densities of the sample $\boldsymbol{X}_{1:n}\triangleq(X_1,\ldots,X_n)$, $n\ge1$, under each of these hypotheses are given by
\begin{align*}
p(\boldsymbol{X}_{1:n}|\mathcal{H}_{\infty})
&=
\prod_{j=1}^n f(X_j)
\;\text{and}\;
p(\boldsymbol{X}_{1:n}|\mathcal{H}_k)
=
\prod_{j=1}^k f(X_j)\prod_{j=k+1}^n
g(X_j)\;\text{for $k<n$},
\end{align*}
assuming $p(\boldsymbol{X}_{1:n}|\mathcal{H}_{\infty})=p(\boldsymbol{X}_{1:n}|\mathcal{H}_k)$ for $k\ge n$. Hence, the corresponding LR is
\begin{align*}
\LR_{1:n,\nu=k}
&\triangleq
\frac{p(\boldsymbol{X}_{1:n}|\mathcal{H}_k)}{p(\boldsymbol{X}_{1:n}|\mathcal{H}_{\infty})}
=\prod_{j=k+1}^n\LR_j,\;\text{for $k<n$},
\end{align*}
where $\LR_n\triangleq g(X_n)/f(X_n)$ is the LR for the $n$-th observation, $X_n$.

We now make an observation that will play an important role in the sequel. Let $P_d^{\LR}(t)\triangleq\Pr_d(\LR_1\le t)$, $t\ge0$, $d=\{0,\infty\}$, denote the cumulative distribution function (cdf) of the LR under measure $\Pr_d$, $d=\{0,\infty\}$, respectively. Since the LR is the Radon--Nikod\'{y}m derivative of measure $\Pr_0$ with respect to measure $\Pr_\infty$ (the two measures are assumed to be mutually absolutely continuous), one can conclude that $dP_0^{\LR}(t)=tdP_\infty^{\LR}(t)$. To be more specific, this can be seen from the following argument:
\begin{align*}
dP_0^{\LR}(t)
&\triangleq d\Pr_0(\LR_1\le t)\\
&=
d\Pr_0(X_1\le \LR_1^{-1}(t))\\
&=
\LR(\LR^{-1}(t))\,d\Pr_\infty(X_1\le \LR_1^{-1}(t))\\
&=
td\Pr_\infty(\LR_1\le t)\\
&=
tdP_\infty^{\LR}(t).
\end{align*}
We will rely on this change-of-measure identity later in Section~\ref{sec:GSR-RL-evaluation} to design our numerical method: the identity and a martingale property of the GSR procedure discussed below will be key to improve the method's accuracy and stability.

\begin{remark}
The above change-of-measure identity can also be derived from Wald's~\citeyearpar{Wald:Book47} likelihood ratio identity; see also, e.g.,~\cite[p.~13]{Siegmund:Book85},~\cite[p.~4]{Woodroofe:Book82}, or~\cite{Lai:SA2004}. Alternatively, since by design $\LR_n\in[0,\infty)$ with probability 1 for all $n\ge1$, and $\EV_\infty[\LR_n]=1$ for all $n\ge1$, one can regard $dP_0^{\LR}(t)=tdP_\infty^{\LR}(t)$ as a size-bias probability-measure transformation; see, e.g.,~\cite{Arratia+Goldstein:arXiv2010}.
\end{remark}

Formally, the SR procedure is defined as the stopping time $\mathcal{S}_A\triangleq\inf\big\{n\ge1\colon R_n\ge A\big\}$, where $A>0$ is a detection threshold that controls the false alarm risk, and
\begin{align}\label{eq:Rn-SR-def}
R_n
&=
\sum_{k=1}^n\LR_{1:n,\nu=k}=\sum_{k=1}^n \prod_{i=k}^n\LR_i,\; n\ge1,
\end{align}
is the SR detection statistic; note the recursion $R_{n}=(1+R_{n-1})\LR_n$ for $n=1,2,\ldots$, where $R_0=0$.

We remark that the detection threshold, $A>0$, can be large, making it unlikely for the detection statistic, $\{R_n\}_{n\ge1}$, to hit (much less to cross) it. As a result, $\{n\ge1\colon R_n\ge A\big\}$ is asymptotically (as $A\to\infty$) an empty set, and therefore the stopping rule, $\mathcal{S}_A$, will never terminate. To account for this, we will assume here and in every definition of a detection procedure to follow that $\inf\{\varnothing\}=\infty$.

Observe that $\{R_n-n\}_{n\ge0}$ is a zero-mean $\Pr_\infty$-martingale, i.e., $\EV_\infty[R_n-n]=0$. From this and the Optional stopping theorem (see, e.g.,~\citealp[Subsection~2.3.2]{Poor+Hadjiliadis:Book08}) one can conclude that $\EV_\infty[R_{\mathcal{S}_A}-\mathcal{S}_A]=0$ so that $\ARL(\mathcal{S}_A)\triangleq\EV_\infty[\mathcal{S}_A]=\EV_\infty[R_{\mathcal{S}_A}]\ge A$. Hence, it is sufficient to set $A=\gamma$ to ensure $\ARL(\mathcal{S}_A)\ge\gamma$ for any desired $\gamma>1$. More precisely,~\cite{Pollak:AS87} established that
\begin{align*}
\ARL(\mathcal{S}_A)
&=
\dfrac{A}{\xi}[1+o(1)],\;\text{as}\;\gamma\to\infty,
\end{align*}
where $\xi\in(0,1)$ is the limiting exponential overshoot. It is model-dependent constant and can be computed using nonlinear renewal theory; see, e.g.,~\cite{Siegmund:Book85} or~\cite{Woodroofe:Book82}. For practical purposes the approximation $\ARL(\mathcal{S}_A)\approx A/\xi$ is known to be extremely accurate, even if the corresponding ARL is in the tens.

We now describe the SR--$r$ procedure. It was introduced by~\cite{Moustakides+etal:SS11} who regarded starting off the original SR procedure at a fixed (but specially designed) $R_0^r=r$, $r\ge0$. This is similar to the idea proposed earlier by~\cite{Lucas+Crosier:T1982} for Page's~\citeyearpar{Page:B54} CUSUM chart. However, as shown by~\cite{Moustakides+etal:SS11} and~\cite{Tartakovsky+etal:TPA2012}, giving the SR procedure a headstart is practically putting it on steroids: the gain in performance far exceeds that observed by~\cite{Lucas+Crosier:T1982} for the CUSUM chart.

Specifically, the SR--$r$ procedure is defined as the stopping time
\begin{align}\label{eq:T-SRr-def}
\mathcal{S}_{A}^r
&=
\inf\{n\ge1\colon R_n^r \ge A\} ,\; A >0,
\end{align}
where
\begin{align}\label{eq:statistic-SRr-def}
R_{n+1}^r
&=
(1+R_{n}^r)\LR_{n+1}\;\;\text{for}\;\; n=1,2,\ldots,\;\;\text{where}\;\;R_0^r=r,
\end{align}
and we remark that $\{R_n^r-n-r\}_{n\ge0}$ is a zero-mean $\Pr_\infty$-martingale, i.e., $\EV_\infty[R_n^r-n-r]=0$. As a result, one can establish that
\begin{align*}
\ARL(\mathcal{S}_A^r)
&\approx
\dfrac{A}{\xi}-r,\;\text{as}\;\gamma\to\infty,
\end{align*}
where $\xi\in(0,1)$ is again the limiting exponential overshoot. This approximation is quite accurate as well, even if the ARL is small.

We now make a comment on the foregoing approximation for $\ARL(\mathcal{S}_A^r)$. For a fixed $A>0$ it may seem as though $\ARL(\mathcal{S}_A^r)$ is a linear function of $r$ with a slope of $-1$ and a constant $y$-intercept (proportional to $A$). However, this is true only if the headstart is not too large. For otherwise $\ARL(\mathcal{S}_A^r)$ would have zero or negative values, which is impossible since by definition $\ARL(\mathcal{S}_A^r)\ge1$ for all $r,A>0$. If $r$ is too large, then $\ARL(\mathcal{S}_A^r)$ is close to $1$, and in fact $\lim_{r\to\infty}\ARL(\mathcal{S}_A^r)=1$. This is illustrated in Figure~\ref{fig:ARL_vs_r}.
\begin{figure}[!htb]
    \centering
    \includegraphics[width=0.9\textwidth]{ARL_vs_r}
    \caption{Typical behavior of $\ARL(\mathcal{S}_A^r)$ as a function of $r\ge0$ for a fixed $A>0$.}
    \label{fig:ARL_vs_r}
\end{figure}
Since $\ARL(\mathcal{S}_A^{r=0})\approx A/\xi>A$ and as a function of $r$ it's a line with a slope of $-1$, the value of the headstart at which $\ARL(\mathcal{S}_A^{r=0})$ starts to curl is also about $r_{\mathrm{crit}}\approx A/\xi$. Therefore, roughly speaking for $r\in[0,A/\xi]$, the behavior of $\ARL(\mathcal{S}_A^{r})$ as a function of $r$ is almost linear with a slope of $-1$. This is a direct consequence of the GSR procedure's martingale property mentioned above, and it will be used to improve the accuracy of our numerical method.

We now make a comment on the foregoing approximation for $\ARL(\mathcal{S}_A^r)$. For a fixed $A>0$ it may seem as though $\ARL(\mathcal{S}_A^r)$ is a linear function of $r$ with a slope of $-1$ and a constant $y$-intercept (linearly proportional to $A$). However, this is true only if the headstart is not too large, in which case $\ARL(\mathcal{S}_A^r)$ is close to $1$, and in fact $\lim_{r\to\infty}\ARL(\mathcal{S}_A^r)=1$. This is illustrated in Figure~\ref{fig:ARL_vs_r}.
Since $\ARL(\mathcal{S}_A^{r=0})\approx A/\xi>A$, the value of the headstart at which $\ARL(\mathcal{S}_A^{r=0})$ starts to curl is also about $r_{\mathrm{crit}}\approx A/\xi$. Therefore, roughly speaking, for $r\in[0,A/\xi]$ the behavior of $\ARL(\mathcal{S}_A^{r})$ as a function of $r$ is almost linear. This is a direct consequence of the GSR procedure's martingale property mentioned above, and it will be used to improve the accuracy of our numerical method.

We have now introduced all of the procedures of interest. Next, we present a numerical method to study their operating characteristics.

\section{Run-Length distribution}
\label{sec:GSR-RL-evaluation}

We develop here a numerical method to study the Run-Length distribution of the GSR procedure; the corresponding stopping time, $\mathcal{S}_A^r$, is given by~\eqref{eq:T-SRr-def} and~\eqref{eq:statistic-SRr-def} above. Specifically, the focus is on the ``in-control'' distribution and its characteristics. The distribution itself is given by $\Pr_\infty(\mathcal{S}_A^r=k)$, $k\ge0$, i.e., the probability mass function (pmf) of the GSR stopping time; we will also consider $\Pr_\infty(\mathcal{S}_A^r>k)$, $k\ge0$, i.e., the corresponding ``survival function''. The distribution's characteristics of interest are\begin{inparaenum}[\itshape a)]\item the first moment (under the $\Pr_\infty$ measure), i.e., the ARL to false alarm defined as $\ARL(\mathcal{S}_A^r)\triangleq\EV_\infty[\mathcal{S}_A^r]$ and \item the distribution's higher moments (all under the $\Pr_\infty$ measure)\end{inparaenum}. The proposed numerical method is a build-up over one previously proposed and applied by~\cite{Tartakovsky+etal:IWSM2009,Moustakides+etal:CommStat09,Moustakides+etal:SS11}.

\subsection{Integral Equations Framework}
\label{ssec:integral-equations-framework}

We begin with notation and assumptions. First recall $\LR_n\triangleq g(X_n)/f(X_n)$, i.e., the ``instantaneous'' LR for the $n$-th data point, $X_n$, and note that $\{\LR_n\}_{n\ge1}$ are iid under both measures $\Pr_d$, $d=\{0,\infty\}$. For simplicity, $\LR_1$ will be assumed absolutely continuous, although at an additional effort the case of non-arithmetic $\LR_1$ can be handled as well. Let $P_d^{\LR}(t)\triangleq\Pr_d(\LR_1\le t)$, $t\ge0$, be the cdf of the LR under the measure $\Pr_d$, $d=\{0,\infty\}$. Also, denote
\begin{align}\label{eq:K-def}
K_d(x,y)
&\triangleq
\frac{\partial}{\partial y}\Pr_d(R_{n+1}^x\le y|R_n^x=x)
=
\frac{\partial}{\partial y}P_d^{\LR}\left(\frac{y}{1+x}\right),
\; d=\{0,\infty\},
\end{align}
the {\em transition probability density kernel} for the (stationary) Markov process $\{R_n^x\}_{n\ge0}$.

We now note that from~\eqref{eq:K-def} and the change-of-measure identity $d{P}_0^{\LR}(t)=t d{P}_{\infty}^{\LR}(t)$, $t\ge0$ established earlier, one can readily deduce that $(1+x){K}_0(x,y)=y{K}_{\infty}(x,y)$. This can be used, e.g., as a ``shortcut'' in deriving the formula for ${K}_0(x,y)$ from that for ${K}_{\infty}(x,y)$, or the other way around -- whichever one of the two is found first. More importantly, this connection between ${K}_0(x,y)$ and ${K}_{\infty}(x,y)$ will allow our numerical method to have greater accuracy and robustness.

We now state the first equation of interest. Let $R_0^x=x\ge-1$ be fixed. For notational brevity, from now on let $\ell(x,A)\triangleq\ARL(\mathcal{S}_A^x)\triangleq\EV_{\infty}[\mathcal{S}_A^x]$; we reiterate that this expectation is conditional on $R_0^x=x$. Using the fact that $\{R_n^x\}_{n\ge0}$ is Markovian, it can be shown that $\ell(x,A)$ satisfies the equation
\begin{align}\label{eq:ARL-int-eqn}
\ell(x,A)
&=
1+\int_0^AK_{\infty}(x,y)\,\ell(y,A)\,dy;
\end{align}
cf.~\cite{Moustakides+etal:SS11}. It is also worth noting here that so long as $R_0^x=x\ge-1$, one can infer from~\eqref{eq:Rn-SR-def} that $R_n^x\ge0$ with probability 1 for all $n\ge1$ under either measure $\Pr_d$, $d=\{0,\infty\}$. This is why the lower limit of integration in the integral in the right-hand side of~\eqref{eq:ARL-int-eqn} is 0. Furthermore, due to the GSR procedure's martingale property discussed earlier, we have $\ell(x,A)\approx A/\xi-x$ at least for $x\le A/\xi$. Any good numerical method to solve the equation for $\ell(x,A)$ is to factor in the expected shape of the sought function.

Next, introduce $\rho_k(x,A)\triangleq\Pr_\infty(\mathcal{S}_A^x>k)$ for a fixed $k\ge0$; note that $\rho_0(x,A)\equiv1$ for all $x,A\in\mathbb{R}$, and in general $0\le\rho_k(x,A)\le1$ for all $x,A\in\mathbb{R}$ and $k\ge0$. We will occasionally refer to $\rho_k(x,A)$ as the ``survival function''. Again, since $\{R_n^x\}_{n\ge0}$ is Markovian, one can establish the recurrence
\begin{align}\label{eq:recursion2}
\rho_{k+1}(x,A)
&=
\int_0^A K_{\infty}(x,y)\,\rho_k(y,A)\,dy,
\end{align}
wherewith one can generate the functional sequence $\{\rho_k(x,A)\}_{k\ge0}$; cf.~\cite{Moustakides+etal:SS11}. Note that this recurrence is repetitive application of the linear integral operator
\begin{align*}
\mathcal{K}_{\infty}\circ u
&=
\int_0^A K_{\infty}(x,y)\,u(y)\,dy,
\end{align*}
where $u(x)$ is assumed to be sufficiently smooth inside the interval $[0,A]$. Temporarily deferring formal discussion of this operator's properties, note that using this operator notation, recurrence~\eqref{eq:recursion2} can be rewritten as $\rho_{k+1}=\mathcal{K}_{\infty}\circ \rho_{k}$, $k\ge0$, or equivalently, as $\rho_{k}=\mathcal{K}_{\infty}^{k}\circ \rho_{0}$, $k\ge0$, where
\begin{align*}
\mathcal{K}_{\infty}^{k}\circ u
&\triangleq
\underbrace{\mathcal{K}_{\infty}\circ\cdots\circ\mathcal{K}_{\infty}}_{\text{$k$ times}}\circ\,u\;\text{for}\;k\ge1,
\end{align*}
and $\mathcal{K}_{\infty}^{0}$ is the identity operator from now on denoted as $\mathbb{I}$, i.e., $\mathcal{K}_{\infty}^{0}\circ u=\mathbb{I}\circ u=u$. Similarly, in the operator form, equation~\eqref{eq:ARL-int-eqn} can be rewritten as $\ell=1+\mathcal{K}_\infty\circ\ell$.

We note that the sequence $\{\rho_k(x,A)\}_{k\ge0}$ can be used to derive many characteristics of the GSR stopping time. For example, since
\begin{align}\label{eq:ARL-Neumann-series}
\ell
&=
\sum_{k\ge0}\rho_k=\sum_{k\ge0}\mathcal{K}_\infty^k\circ\rho_0
=
\left(\,\sum_{k\ge0}\mathcal{K}_\infty^k\right)\circ\rho_0
=
(\,\mathbb{I}-\mathcal{K}_\infty)^{-1}\circ\rho_0,
\end{align}
one obtains equation~\eqref{eq:ARL-int-eqn} recalling that $\rho_0(x,A)=1$ for all $x,A\in\mathbb{R}$. The implicit use of the geometric series convergence theorem to reduce the last infinite sum to the operator $(\,\mathbb{I}-\mathcal{K}_\infty)^{-1}$ is justified by the fact that the spectral radius of the operator $\mathcal{K}_\infty$ is strictly less than 1. This is shown, e.g.,~\cite{Moustakides+etal:SS11}.

More generally, through the sequence $\{\rho_k(x,A)\}_{k\ge0}$, one can derive an equation for the characteristic function of the GSR stopping time, and then use it to obtain an equation for any moment of the GSR stopping time (under the $\Pr_\infty$ measure). Specifically, denote $\Psi(s,x,A)\triangleq\EV_\infty[e^{\imath s\mathcal{S}_A^x}]$, where hereafter $\imath=\sqrt{-1}$. Similarly to~\eqref{eq:ARL-Neumann-series}, it can be shown that $\Psi(s,x,A)$ solves the equation
\begin{align}\label{eq:GSR-char-fcn-eqn}
\Psi(s,x,A)
&=
e^{\imath s}[1-\rho_1(x,A)]
+e^{\imath s}\int_0^A K_\infty(x,y)\,\Psi(s,y,A)\,dy,
\end{align}
where $\rho_1(x,A)$ is as in~\eqref{eq:recursion2} above, and $\Psi(0,x,A)\equiv1$ for all $x,A\in\mathbb{R}$.

To illustrate the power of equation~\eqref{eq:GSR-char-fcn-eqn}, consider the second moment of the GSR stopping time (under the $\Pr_\infty$ measure). Let $\mu_2(x,A)\triangleq\EV_\infty[(\mathcal{S}_A^x)^2]$, and note that $\mu_2(x,A)=-\partial^2 \Psi(s,x,A)/\partial s^2$ evaluated at $s=0$. Hence, from differentiating equation~\eqref{eq:GSR-char-fcn-eqn} twice with respect to $s$, and then setting $s=0$, we obtain
\begin{align}\label{eq:mu2-int-eqn}
\mu_2(x,A)
&=
2\ell(x,A)-1
+\int_0^A K_\infty(x,y)\,\mu_2(y,A)\,dy,
\end{align}
where $\ell(x,A)$ is as in~\eqref{eq:ARL-int-eqn}. Respective equations for the third and higher moments of the GSR stopping time (under the $\Pr_\infty$ measure), i.e., $\mu_n(x,A)\triangleq\EV_\infty[(\mathcal{S}_A^x)^n]$, $n\ge3$, can be obtained in an analogous manner by evaluating $\mu_n(x,A)=(-\imath)^n\partial^n \Psi(s,x,A)/\partial s^n$ at $s=0$.

We remark parenthetically that equation~\eqref{eq:GSR-char-fcn-eqn} resembles that for the characteristic function of the geometric distribution (over the set $\{1,2,\ldots\}$). One may therefore deduce that the $\Pr_\infty$-distribution of the GSR stopping time, $\mathcal{S}_A^x$, is geometric. This is, in fact, true, though generally only in the limit, as $A\to\infty$, and is a special case of the more general result of~\cite{Pollak+Tartakovsky:TVP09,Tartakovsky+etal:IWAP08}. Under certain mild conditions, they showed that the first exit time of a nonnegative recurrent monotone Markov process from the strip $[0,B]$, $B>0$, is asymptotically (as $B\to\infty$) geometrically distributed; the ``probability of success'' is the reciprocal of the first exit time's expectation.

The series $\{\rho_k(x,A)\}_{k\ge0}$ can also be used to obtain the $\Pr_\infty$-pmf of the GSR's stopping time. To this end, it suffices to note that $\Pr_\infty(\mathcal{S}_A^x=k)=\Pr_\infty(\mathcal{S}_A^x>k-1)-\Pr_\infty(\mathcal{S}_A^x>k)=\rho_{k-1}(x,A)-\rho_{k}(x,A)$, $k\ge1$, whence
\begin{align*}
\Pr_\infty(\mathcal{S}_A^x=k)
&=
(\mathbb{I}-\mathcal{K}_\infty)\circ\rho_{k-1}
=
(\mathbb{I}-\mathcal{K}_\infty)\circ\mathcal{K}_\infty^{k-1}\circ\rho_0
=
\mathcal{K}_\infty^{k-1}\circ(\mathbb{I}-\mathcal{K}_\infty)\circ\rho_0,\;k\ge1,
\end{align*}
since the operators $\mathcal{K}_\infty$ and $\mathbb{I}-\mathcal{K}_\infty$ commute with one another.

We have now obtained a complete set of integral equations and relations to compute any of the desired performance characteristics of the GSR stopping time. The question to be considered next is that of computing these characteristics in practice.

\subsection{The Numerical Method and Its Accuracy}
\label{ssec:numerical-method+accuracy}

We now turn attention to the question of solving the equations presented in the preceding subsection. To this end, observe first that equations~\eqref{eq:ARL-int-eqn} and~\eqref{eq:mu2-int-eqn} are both renewal-type equations of the general form:
\begin{align*}
u(x)
&=
\upsilon(x)+\int_0^A K_\infty(x,y)\,u(y)\,dy,
\end{align*}
where $\upsilon(x)$ is a given (known) function, $K_\infty(x,y)$ is as in~\eqref{eq:K-def}, and $u(x)$ is the unknown to be found; note that while $u(x)$ does depend on the upper limit of integration, $A$, for notational simplicity we will no longer emphasize that, and use $u(x)$ instead of $u(x,A)$.

To see that the above equation is an ``umbrella'' equation for the equations of interest, observe that, e.g., to obtain equation~\eqref{eq:ARL-int-eqn} on the ARL to false alarm, it suffices to set $\upsilon(x)\equiv 1$ for any $x\in\mathbb{R}$. Similarly, choosing $\upsilon(x)=2\ell(x,A)-1$, where $\ell(x,A)$ is as in~\eqref{eq:ARL-int-eqn}, will yield equation~\eqref{eq:mu2-int-eqn}, which governs the second $\Pr_\infty$-moment of the GSR stopping time. Thus, solving the above equation is the same as solving~\eqref{eq:ARL-int-eqn} and~\eqref{eq:mu2-int-eqn}. The problem, however, is that the general equation is a Fredholm integral equation of the second kind, and such equations rarely allow for an analytical solution. Hence, a numerical approach is in order.

We first set the underlying space for the problem. Let $\mathcal{X}=\mathbb{C}[0,A]$ be the space of continuous functions over the interval $[0,A]$. Equip $\mathcal{X}$ with the usual uniform norm. We will assume that ${P}_\infty^{\LR}(t)$ and the unknown function $u(x)$ are both differentiable as far as necessary. Under these assumptions $\mathcal{K}_\infty$ is a bounded linear operator from $\mathcal{X}$ into $\mathcal{X}$, equipped with the usual $\mathrm{L}_\infty$ norm, $\|u\|_\infty\triangleq\max_{x\in[0,A]}|u(x)|$. Specifically,
\begin{align*}
\|\mathcal{K}_\infty\|_\infty
&\triangleq
\sup_{x\in[0,A]}\int_0^A\abs{K_\infty(x,y)}dy < 1;
\end{align*}
cf.~\cite{Moustakides+etal:SS11}. An important implication of this result is that one can apply the Fredholm alternative (see, e.g.,~\citealp[Theorem~2.8.10]{Atkinson+Han:Book09}) and conclude that all of the equations of interest do have a solution and it is unique. The same conclusion can be reached from the Banach fixed-point theorem, proved by Banach in his Ph.D. thesis in 1920 (published in 1922).

We now define a projection space within which our collocation solution, $u(x)$, will then be sought. Under this method $u(x)$ is approximated as
\begin{align*}
u_N(x)
&=
\sum_{j=1}^N u_{j,N}\,\phi_j(x),
\end{align*}
where $\{u_{j,N}\}_{1\le j\le N}$ are constant coefficients to be determined, and $\{\phi_j(x)\}_{1\le j\le N}$ are suitably chosen (known) basis functions.

We now discuss a strategy for determining the coefficients $\{u_{j,N}\}_{1\le j\le N}$. For any fixed set of these coefficients, substitution of $u_N(x)$ into the equation will give a residual $r_N=u_N-\mathcal{K}_{\infty}\circ u_N-\upsilon$. Unless the true solution $u(x)$ itself is a linear combination of the basis functions $\{\phi_j(x)\}_{1\le j\le N}$, no choice of the coefficients $\{u_{j,N}\}_{1\le j\le N}$ will make the residual identically zero uniformly for all $x\in[0,A]$. However, by requiring $r_N(x)$ to be zero at some $\{z_j\}_{1\le j\le N}$, where $z_j\in[0,A]$ for all $j=1,2,\ldots,N$, one can achieve a certain level of proximity of the residual to zero. As a result, one readily arrives at the following system of $N$ algebraic equations on the coefficients $u_{j,N}$
\begin{align}\label{eq:fredholm-lin-system}
\boldsymbol{u}_N
&=
\boldsymbol{\upsilon}
+
\mathcal{\boldsymbol{K}}_\infty \boldsymbol{u}_N,
\end{align}
where $\boldsymbol{u}_N=[u_{1,N},\ldots,u_{N,N}]^\top$, $\boldsymbol{\upsilon}=[\upsilon(z_1),\ldots,\upsilon(z_N)]^\top$, and  $\boldsymbol{K}_\infty$ is a matrix of size $N$-by-$N$ such that
\begin{align}\label{eq:def-matrix-K}
(\boldsymbol{K}_\infty)_{i,j}
&=
\int_0^A K_\infty(z_i,y)\,\phi_j(y)\,dy,\;\;1\le i,j\le N.
\end{align}
The points $\{z_j\}_{1\le j\le N}$ are referred to as the collocation points.

We now remark that for the system of linear equations~\eqref{eq:fredholm-lin-system} to be consistent, the functions $\{\phi_j(x)\}_{1\le j\le N}$ need to be chosen so as to form a basis in the appropriate functional space, i.e., in particular, $\{\phi_j(x)\}_{1\le j\le N}$ need to be linearly independent. This is equivalent to introducing an interpolating projection operator, $\pi_N$, that projects the sought function $u(x)$ onto the span of these basis functions. Specifically, this operator is defined as $\pi_N\circ u=\sum_{j=1}^N u_{j,N} \phi_j(x)$, and
\begin{align*}
\|\pi_N\|_\infty
&=
\max_{0\le x\le A}\sum_{j=1}^N|\phi_j(x)|\ge1.
\end{align*}

It is also worth noting that, by design, this method is most accurate at the collocation points, $\{z_j\}_{1\le j\le N}$, i.e., at the points at which the residual is zero. For an arbitrary point $x$ the unknown function can be evaluated as
\begin{align}\label{eq:int-eqn-iter-sol}
\begin{aligned}
\widetilde{u}_N(x)
&=
\upsilon(x)+\int_0^A K_\infty(x,y)\,u_N(y)\,dy\\
&=
\upsilon(x)+\sum_{j=1}^Nu_{j,N}\int_0^A K_\infty(x,y)\,\phi_j(y)\,dy.
\end{aligned}
\end{align}
This technique is known as the iterated solution; see, e.g.,~\cite[Subsection~12.3.2]{Atkinson+Han:Book09}. Also note that $\widetilde{u}_N(z_j)=u_N(z_j)=u_{j,N}$, $1\le j\le N$.

We now consider the question of accuracy. To that end, it is apparent that the choice of the basis functions, $\{\phi_i(x)\}_{1\le i\le N}$, must play a critical role. This is, in fact, the case, as may be concluded from, e.g.,~\cite[Theorem~12.1.12,~p.~479]{Atkinson+Han:Book09}. Specifically, using $\|u-\widetilde{u}_N\|_\infty$ as a sensible measure of the method's error and applying~\cite[Formula~12.3.21,~p.~499]{Atkinson+Han:Book09}, we obtain
\begin{align}\label{eq:err-bound-1}
\begin{aligned}
\|u-\widetilde{u}_N\|_\infty
&\le
\|(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty\|\mathcal{K}_\infty\circ(\mathbb{I}-\pi_N)\circ u\|_\infty\\
&\le
\|(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty\|\mathcal{K}_\infty\|_\infty\|(\mathbb{I}-\pi_N)\circ u\|_\infty,
\end{aligned}
\end{align}
whence one can see that the method's error is determined by two factors:\begin{inparaenum}[\itshape a)]\item $\|(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty$ and \item $\|(\mathbb{I}-\pi_N)\circ u\|_\infty$, which is the corresponding interpolation error\end{inparaenum}. The latter can be found for each particular choice of $\pi_N$. The bigger problem is to upperbound $\|(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty$. To this end, the standard result
\begin{align*}
\|(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty
&\le
\dfrac{1}{1-\|\mathcal{K}_\infty\|_\infty}
\end{align*}
is applicable since $\|\mathcal{K}_\infty\|_\infty<1$. However, it is well-known that this is a very generic result, and as an upper bound is often far off. Consequently, this loosens the upper bound on $\|u-\widetilde{u}_N\|_\infty$ and diminishes its practical potential. However, since in our particular case ${K}_\infty(x,y)$ is a transition probability kernel, a tighter (in fact, exact) bound is possible to obtain. We now state the corresponding result.
\begin{lemma}\label{lem:ARL-inv-bound}
$\|\,(\mathbb{I}-\mathcal{K}_\infty)^{-1}\|_\infty=\|\,\ell\,\|_\infty$.
\end{lemma}
\begin{proof}
The desired result can be shown using the argument akin to one given to prove, e.g.,~\cite[Theorem~5, part (b)]{Fan:MfM1958} for square matrices with non-negative elements. Specifically, even though $\mathcal{K}_\infty$ and $(\mathbb{I}-\mathcal{K}_\infty)^{-1}$ are not matrices, Fan's proof for matrices extends easily to $\mathcal{K}_\infty$ and $(\mathbb{I}-\mathcal{K}_\infty)^{-1}$, as either operator is bounded, linear and non-negative-valued, since $K_\infty(x,y)\ge0$ for all $x,y\ge0$.
\end{proof}

This lemma allows to find \emph{exactly} the bound for the proposed method's error. We now state the corresponding result for equation~\eqref{eq:ARL-int-eqn}.
\begin{theorem}\label{thm:error-bound}
Assuming $N\ge1$ is sufficiently large
\begin{align*}
\|\ell-\widetilde{\ell}_N\|_\infty
&<
\|\ell\|_\infty\|\ell_{xx}\|_\infty\frac{h^2}{8},\;\text{where}\; \ell_{xx}\triangleq\dfrac{\partial^2}{\partial x^2}\ell(x,A);
\end{align*}
note that the inequality is strict.
\end{theorem}
\begin{proof}
It suffices to recall the standard bound on the polynomial interpolation error
\begin{align*}
\|(\mathbb{I}-\pi_N)\circ \ell\|_\infty
&\leq
\|\ell_{xx}\|_\infty\frac{h^2}{8};
\end{align*}
see, e.g.,~\cite[Formula~3.2.9, p.~124]{Atkinson+Han:Book09}. The desired inequality can then be readily obtained from~\eqref{eq:err-bound-1}, Lemma~\ref{lem:ARL-inv-bound}, and the fact that $\|\mathcal{K}\|_\infty<1$.
\end{proof}

We propose to use piecewise linear polynomial space. To this end, for any positive integer $N\ge2$, let $\Pi_N\colon 0\triangleq x_0<x_1<\ldots<x_{N-1}=A$ denote a partition of the interval $[0,A]$, and for $i=1,\ldots,N-1$ set $I_i^N\triangleq(x_{i-1}, x_i)$, $h_i\triangleq x_i-x_{i-1}(>0)$, and $h\triangleq h(N)=\max_{1\le i\le N-1} h_i$; assume also that $h\to0$ as $N\to\infty$.
Next, set $z_j=x_{j-1}$, $1\le j\le N$ and use the ``hat functions''
\begin{empheq}
    left=
        \phi_i(x)
    \empheqlbrace}]{align}\nonumber
&\frac{x-x_{i - 2}}{h_{i-1}},\quad\text{if $x\in I_{i-1}^N, i>1$;}\label{eq:lin-basis}\\
&\frac{x_i-x}{h_{i}},\quad\text{if $x\in I_{i}^N, i<N$;}\\\nonumber
&0,\quad\text{otherwise},
\end{empheq}
where $1\le i\le N$. These functions are clearly linearly independent. One important remark is now in order to justify this choice. The error bound given in Theorem~\ref{thm:error-bound} is tight, and can be seen to be directly proportional to the magnitude of the solution, i.e., to $\ell(x,A)=\ARL(\mathcal{S}_A^x)$. Worse yet, it is also proportional to the detection threshold squared lurking in $h^2$. Since $\ARL(\mathcal{S}_A^x)\approx A/\xi-x$, and $\xi\in(0,1)$, one can roughly set $A\approx\ell(x,A)$ and conclude that the error bound is roughly proportional to the magnitude of the solution cubed. This may seem to drastically offset the second power of $N$ in the denominator buried in $h^2$. However, as we will demonstrate experimentally, due to the almost linearity of $\ell(x,A)$ in $x$ at least for $x\in[0,A]$, this does not happen. Specifically, since $\ell(x,A)\approx A/\xi-x$ at least for $x\in[0,A]$, we have:
\begin{align*}
\frac{\partial^2}{\partial x^2}
\ell(x,A)
&\approx 0,\;\text{at least for}\; x\in[0,A].
\end{align*}
This makes the obtained error bound extremely close to zero, even for relatively small $N$. Consequently, the method is rather accurate and does not require $N$ to be large. This is precisely the reason why we suggested to use the piecewise linear basis.

The only question left to discuss is that of the role of the change-of-measure ploy in all this. To this end, recall that to implement the numerical method, one has to compute the matrix~\eqref{eq:def-matrix-K}, which involves finding the corresponding integrals. At first, it may seem to be an additional source of errors, since the integrals would have to be evaluated numerically as well. However, due to the change-of-measure identity, $(1+x){K}_0(x,y)=y{K}_\infty(x,y)$, the integrals involved in~\eqref{eq:def-matrix-K} can be computed {\it exactly}. Specifically, using~\eqref{eq:def-matrix-K} and~\eqref{eq:lin-basis}, and recalling that $z_j=x_{j-1}$, $1\le j\le N$, the corresponding formula is as follows:
\begin{align*}
(\boldsymbol{K}_\infty)_{i,j}
&=
\int_0^A{K}_\infty(x_i,y)\,\phi_j(y)\,dy\\
&=
\dfrac{1}{h_{j-1}}\left\{(1+x_i)\left[{P}_0^{\LR}\left(\dfrac{x_{j-1}}{1+x_i}\right)-{P}_0^{\LR}\left(\dfrac{x_{j-2}}{1+x_i}\right)\right]\right.-\\
&\qquad\qquad\qquad\qquad\qquad x_{j-2}\left.\left[{P}_\infty^{\LR}\left(\dfrac{x_{j-1}}{1+x_i}\right)-{P}_\infty^{\LR}\left(\dfrac{x_{j-2}}{1+x_i}\right)\right]\right\}\indicator{j>1}+\\
&\qquad\dfrac{1}{h_{j}}\left\{x_j\left[{P}_\infty^{\LR}\left(\dfrac{x_{j}}{1+x_i}\right)-{P}_\infty^{\LR}\left(\dfrac{x_{j-1}}{1+x_i}\right)\right]\right.-\\
&\qquad\qquad\qquad\qquad\qquad (1+x_i)\left.\left[{P}_0^{\LR}\left(\dfrac{x_{j}}{1+x_i}\right)-{P}_0^{\LR}\left(\dfrac{x_{j-1}}{1+x_i}\right)\right]\right\}\indicator{j<N}
\end{align*}
for $1\le i,j\le N$.

To wrap this subsection, note that the developed numerical framework can also be used to assess the accuracy of the popular Markov chain approach, introduced by~\cite{Brook+Evans:B1972}, and later extended, e.g., by~\cite{Woodall:T1983}. To this end, as noted by~\cite{Champ+Rigdon:CommStat1991}, the Markov chain approach is equivalent to the integral-equations approach if the integral is approximated via the product midpoint rule. This, in turn, is equivalent to choosing the basis functions, $\{\phi_i(x)\}_{1\le j\le N}$, as piecewise constants on $\Pi_{N+1}$, i.e., $\phi_j(x)=\indicator{x\in I_j^{N+1}}$, and equating the residual to zero at the midpoints of the intervals $I_j^{N+1}$, i.e., setting $z_j=(x_{j-1}+x_j)/2$, $1\le j\le N$. In this case the $(i,j)$-th element of the matrix $\boldsymbol{K}$ defined by~\eqref{eq:def-matrix-K} is as follows:
\begin{align*}
(\boldsymbol{K}_\infty)_{i,j}
&=
P_\infty^{\LR}\left(\dfrac{x_j}{1+z_i}\right)
-
P_\infty^{\LR}\left(\dfrac{x_{j-1}}{1+z_i}\right),\;1\le i,j\le N.
\end{align*}
It can be shown that this approach exhibits a superconvergence effect: the rate is also quadratic, even though the interpolation is based on using polynomials of degree zero (i.e., constants). This was first observed by~\cite{Kryloff+Bogoliubov:CRA-URSS1929}. See also~\cite[pp.~130--135]{Kantorovich+Krylov:Book1958} for a different proof. However, in spite of the quadratic convergence, the corresponding constant in the error bound is large, and, as a result, the partition size required by the method ends up being substantial. In fact, this method was employed, e.g., by~\cite{Moustakides+etal:CommStat09}, to compare the CUSUM chart and the original SR procedure, and the partition size they used consisted of thousands of points to ensure reasonable accuracy. In the next section we will compare this method against ours, and show that our method is superior.

Next, we apply the proposed numerical method to a particular example to study the GSR procedure's pre-change Run-Length distribution. We note that, to the best of our knowledge, this is the first time the GSR procedure's actual pre-change Run-Length distribution is looked at.

\section{A case study}
\label{sec:case-study}

Consider a scenario where the observations are independent Gaussian with mean zero pre-change and $\theta\neq0$ post-change; the variance is 1 (known) and does not change. Formally, the pre- and post-change densities in this case are
\begin{align*}
f(x)
&=
\dfrac{1}{\sqrt{2\pi}}\exp\left\{-\dfrac{x^2}{2}
\right\}\;\text{and}\;
g(x)=\dfrac{1}{\sqrt{2\pi}}\exp\left\{-\dfrac{(x-\theta)^2}{2}\right\},
\end{align*}
respectively, where $x\in\mathbb{R}$ and $\theta\neq0$.

Next, it is straightforward to see that
\begin{align*}
\LR_n
&\triangleq
\dfrac{g(X_n)}{f(X_n)}
=\exp\left\{\theta X_n-\frac{\theta^2}{2}\right\},\;n\ge1,
\end{align*}
and subsequently, $P_d^{\LR}(t)\triangleq\Pr_d(\LR_1\le t)$ is log-normal with mean $-\theta^2/2$ and variance $\theta^2$ under $d=\infty$, and with mean $\theta^2/2$ and variance $\theta^2$ under $d=0$. Hence, one obtains
\begin{align*}
K_\infty(x,y)
&\triangleq
\dfrac{\partial}{\partial y}P_\infty^{\LR}\left(\dfrac{y}{1+x}\right)
=
\dfrac{1}{y\sqrt{2\pi\theta^2}}\exp\left\{-\dfrac{1}{2\theta^2}\left(\log\dfrac{y}{1+x}+\dfrac{\theta^2}{2}\right)^2\right\}\indicator{y/(1+x)\ge0},
\end{align*}
and one can see that the kernel is the same regardless of whether $\theta<0$ or $\theta>0$. We therefore, without any loss of generality, will consider only the former case, i.e., assume from now on that $\theta>0$. As $(1+x)K_0(x,y)=y K_\infty(x,y)$, one could quickly obtain the formula for $K_0(x,y)$, if it were necessary. However, it isn't, since the change-of-measure identity allows to express everything in terms of $K_\infty(x,y)$ only.

We now employ the proposed numerical method to evaluate the characteristics of the GSR procedure. The method was implemented in MATLAB. To test the accuracy of the method, we will perform our analysis for a broad range of changes $\theta=0.01, 0.1, 0.5$ and $1.0$, i.e., from very faint, to small, to moderate, to very contrast, respectively. We will also consider a wide range of values of the ARL to false alarm: $\gamma=10^2, 10^3, 10^4$ and $10^5$, i.e., from large risk of false alarm, to moderate, to low, respectively (the false alarm risk is inversely proportional to the ARL to false alarm).

Since the accuracy of our numerical method is determined by the accuracy of the underlying piecewise linear polynomial interpolation, we recall that forming the partition of the interval of integration, $[0,A]$, by the Chebyshev nodes (i.e., roots of the Chebyshev polynomials of the first kind) will result in the smallest possible interpolating error. Subsequently, the overall accuracy of the method will improve as well. Specifically, we partition the interval $[0,A]$ into $N-1$, $N\ge2$, non-overlapping subintervals joint at the shifted Chebyshev nodes of the form:
\begin{align*}
x_{N-i}
&=
\frac{A}{2}\left\{1+\cos\left[(2i - 1)\dfrac{\pi}{2N}\right]/\cos\left(\dfrac{\pi}{2N}\right)\right\},\;1\le i\le N.
\end{align*}
The shift is to make sure the left and right end nodes coincide with the left and right end points of the interval $[0,A]$, i.e., $x_0=0$ and $x_{N-1}=A$. To evaluate the solution at a point different from one of $x_j$'s, we will use the iterated solution~\eqref{eq:int-eqn-iter-sol}. As we will see below, the method's accuracy is high even if $N$ is small, and the accuracy remains high for a large range of $\theta$'s and for a large range of values of the ARL to false alarm. To measure the rate of convergence of the method we will rely on the common Richardson extrapolation technique: if $\ell_{2N}$, $\ell_N$ and $\ell_{N/2}$ are the solutions assuming the partition size is $2N$, $N$ and $N/2$, respectively, then the corresponding rate of convergence, $p$, can be estimated as
\begin{align*}
2^{-p(N)}
&\approx
\dfrac{\|\ell_{2N}-\ell_N\|_\infty}{\|\ell_{N}-\ell_{N/2}\|_\infty},\;\text{so that}\;
p(N)\approx-\log_2\dfrac{\|\ell_{2N}-\ell_N\|_\infty}{\|\ell_{N}-\ell_{N/2}\|_\infty}.
\end{align*}
We will try $N=2^i$ for $i=1,2,\ldots,12$. Also, the actual error can be estimated by using $\|\ell-\ell_N\|_\infty\approx2^{-p(N)}\|\ell_{N}-\ell_{N/2}\|_\infty$.

The first set of numerical results from the convergence analysis is summarized in Tables~\ref{tab:ARL__theta__0_01},~\ref{tab:ARL__theta__0_1},~\ref{tab:ARL__theta__0_5}, and~\ref{tab:ARL__theta__1_0}. Specifically, each table is for a specific value of $\theta$, and reports the estimated convergence rate for various values of the ARL to false alarm for two methods: one proposed in this work and one proposed in the work of~\cite{Moustakides+etal:SS11}. For the latter, the corresponding results are shown in brackets with [NaN]'s indicating the method's failure. We observe that, as expected from Theorem~\ref{thm:error-bound}, the rate of convergence of our method is quadratic. More importantly, it is attained almost right away, regardless of the magnitude of the change or the value of the ARL to false alarm. The reason is the use of the linear interpolation to approximate the solution of the corresponding integral equation, which is almost linear with respect to the headstart. Hence, one would expect our numerical scheme to work well even for small $N$. The results presented in the tables do confirm that. On the other hand, the method of~\cite{Moustakides+etal:SS11} can be seen to converge much slower, requiring $N$ to be in the thousands or higher to deliver decent accuracy.
\begin{sidewaystable}
    \centering
    \begin{tabularx}{\textwidth}{c *{9}{Y}}
    \toprule

     & \multicolumn{2}{c}{$\gamma=10^2$ ($A=99.2$)}
     & \multicolumn{2}{c}{$\gamma=10^3$ ($A=994.2$)}
     & \multicolumn{2}{c}{$\gamma=10^4$ ($A=9941.9$)}
     & \multicolumn{2}{c}{$\gamma=10^5$ ($A=99419.0$)}\\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
          $N$ & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate\\
    \midrule
        \multirow{2}{*}{2}&100.48747&-&1,002.54172&-&10,021.90967&-&100,215.7892&-\\
        &[NaN]&-&[NaN]&-&[NaN]&-&[NaN]&-\\
        \midrule
        \multirow{2}{*}{4}&100.48747&0.37551&1,002.54172&0.68221&10,021.90967&0.77378&100,215.7892&0.34451\\
        &[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{8}&100.48747&-34.82763&1,002.54172&-32.94567&10,021.90967&-38.09409&100,215.7892&-27.23368\\
        &[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{16}&100.22051&-0.72003&1,001.72125&-1.39624&10,014.54917&-1.44092&100,142.96209&-1.44092\\
        &[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{32}&100.10203&2.81168&1,000.52457&2.57269&10,002.81619&2.55093&100,025.87105&2.54878\\
        &[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{64}&100.08052&2.09695&1,000.33233&2.00637&10,000.90452&1.99517&100,006.76431&1.99403\\
        &[89.6219]&-&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{128}&100.07523&2.02038&1,000.28277&1.99916&10,000.4097&1.99661&100,001.81661&1.99635\\
        &[104.01353]&[10.96141]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{256}&100.07391&2.00506&1,000.27031&1.99966&10,000.28518&1.99901&100,000.57131&1.99894\\
        &[99.57944]&[-6.54683]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{512}&100.07358&2.00126&1,000.2672&1.99991&10,000.25399&1.99974&100,000.25943&1.99973\\
        &[100.11896]&[8.90420]&[995.50653]&-&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{1024}&100.0735&2.00032&1,000.26642&1.99998&10,000.24619&1.99993&100,000.18142&1.99993\\
        &[100.07442]&[-2.61104]&[1,000.52276]&[9.90705]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{2048}&100.07348&2.00008&1,000.26622&1.99999&10,000.24424&1.99998&100,000.16191&1.99999\\
        &[100.07813]&[0.38884]&[1,000.3918]&[1.47489]&[NaN]&[NaN]&[NaN]&[NaN]\\
        \midrule
        \multirow{2}{*}{4096}&100.07347&-&1,000.26617&-&10,000.24375&-&100,000.15704&-\\
        &[100.07324]&-&[1,000.31815]&-&[NaN]&-&[NaN]&-\\
    \bottomrule
    \end{tabularx}
    \caption{Convergence analysis for $\ARL(\mathcal{S}_A)$ assuming $\theta=0.01$.}
    \label{tab:ARL__theta__0_01}
\end{sidewaystable}
\begin{sidewaystable}
    \centering
    \begin{tabularx}{\textwidth}{c *{9}{Y}}
    \toprule

     & \multicolumn{2}{c}{$\gamma=10^2$ ($A=94.34$)}
     & \multicolumn{2}{c}{$\gamma=10^3$ ($A=943.41$)}
     & \multicolumn{2}{c}{$\gamma=10^4$ ($A=9434.08$)}
     & \multicolumn{2}{c}{$\gamma=10^5$ ($A=94340.5$)}\\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
          $N$ & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate\\
    \midrule
        \multirow{2}{*}{2}&102.58157&-&1,022.1717&-&10,218.14169&-&102,177.54835&-\\
        &[$\approx 4\times 10^{10}$]&-&[$\approx 5.2\times 10^{11}$]&-&[$\approx 6.8\times 10^{11}$]&-&[$\approx 6.9\times 10^{11}$]&-\\
        \midrule
        \multirow{2}{*}{4}&102.55852&-7.36401&1,022.00734&-7.69785&10,216.55311&-7.73156&102,161.71661&-7.73494\\
        &[$\approx 2.9\times 10^{9}$]&[8.39767]&[$\approx 3.9\times 10^{11}$]&[2.50739]&[$\approx 6.7\times 10^{11}$]&[-0.22883]&[$\approx 7\times 10^{11}$]&[-0.72388]\\
        \midrule
        \multirow{2}{*}{8}&100.72368&2.24005&1,004.66098&2.19119&10,044.04284&2.18632&100,437.56746&2.18583\\
        &[$\approx 3.3\times 10^{7}$]&[13.16679]&[$\approx 2.2\times 10^{11}$]&[0.81472]&[$\approx 6.4\times 10^{11}$]&[-1.53456]&[$\approx 7.2\times 10^{11}$]&[-0.97868]\\
        \midrule
        \multirow{2}{*}{16}&100.39247&1.96637&1,001.3746&1.95635&10,011.20082&1.95538&100,109.16953&1.95529\\
        &[$\approx 4.1\times 10^4$]&[7.54089]&[$\approx 6.9\times 10^{10}$]&[2.01676]&[$\approx 5.9\times 10^{11}$]&[0.12059]&[$\approx 7.5\times 10^{11}$]&[-1.02206]\\
        \midrule
        \multirow{2}{*}{32}&100.3113&2.00152&1,000.55799&1.99687&10,003.02932&1.99639&100,027.44935&1.99635\\
        &[128.34953]&[2.03617]&[$\approx 8\times 10^{9}$]&[5.07392]&[$\approx 4.9\times 10^{11}$]&[3.77045]&[$\approx 8\times 10^{11}$]&[-0.93099]\\
        \midrule
        \multirow{2}{*}{64}&100.29088&2.00006&1,000.35206&1.99884&10,000.96814&1.99872&100,006.83578&1.9987\\
        &[99.87488]&[-3.6633]&[$\approx 1.8\times 10^{8}$]&[10.40265]&[$\approx 3.2\times 10^{11}$]&[-1.53804]&[$\approx 8.8\times 10^{11}$]&[-0.63049]\\
        \midrule
        \multirow{2}{*}{128}&100.28576&1.99999&1,000.30045&1.99968&10,000.45155&1.99965&100,001.66941&1.99965\\
        &[100.24439]&[3.23903]&[$\approx 4.3\times 10^5$]&[14.55815]&[$\approx 1.3\times 10^{11}$]&[1.01609]&[$\approx 10^{12}$]&[-0.01633]\\
        \midrule
        \multirow{2}{*}{256}&100.28448&2.0&1,000.28754&1.99992&10,000.32232&1.99991&100,000.37698&1.99991\\
        &[100.3111]&[2.2709]&[1,249.19154]&[3.06013]&[$\approx 2\times 10^{10}$]&[3.82822]&[$\approx 1.1\times 10^{12}$]&[7.72911]\\
        \midrule
        \multirow{2}{*}{512}&100.28416&2.0&1,000.28431&1.99998&10,000.29001&1.99998&100,000.05382&1.99998\\
        &[100.28414]&[9.51626]&[1,000.05942]&[-5.63427]&[$\approx 7.6\times 10^{8}$]&[8.34186]&[$\approx 9.2\times 10^{11}$]&[-4.70896]\\
        \midrule
        \multirow{2}{*}{1024}&100.28408&2.0&1,000.2835&1.99999&10,000.28193&1.99999&99,999.97303&2.00001\\
        &[100.28415]&[-1.36052]&[1,000.14412]&[1.30463]&[$\approx 3.8\times 10^{6}$]&[15.53288]&[$\approx 4.5\times 10^{11}$]&[0.26133]\\
        \midrule
        \multirow{2}{*}{2048}&100.28406&2.0&1,000.2833&2.00000&10,000.27991&2.00000&99,999.95283&1.99995\\
        &[100.28408]&[2.0]&[1,000.23442]&[1.93052]&[12,725.65581]&[9.49778]&[$\approx 7.4\times 10^{10}$]&[2.79838]\\
        \midrule
        \multirow{2}{*}{4096}&100.28406&-&1,000.28325&-&10,000.27941&-&99,999.94779&-\\
        &[100.28406]&-&[1,000.28103]&-&[10,003.04331]&-&[$\approx 3\times 10^{9}$]]&-\\
    \bottomrule
    \end{tabularx}
    \caption{Convergence analysis for $\ARL(\mathcal{S}_A)$ assuming $\theta=0.1$.}
    \label{tab:ARL__theta__0_1}
\end{sidewaystable}
\begin{sidewaystable}
    \centering
    \begin{tabularx}{\textwidth}{c *{9}{Y}}
    \toprule

     & \multicolumn{2}{c}{$\gamma=10^2$ ($A=74.76$)}
     & \multicolumn{2}{c}{$\gamma=10^3$ ($A=747.62$)}
     & \multicolumn{2}{c}{$\gamma=10^4$ ($A=7476.15$)}
     & \multicolumn{2}{c}{$\gamma=10^5$ ($A=74761.5$)}\\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
          $N$ & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate\\
    \midrule
        \multirow{2}{*}{2}&112.08343&-&1,115.94875&-&11,154.51566&-&111,540.26117&-\\
        &[40.48187]&-&[50.02797]&-&[51.15832]&-&[51.27337]&-\\
        \midrule
        \multirow{2}{*}{4}&102.84839&2.23395&1,024.79306&2.20209&10,244.14193&2.19892&102,437.69936&2.1986\\
        &[70.3026]&[0.17979]&[109.88631]&[-0.9552]&[115.49826]&[-1.07609]&[116.08141]&[-1.08829]\\
        \midrule
        \multirow{2}{*}{8}&100.9572&2.18209&1,005.64335&2.18298&10,052.41311&2.18308&100,520.1782&2.18309\\
        &[96.62922]&[1.81075]&[225.94273]&[-0.75784]&[251.147]&[-1.00629]&[253.8778]&[-1.03152]\\
        \midrule
        \multirow{2}{*}{16}&100.57276&2.0067&1,001.74986&2.0052&10,013.43&2.00504&100,130.29863&2.00502\\
        &[104.13341]&[2.86321]&[422.18974]&[-0.45992]&[523.63068]&[-0.95905]&[535.55853]&[-1.01016]\\
        \midrule
        \multirow{2}{*}{32}&100.47686&2.00153&1,000.77749&2.0011&10,003.69296&2.00106&100,032.91489&2.00106\\
        &[101.45615]&[2.02681]&[692.12063]&[0.10653]&[1,053.34664]&[-0.89331]&[1,102.90019]&[-0.99597]\\
        \midrule
        \multirow{2}{*}{64}&100.45288&2.00037&1,000.53434&2.00026&10,001.25812&2.00025&100,008.56303&2.00025\\
        &[100.52541]&[6.07368]&[942.83838]&[1.37428]&[2,037.25903]&[-0.77560]&[2,234.4169]&[-0.98059]\\
        \midrule
        \multirow{2}{*}{128}&100.44689&2.00009&1,000.47355&2.00007&10,000.64937&2.00006&100,002.47469&2.00006\\
        &[100.44801]&[1.84739]&[1,039.55157]&[3.70162]&[3,721.61729]&[-0.54838]&[4,467.20008]&[-0.95572]\\
        \midrule
        \multirow{2}{*}{256}&100.44539&2.00002&1,000.45836&2.00002&10,000.49718&2.00002&100,000.95257&2.00002\\
        &[100.44504]&[3.9197]&[1,016.14038]&[0.72821]&[6,184.8977]&[-0.0999]&[8,797.78196]&[-0.90889]\\
        \midrule
        \multirow{2}{*}{512}&100.44501&2.00001&1,000.45456&2.0&10,000.45913&2.0&100,000.57205&2.0\\
        &[100.44497]&[0.61014]&[1,001.12312]&[4.70144]&[8,824.79151]&[0.84811]&[16,928.882]&[-0.81709]\\
        \midrule
        \multirow{2}{*}{1024}&100.44492&2.0&1,000.45361&2.0&10,000.44962&2.0&100,000.4769&2.00002\\
        &[100.44491]&[1.9999]&[1,000.4463]&[2.95142]&[10,291.28225]&[7.04837]&[31,254.63194]&[-0.63583]\\
        \midrule
        \multirow{2}{*}{2048}&100.4449&2.0&1,000.45337&2.0&10,000.44724&2.0&100,000.4531&1.99992\\
        &[100.44489]&[2.0]&[1,000.4529]&[5.77463]&[10,257.2434]&[-2.68232]&[53,514.38599]&[-0.27727]\\
        \midrule
        \multirow{2}{*}{4096}&100.44489&-&1,000.45331&-&10,000.44665&-&100,000.44718&-\\
        &[100.44489]&-&[1,000.4533]&-&[10,023.86256]&-&[80,490.89164]&-\\
    \bottomrule
    \end{tabularx}
    \caption{Convergence analysis for $\ARL(\mathcal{S}_A)$ assuming $\theta=0.5$.}
    \label{tab:ARL__theta__0_5}
\end{sidewaystable}
\begin{sidewaystable}
    \centering
    \begin{tabularx}{\textwidth}{c *{9}{Y}}
    \toprule
     & \multicolumn{2}{c}{$\gamma=10^2$ ($A=56.0$)}
     & \multicolumn{2}{c}{$\gamma=10^3$ ($A=560.0$)}
     & \multicolumn{2}{c}{$\gamma=10^4$ ($A=5603.5$)}
     & \multicolumn{2}{c}{$\gamma=10^5$ ($A=56037.0$)}\\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
          $N$ & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate & $\ARL(\mathcal{S}_A)$ & Rate\\
    \midrule
        \multirow{2}{*}{2}&126.30518&-&1,255.83903&-&12,558.81029&-&125,585.16094&-\\
        &[18.90254]&-&[21.19442]&-&[21.4518]&-&[21.47784]&-\\
        \midrule
        \multirow{2}{*}{4}&102.91218&3.73911&1,021.53276&3.80202&10,214.35752&3.80672&102,139.87197&3.80718\\
        &[36.13905]&[-0.44375]&[46.03212]&[-0.96967]&[47.27734]&[-1.03168]&[47.40492]&[-1.038]\\
        \midrule
        \multirow{2}{*}{8}&101.36866&1.73324&1,007.01242&1.58154&10,069.74031&1.56622&100,694.31777&1.56474\\
        &[59.58298]&[0.04205]&[94.67408]&[-0.88292]&[100.07532]&[-1.00644]&[100.64298]&[-1.01923]\\
        \midrule
        \multirow{2}{*}{16}&100.8784&2.04648&1,001.71281&2.13084&10,016.23849&2.14898&100,158.83358&2.15053\\
        &[82.35344]&[0.74187]&[184.37489]&[-0.74321]&[206.14337]&[-0.98293]&[208.5477]&[-1.00859]\\
        \midrule
        \multirow{2}{*}{32}&100.76017&2.00161&1,000.52849&1.98007&10,004.47337&1.96408&100,041.24097&1.96265\\
        &[95.96936]&[1.6784]&[334.52493]&[-0.49879]&[415.78372]&[-0.94804]&[425.64534]&[-0.99915]\\
        \midrule
        \multirow{2}{*}{64}&100.73062&2.00075&1,000.22665&2.00270&10,001.43135&2.01008&100,010.79695&2.0115\\
        &[100.22338]&[2.99676]&[546.69114]&[-0.09267]&[820.23101]&[-0.8899]&[859.58367]&[-0.99075]\\
        \midrule
        \multirow{2}{*}{128}&100.72324&2.00019&1,000.15136&2.00010&10,000.67756&1.99883&100,003.2628&1.99767\\
        &[100.75632]&[5.36654]&[772.93304]&[0.5279]&[1,569.69277]&[-0.78224]&[1,721.91099]&[-0.97816]\\
        \midrule
        \multirow{2}{*}{256}&100.72139&2.00005&1,000.13254&2.00004&10,000.4889&2.00011&100,001.37516&2.00056\\
        &[100.73537]&[1.02544]&[929.84595]&[1.38532]&[2,858.62024]&[-0.58435]&[3,420.64864]&[-0.95481]\\
        \midrule
        \multirow{2}{*}{512}&100.72093&2.00001&1,000.12783&2.00001&10,000.44174&2.00001&100,000.90348&1.99996\\
        &[100.72228]&[3.26059]&[989.91309]&[2.54712]&[4,791.19095]&[-0.24044]&[6,713.34755]&[-0.90975]\\
        \midrule
        \multirow{2}{*}{1024}&100.72082&2.0&1,000.12666&2.0&10,000.42995&2.0&100,000.78555&2.00001\\
        &[100.72085]&[4.39192]&[1,000.19036]&[6.04614]&[7,074.24421]&[0.30588]&[12,899.39463]&[-0.82298]\\
        \midrule
        \multirow{2}{*}{2048}&100.72079&2.0&1,000.12636&2.0&10,000.427&1.99999&100,000.75607&1.99996\\
        &[100.72078]&[4.19398]&[1,000.34589]&[0.63957]&[8,921.11562]&[1.08545]&[23,842.84271]&[-0.66023]\\
        \midrule
        \multirow{2}{*}{4096}&100.72078&-&1,000.12629&-&10,000.42626&-&100,000.7487&-\\
        &[100.72078]&-&[1,000.15503]&-&[9,791.44373]&-&[41,137.15155]&-\\
    \bottomrule
    \end{tabularx}
    \caption{Convergence analysis for $\ARL(\mathcal{S}_A)$ assuming $\theta=1.0$.}
    \label{tab:ARL__theta__1_0}
\end{sidewaystable}
\begin{sidewaystable}[p]
    \centering
    \begin{tabularx}{\textwidth}{c *{10}{Y}}
    \toprule
     &
     & \multicolumn{2}{c}{$\gamma=10^2$}
     & \multicolumn{2}{c}{$\gamma=10^3$}
     & \multicolumn{2}{c}{$\gamma=10^4$}
     & \multicolumn{2}{c}{$\gamma=10^5$}\\
    \cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8} \cmidrule(lr){9-10}
          $\theta$ & $R_0^r=r$ & $\ARL(\mathcal{S}_A^r)$ & $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ & $\ARL(\mathcal{S}_A^r)$ & $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ & $\ARL(\mathcal{S}_A^r)$ & $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ & $\ARL(\mathcal{S}_A^r)$ & $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$\\
    \midrule
        \multirow{4}{*}{0.01}&0&100.27&5.86&1,000.27&176.65&10,000.24&4,565.16&100,000.16&78,476.25\\\cmidrule(lr){2-10}
        &$10^2$&1.08&0.42&900.27&3.81767&9,900.24&4,565.15763&99,900.16&78,476.24\\\cmidrule(lr){2-10}
        &$10^3$&1.0&0.0&4.11&17.51&9,000.24&4,561.56068&99,000.16&78,475.93\\\cmidrule(lr){2-10}
        &$10^4$&1.0&0.0&1.0&0.0&35.49&438.74&90,000.16&78,336.91\\
    \midrule
        \multirow{4}{*}{0.1}&0&100.28&45.92&1,000.28&783.89&10,000.28&9,470.47&99,999.95&99,060.19\\\cmidrule(lr){2-10}
        &$10^2$&4.13&13.58&900.28&782.54&9,900.28&9,470.36&99,899.95&99,060.18\\\cmidrule(lr){2-10}
        &$10^3$&1.0&0.0&35.52&218.63&9,000.28&9,437.96&98,999.95&99,057.08\\\cmidrule(lr){2-10}
        &$10^4$&1.0&0.0&1.0&0.0&349.47&2,525.59&89,999.95&98,606.29\\
    \midrule
        \multirow{4}{*}{0.5}&0&100.44&87.69&1,000.45&973.27&10,000.45&9,956.0&100,000.45&99,937.89\\\cmidrule(lr){2-10}
        &$10^2$&18.11&51.07&900.45313&969.33&9,900.45&9,955.62&99,900.45&99,937.85\\\cmidrule(lr){2-10}
        &$10^3$&1.00001&0.05&173.96&552.33&9,000.44&9,908.34&99,000.45&99,933.15\\\cmidrule(lr){2-10}
        &$10^4$&1.0&0.0&1.0&0.52&1,732.4&5,608.77&90,000.43&99,440.75\\
    \midrule
        \multirow{4}{*}{1.0}&0&100.72&95.72&1,000.13&991.03&10,000.43&9,986.83961&100,000.76&99,982.6\\\cmidrule(lr){2-10}
        &$10^2$&34.92&72.78&899.83&986.41496&9,900.43&9,986.39&99,900.76&99,982.55\\\cmidrule(lr){2-10}
        &$10^3$&1.55&10.15&339.33&745.3&8,997.51&9,937.25&99,000.75&99,977.68\\\cmidrule(lr){2-10}
        &$10^4$&1.0&0.13&6.39&103.18&3,387.5&7,494.37&89,971.64&99,479.67\\
    \bottomrule
    \end{tabularx}
    \caption{Selected values of $\ARL(\mathcal{S}_A^r)$ and $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ vs. $R_0^r$, $A$, and $\theta$.}
    \label{tab:ARL+Var_vs_R_A_theta}
\end{sidewaystable}

We now proceed to computing the higher moments of the distribution of the GSR procedure's Run-Length. Specifically, consider the GSR procedure's stopping time's standard deviation: $\sqrt{\Var_\infty(\mathcal{S}_A^r)}\triangleq\sqrt{\EV_\infty[(\mathcal{S}_A^r)^2]-(\ARL(\mathcal{S}_A^r))^2}$. Table~\ref{tab:ARL+Var_vs_R_A_theta} presents selected values of $\ARL(\mathcal{S}_A^r)$ and $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ for different values of $R_0^r$, $A$, and $\theta$. We note that since the GSR procedure's stopping rule is asymptotically (as $A\to\infty$) geometric (see, e.g.,~\citealp{Pollak+Tartakovsky:TVP09,Tartakovsky+etal:IWAP08}), one would expect $\ARL(\mathcal{S}_A^r)$ and $\sqrt{\Var_\infty(\mathcal{S}_A^r)}$ to be close to one another. As may be seen from Table~\ref{tab:ARL+Var_vs_R_A_theta} this is, in fact, the case. However, the asymptotic geometric distribution is attained quicker for lower values of the headstart (i.e., for small $R_0^r$) and higher magnitudes of the change (i.e., for large $\theta$).

We conclude this section with a set of brief analysis of $\Pr_\infty(\mathcal{S}_A>k)$, $k\ge1$, i.e., the survival function of the SR stopping time, and consider various values of $\theta$ and $\ARL(\mathcal{S}_A)\approx\gamma>1$. Recall that that asymptotically (as $\gamma\to\infty$) the $\Pr_\infty$-distribution of $\mathcal{S}_A$ is geometric, and the parameter is the reciprocal of the ARL to false alarm; cf.~\cite{Pollak+Tartakovsky:TVP09,Tartakovsky+etal:IWAP08}. Hence, $\Pr_\infty(\mathcal{S}_A>k)$ should be close to $(1-1/\gamma)^k$ for sufficiently large $\gamma$. We therefore will focus on $\log\Pr_\infty(\mathcal{S}_A>k)$, which as a function of $k$ should be asymptotically (as $A\to\infty$) linear with a slope of $\log(1-1/\gamma)$.

Figure~\ref{fig:logPMF__ARL_100} shows the results for the case of $\gamma=10^2$ for $\theta=0.01,0.1,0.5$ and $1.0$. The dashed line corresponds to the geometric distribution expected in the limit. Note that $\gamma=10^2$ is generally too low of an ARL to false alarm for the asymptotic distribution to kick in. Hence, for $\theta=0.01$ and $0.1$ we observe a substantial deviation of the actual distribution of the GSR stopping time from the limiting geometric one. Yet for $\theta=0.5$ and $1.0$ the distribution is reasonably close to the geometric. Figure~\ref{fig:logPMF__ARL_1000} shows the results for the case of $\gamma=10^3$, also for $\theta=0.01,0.1,0.5$ and $1.0$. In this case the distribution for $\theta=0.5$ and $1.0$ is almost indistinguishable from geometric, and that for $\theta=0.1$ is much closer to being geometric than when $\gamma=10^2$; that said, $\theta=0.01$ is still a problem. Figure~\ref{fig:logPMF__ARL_10000} corresponds to the case when $\gamma=10^4$. In this case the distribution corresponding to $\theta=0.1$ is almost geometric, and that corresponding to $\theta=0.01$ is still considerably different from the limiting geometric. One conclusion to draw from this analysis is that the rate of convergence to the geometric distribution heavily depends on the magnitude of the change. For faint changes ($\theta=0.01$ and $0.1$), the rate is extremely slow, even if the ARL to false alarm is large. However, for moderate and contrast changes ($\theta=0.5$ and $1.0$), the distribution is very close to geometric already for $\gamma=10^2$.
\begin{sidewaysfigure}[p]
\begin{subfigure}[b]{0.35\textheight}
    \centering
    \includegraphics[width=0.35\textheight]{logPMF_ARL_100}
    \caption{$\gamma=10^2$.}
    \label{fig:logPMF__ARL_100}
\end{subfigure}
\begin{subfigure}[b]{0.35\textheight}
    \centering
    \includegraphics[width=0.35\textheight]{logPMF_ARL_1000}
    \caption{$\gamma=10^3$.}
    \label{fig:logPMF__ARL_1000}
\end{subfigure}
\begin{subfigure}[b]{0.35\textheight}
    \centering
    \includegraphics[width=0.35\textheight]{logPMF_ARL_10000}
    \caption{$\gamma=10^4$.}
    \label{fig:logPMF__ARL_10000}
\end{subfigure}
\caption{$\log\Pr(\mathcal{S}_A>k)$ as a function of $k=1,2,\ldots,\gamma$.}
\end{sidewaysfigure}

\section{Further discussion}
\label{sec:further-discussion}

The proposed numerical method (including the change-of-measure trick) can be extended to other performance measures as well as to other detection procedures. For example, note that throughout the paper we used the usual ARL to false alarm as a capture of the risk of false detection. An alternative, more exhaustive measure of he risk of false detection is based on the ``worst'' $\Pr_\infty$-probability of sounding a false alarm (PFA) within a batch of $m\ge1$ successive observations indexed $k+1$ through $k+m$ (inclusive), where $k\ge0$. To define this probability, one can either use $\sup_{0\le k}\Pr_\infty(k<\T\le k+m)$ or $\sup_{0\le k}\Pr_\infty(k<\T\le k+m|\T> k)$; see, e.g.,~\cite{Lai:IEEE-IT1998}. The constraint $\ARL(\T)\ge\gamma$, where $\gamma>1$, is then replaced with $\sup_{0\le k}\Pr_\infty(k<\T\le k+m)<\alpha$ or $\sup_{0\le k}\Pr_\infty(k<\T\le k+m|\T>k)<\alpha$, respectively, where $\alpha\in(0,1)$. As shown by~\cite{Lai:IEEE-IT1998}, either of these new constraints is stronger than the original $\ARL(\T)\ge\gamma$, assuming a properly picked $m\ge1$. Following~\cite{Tartakovsky:IEEE-CDC05,Tartakovsky:SA08-discussion}, let us focus on the conditional PFA and denote $\PFA_k^m(\T)\triangleq\Pr_\infty(k<\T\le k+m|\T>k)$. Our numerical method can be used to evaluate this probability for the SR--$r$ procedure using
\begin{align*}
\PFA_k^m(\mathcal{S}_A^x)
&=
1-\frac{\rho_{k+m}(x,A)}{\rho_k(x,A)},
\end{align*}
where $\rho_0(x,A)\equiv1$ for all $x,A\in\mathbb{R}$, and $\rho_k(x,A)$, for $k=1,2\ldots$ are given by~\eqref{eq:recursion2}.

We now show how to generalize the method to other procedures. Consider a generic detection procedure described by the stopping time $\mathcal{T}_A^s=\inf\{n\ge1\colon V_n^s\ge A\}$
where $A>0$ is a detection threshold, and $\{V_n^s\}_{n\ge0}$ is a generic Markovian detection statistic that admits the recursion $V_{n}^s=\psi(V_{n-1}^s)\LR_{n}$ for $n=1,2,\ldots$ with $V_0^s=s$, where $\psi(x)$ is a non-negative-valued function, and the headstart $s$ is fixed. This generic stopping time was considered by~\cite{Moustakides+etal:SS11} and it describes a rather broad class of detection procedures; in particular, the GSR procedure given by~\eqref{eq:T-SRr-def} and~\eqref{eq:statistic-SRr-def} is a member of this class with $\psi(x)=1+x$. Also, for the CUSUM chart (not considered here) it is enough to take $\psi(x)=\max\{1,x\}$. This universality of $\mathcal{T}_A^s$ enables one to evaluate practically any procedure that is a special case of $\mathcal{T}_A^s$, merely by picking the right $\psi(x)$.

The integral equations framework (including the numerical method) developed above can be easily extended to $\mathcal{T}_A^s$. It suffices to let
\begin{align*}
\mathcal{K}_d(x,y)
&=
\dfrac{\partial}{\partial y}\Pr_d(V_{n+1}^s\le y|V_n^s=x)
=
\dfrac{\partial}{\partial y}P_d^{\LR}\left(\frac{y}{\psi(x)}\right),
\; d=\{0,\infty\}
\end{align*}
to denote the transition probability density kernel for the Markov process $\{V_n^s\}_{n\ge1}$. Note that $dP_0^{\LR}(t)=tdP_{\infty}^{\LR}(t)$ is still true, and therefore, so is $\psi(x)K_0(x,y)=yK_{\infty}(x,y)$. This is a generalization of the $(1+x)K_0=yK_\infty(x,y)$ identity used earlier for the GSR procedure.

However, in general, while the change-of-measure trick can be used to develop numerical methods to evaluate other detection procedures, the improved convergence rate will not take effect as fast as for the GSR procedure. This is because of the GSR procedure's martingale property that the numerical method exploits.

\section{Conclusion}
\label{sec:conclusion}

We proposed a numerical method to compute the Generalized Shiryaev--Roberts (GSR) procedure's pre-change Run-Length distribution and its moments. The GSR procedure is an umbrella term for the original SR procedure and its extension -- the SR--$r$ procedure. The proposed method is based on the integral-equations approach, and uses the collocation framework with the basis functions selected so as to exploit a certain change-of-measure gimmick and a specific martingale property of the GSR procedure's detection statistic. This resulted in substantial improvement of the method's accuracy and overall stability. We also carried out a complete accuracy analysis of the method, and provided a tight upper bound on the method's error as well as showed the theoretical rate of convergence to be quadratic. By testing the method in a specific change-point scenario we confirmed that the accuracy is high for a wide range of changes (faint, moderate, contrast) and for a large range of values of the ARL to false alarm. We also confirmed that the method's accuracy remains high even if the partition is rough.

\section*{Acknowledgements}

The authors are thankful to Shelemyahu~Zacks (SUNY Binghamton), Shyamal~K.~De (SUNY Binghamton), Sven~Knoth (Helmut Schimdt University, Hamburg, Germany), and to George~V.~Moustakides (University of Patras, Greece) for reading this work's preliminary draft and for providing valuable feedback that helped improve the paper. The authors are also grateful to the Editor-in-Chief, Nitis~Mukhopadhyay (University of Connecticut-Storrs), and to the anonymous reviewers whose comments helped to improve the manuscript further.

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