Photodynamic therapy is a non-invasive skin treatment used to treat sun damaged skin, moderate to severe cystic acne, and superficial skin cancers. Photodynamic therapy uses intense pulsed light therapy (IPL) to increase cell turnover and stimulate collagen production.
Levulan photodynamic therapy involves the application of an amino acid that reacts to blue light. This treatment has been approved by the FDA for actinic keratosis, but is sometimes prescribed off-label for acne.


Let me take you back to my childhood for just a bit.
I'm my early years of falling in love with hip hop, it was a male dominated arena that featured next to no female acts. Sure there were the Lauryn Hill's, Missy Elliot's and – of course – Lil Kim's of the world. But no one quite made me feel the way that Trina did.
I mean, I was a teenage boy. She was Trina. What more needs to be said?
Fast forward to today, and Trina is still in the mix. And yes, she still fine as hell.
Anyway check out this clip from an upcoming epsisode of Love and Hip Hop: Miami. Things look to be getting kinda spicy!
These two won't be collaborating anytime soon.
Sorry to all the fans out there who were hoping for a Trick Daddy and Trina record, because it doesn't seem like these two will be collaborating anytime soon. The latest Love & Hip Hop Miami season has followed these two rappers and longtime friends as they attempted to recreate their chart-topping success from their 1990s "Nann" days. However, it was obvious that they were finding it difficult to work with each other decades later and reality television cameras captured their tense moments.


WHO WE ARE: We raise gentle, productive, mite-resistant, winter-hardy queen bees in western South Dakota. In addition to our bees, we also offer honey and free range eggs for sale. Available for pick up at the farm only.
WHERE: Our farm is located in the pine foothills of the Black Hills about 5 miles west of Hermosa, South Dakota, south of Highway 40.
Canyon Rim Honey Bees aspires to produce small numbers of high-quality queen bees which are locally adapted to the conditions of western South Dakota and surrounding regions.
We don't ship our bees south for the winter or west for pollination.
The queens we produce are those which have exactly the same characteristics that we ourselves desire in our bees: we want our bees to be gentle, productive, mite tolerant and winter hardy.


text:
Q: html videos not loading on iphone but load on desktop web and android I am trying to embed a video in my html page but it is not loading on the iPhone (all I get is a crossed out play button). It loads fine on the desktop and Android. It is not a browser issue because I get the same error using both Chrome and Safari on my iPhone.
<div id="video_container">
    <video width="400" height="400" controls preload="none" poster="/static/img/tag_logo.png">
        <source src="/static/videos/movie.mp4" type="video/mp4"></source>
    </video>
</div>

Interestingly I have tested out my code with the video from the video.js sample project. If I replace my current source tag with this the video loads:
<source src="http://video-js.zencoder.com/oceans-clip.mp4" type="video/mp4"></source>

However, I also downloaded the video from the video.js site, but it does not load on my iPhone
<source src="/static/videos/oceans-clip.mp4" type="video/mp4></source>

Codecs for the videos I am trying out all have AAC and H.264 codecs and are mp4s
I have also tried converting my video files to iPhone ready mp4 files using Miro Video Converter but still no luck with loading on the webpage
I am testing on localhost and am using nginx and flask
After more testing: according to the Chrome inspector I am getting response headers back for my video url GET request which explains why it plays on the desktop web. However, for iPhone, the Safari inspector says that I am not getting response headers back. And on Android, I do not get response headers back either, but for some reason the video is still able to play in the webpage on my Android device.
Any help would be appreciated

A: It could be your host server setup. For example when I use Go Daddy, SVG files won't display on my site. You can either check out the config or add a file that allows the MIME type. More on MIME types here: http://blogs.iis.net/bills/archive/2008/03/25/how-to-add-mime-types-with-iis7-web-config.aspx

A: Just change the video extension to .mov.

redpajama_set_name: RedPajamaStackExchange


Have you heard of Michael Phelps? He's the most decorated Olympian of all time. Present-day Olympic competitors might be shaking in their pools: Phelps has come out of retirement for the Summer Olympics in Rio.
Phelps will be making a big splash come night one: He's Team USA's flag bearer for the opening ceremony.
But the weight of the American flag won't be all he bears this month — Phelps will have to defend the gold in the 100-meter butterfly, which he last earned at the 2012 games in London.
1. Phelps has a lot of money.
The 31-year-old has a net worth of $55 million, the Gazette Review reported. His skills in the pool have earned him some serious sponsorships, including one with the athletic brand Under Armour.
2. He's off the market.
Phelps and his partner Nicole Johnson haven't officially said "I do," but the couple is intent on getting married: They've been engaged since February 2015, according to Us Magazine. Phelps and Johnson, who was Miss California USA 2010, are parents to son Boomer Robert, who was born May 5, People reported.
3. He is a true champion.
Phelps has 22 medals — including 18 gold — from three separate Olympics, making him the most decorated Olympian of all time. If he places at all of his events in Rio, Phelps will amass 28 medals.
4. Phelps no longer eats that stunning 12,000 calories a day.
While training for the Beijing Olympics in 2008, Phelps' colossal diet made headlines (and induced nausea) worldwide. He consumed one pound of pasta, an entire pizza and a host of energy drinks, the Washington Post reported. And that was just dinner.
With age comes wisdom — and a slower metabolism (but probably not for Phelps). He told USA Today that eating has become a job for him in his training.
"I am eating just to maintain a weight, trying to stay between 200 and 205 pounds," he said to the Associated Press in March. I think that's kind of my ideal fighting weight. Every now and then I take what I want, but I don't go too overboard on sweets or inappropriate fuel."
5. Some say Phelps has a physiological advantage over other swimmers.
His extra large wingspan, size 14 feet and double-jointedness are often attributed to Phelps' great swimming success. While some say his natural-born flippers make him faster in the water, former team physician H. Richard Weiner told Scientific American it's unlikely any of his special physical features have given the medalist a particular edge.


This, is of course, not the case.
Appearances can be so deceptive. Appearances can reveal more about the viewer than the viewee. The whiskey bottle is merely being reused as a water bottle, which he has filled from a well by the side of the lane. Pulled over, leaned under some foliage and filled with the cool, clear H20 that has never been anywhere near a tap, let alone a fluoridation treatment plant. The huge pile of cocaine, while contraband, isn't in fact narcotic. It's merely goat's cheese he's just taken delivery of, whiter and purer than the driven snow. Unpasteurized goat's milk and cheese, the sale of which is illegal in Ireland, he instead uses as currency amongst his pals to trade for goods and services. The goats, owned by a friend, just eat grass and flowers in the field, no feed, ever, to produce the lightest, purest tasting cheese you could imagine. In France, they might call it caillé. But this isn't France, this is Ireland.
And this is all part of Fergal Smith's big idea.
I decided to stop with the flying, stop getting on planes. I just do pick-ups now.
Rewind to a few days earlier and a chill, blustery early April night, Fergal picks me up from Shannon airport, wearing wellies. He's a little bit late, for which he's sorry, he's tired from a big a day in the garden. We get in his Transit and rattle towards the car park exit, there is untold amounts of crap piled up on the dash, the seat and the floor. Books, clothing, bits of tools, driftwood, glass bottles, flasks, tupperware, packets of seed, sweets, a tin whistle, a 5 month old black collie lab named Bua – Gaelic for to win. It is clear from the man's dashboard and passenger seat that he does not suffer from OCD or other cleanfreak disorders.
'Garden' is something of an understatement. The garden is not just two tracts of land, roughly an acre each, loaned to him by a wealthy benefactor. One, the community garden at Moy Hill, the other, a field for food crops at Moy House. The garden is an idea, a vision. A force for change, much more about people than about plants. It's about getting folk involved, showing them how they can change they way they live, what and how they eat. It's organic farming, just as his dad has being doing commercially for the past two decades, only as yet, no real money is involved. "Folk just pitch in and give us hours, and in return we feed them or give em somewhere to stay. It sorta runs on goodwill, good vibes." The idea is a place for people to work, learn, hang out, eat and play music. It's not a business, and for the moment Fergal does not want to be tied in to being a producer, rather, a grower. To show people what they themselves can each do, with a bit of know how. A tinge of airy fairy idealism maybe to the cynical ear except… it's actually working. Working incredibly well. He has seen a farm up for sale nearby, 80 acres of land that could one day make an ideal business, but for now wants to see how well things run on goodwill before large sums of money and loans become involved. Walk before run, garden before farm, so to speak. And for now the garden takes all priority, except of course when the surf is pumping.
Fergal has been living with Cornish transplants Mitch Corbett and Matt Smith in a old rented farmhouse at Lackamoor, but recently moved in with his 'blade', Sally. He offers me his room, admitting that the sheets are as yet unchanged. If I'm honest, the house is a mess. Sinister black damp grows in the corner of every bit of ceiling, bits of cardboard line the floor. The kitchen is stocked with all manner of dried beans, pulses, grains, spices, 5 kilo sacks of bulk bought health foods, but also looks like it hasn't seen much in the way of mop, cloth or detergent for several moons. I think it was Charles Bukowski who wrote, 'Show me a man with a clean kitchen, and nine times out of ten I'll show you a man with detestable spiritual qualities'. If that is true, it seems the boys' spirits are intact. There is one warm room with a coal burning stove, but tar drips down the chimney and coats the wall in an evil sludge. There's no TV, internet and phone reception are part time. It is dark, cramped, drafty, but I can't think of anywhere I'd rather be staying.
I read on The Inertia that Ferg wanted to be 'a positive force for change', and I'll admit to scoffing at the self- righteousness of it from a distance. But in less than 24 hours with the lad, I'm a convert. Partly because of his demeanour, which has none of the sanctimonious holier-than-thou approach 'eco-warriors' can tend to have. Perhaps more than that, he's not selling you something, he's just doing it. "I never had Facebook but used to put all that stuff up on my site about Monsanto, etc, like everyone, but at some point I asked myself, what am I actually doing about it? You can go on and on listing evils, but why not just do something positive instead?" Indeed, harping on about evil agribusiness or GM or palm oil or Fukushima can be a bit like the competitive grief phenom social media has wrought upon us. You can do so much on your 'device', but Ferg figured you could do much, much more with a spade. We shovel a great mound of sand onto the trailer attached to the back of his "totally illegal, like" van, and talk some more surf.
"Sorry, I don't buy Surf Europe" he says (about the last one). He's not really sorry. "But I've got a subscription to this, it's the best magazine in the world." I must admit, it's pretty good.
"To hell or to Connacht" was the grim ultimatum Oliver Cromwell famously put to the Irish in 1650, in violent attempt to settle by force the barren western coastal province. Fergal's dad, a native of Wembley, north London opted for the latter, a few years after marrying Fergal's mother, a native of Catholic West Belfast. With Fergal's paternal grandad being a C of E vicar, and this being during the grim '80s Troubles, was crossing the sectarian divide a problem with her family? "I'd imagine a few things were probably said at the time," Fergal observes. Having something of an atypical background gave Fergal and brother Kevin a relatively unique dialectic outlook on Ireland's situation. "We'd go back to Belfast with mum and the British Army would stop us and checkpoints, ask to see ID and search the car, then when we'd go with dad and they'd chat about London and wave us on. We've seen both sides. We went to protestant primary school and catholic secondary, so we're sort of both and neither." When the family first moved to Westport, Co Mayo and Fergal's dad set up one of the west coast's first organic farms, his other unusual habits tended to draw attention from the locals. "Dad used to run marathons and train around Mayo, and people would see him and stop and go 'Where's the fire?' Running along roads wasn't something people did for fun. Then they'd see him doing yoga and thought he was worshipping the devil," chuckles Ferg.
In terms of recent output, Fergal's clip on his website makes compelling viewing. Compelling because of the quality, but also compelling in that you are compelled to go to his website to watch it. "It took us two years to make a four minute clip. Originally it was supposed to go in the Analog movie, but they wanted just the footage and to edit it with their own music themselves. I wasn't really into that, so I kept it." The four minute clip was entered into a surf film festival in Ireland, and won, and the website sponsoring the festival (a UK- based surf forecasting site) wanted to host the clip on their site, as you might expect. Fergal however, took a different view. "I don't really like surfing websites," he says with a grin that belies how genuine the sentiment is. "They basically use stuff for their own gain. They weren't involved in making the clip, they weren't here. There's a perception that I'm obliged to hand it over to get views. I'm not bothered about millions of hits. I'd rather fewer people looked at it on my site, but people who are actually interested enough to see what I'm doing. Through that, I've meet loads of cool folk who come and look at the blog and learn about the garden and have got in touch, with something to contribute. So anyway, I kinda had a fall out with them (the website).
The surf forecast looks promising for the few days after I leave, but for the moment, we're in the community garden, planting out shrubs and herbs, making plans for a rebuild on the shed that fell victim to the big winter storm (nobody actually calls it 'Hercules' out loud, do they?). The garden is coming along just fine. Since Holly the pig cleared out the bramble roots, fruit trees have been planted, a central eating area with cob oven is under construction, and all manner of flowers, herbs, fruit bearing shrubs and perennials planted out to give bring to life an area that was for years an overgrown patch of wasteland. The vision is a place where kids can come and learn about growing, a central focus for the community to come and enjoy the bounty of their collective travails. A grant has been awarded for the reconstruction of the shed, not an insignificant sum, and thoughts are being banded around from haybale walls to grass roofs to classroom and cooking areas. The garden has its own well up under some bushes, and Fergal is mid way through explaining the evils of fluoride (he only drinks well water, never mains) when Irene, a lady from across the road appears with plants to donate (roses, blackcurrants, daffs) and introduces herself with words of encouragement.
"It's great that you lads are sanctifying this place," she says, "It used to be a gibbet." A gibbet is a place of public execution, often on high ground overlooking a town, bodies of the executed would hang in metal cages for weeks to serve as a warning to the population. Looking at the lie of the land, it's true, the cul de sac of houses opposite the garden is on the edge of a hill overlooking Lahinch, an ideal execution spot. She tells us about the hand. The hand that comes and taps her and family members on the shoulder in the dead of night. The firm, icy hand of the executioner. And if the community garden helps exorcise the hand, well then more power to it, she thinks. But as a fierce westerly blows in from the low end of the slope, Fergal's main concern, rather than the spirit world, is whether or not the aromatic herbs have enough shelter in the spot earmarked for them. Assuming that they do, some point later this year, they will be flavouring food grown down at Moy House eaten here by someone who pitched in a bit of time, between surfs, to donate labour to the cause. Labour of course being the one thing organic farming relies on, the thing that makes it largely unprofitable. Much like surfing, it requires serious hours to be put it. But similarly to surfing, the rewards are in the process of putting those hours in, rather than just the end product, be that the first tube ride, or first fork pulling of new spuds.
Irene, so impressed with the garden at her first proper look at it has invited us to her cottage over the road to collect more plants she wants to give to the cause. From the vantage point of her yard, you can see across the town of Lahinch and the hinterland to the east where the 80 acres Fergal has had his eye on lies, still on the market after a few years, still unlikely to be sold anytime soon. The family who owns it aren't in a rush, he explains. They're waiting for the right people at the right time. He's hoping his project might be a means to that end, but not pinning his hopes on it. In the meantime, he'll reap reward enough from what he's doing day to day, and from surfing the reefs in between.
And yet while Ferg goes about realising his big idea, I can't but wonder if he'd be tempted back into full-time surfing by a massive contract. After all, the way and waves he surfs are box office, he's an interesting, distinctive character in a sea of bland. What if some brand called you up and said 'Ferg, we want to pay you loads of money to stay home, get tubed off your head in Ireland, make clips, grow your veg and be cool,' I ask.
He grins and thinks about it for a moment.


text:
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.. include:: ./text.html
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.. include:: ./hello.md

.. include:: ../rst_toc.txt

redpajama_set_name: RedPajamaGithub


Mitchella repens, commonly called partridge berry (also partridgeberry), is an herbaceous, woodland, mat-forming, evergreen perennial that is indigenous to eastern North America. This is a prostrate plant that reaches only 2" tall but will spread by trailing stems to 12" wide or more.
This evergreen ground cover displays interesting foliage, flowers, and fruit. Plant in woodland/shade gardens, under trees and in part shade areas of border fronts and rock gardens. Also effective around small ponds. Many gardeners believe this ground cover is not appropriately aggressive for large areas and is best grown in smaller sites.
Wildlife Value: This plant is moderately resistant to damage from deer. Its fruits are eaten by songbirds, ruffed grouse, wild turkeys, quail, and small mammals.
Partridgeberry has oval to round, opposite, glossy, dark green leaves (to 3/4" long) with whitish veins appear in pairs along the stems. The leaves are 0.5-0.7 in. long. This plant has 6-12 in. slender, trailing stems.
Partridgeberry features fragrant, four-lobed, funnel-shaped, white flowers (to 1/2" long) which are lightly flushed with pink. The flowers bloom in spring in pairs mostly at the stem ends. They are subsequently followed by round, bright red berries that typically ripen in late summer. A pair of flowers yields one berry, hence the additional common name of twinberry for this plant. The berries, which have a tiny dimple and star-shaped marks, are edible but rather tasteless. The berries often persist on the plants throughout winter.
Partridgeberry is best grown in organically rich, medium moisture, well-drained soils in part shade to full shade. It tolerates some drought, but best with consistent watering. The stems will root at the nodes to form colonies in optimum growing conditions. Plantings are sometimes difficult to establish.


Faculty in the Department of Pharmaceutical Sciences (PSC) are recognized leaders in the field of drug discovery and development.
Faculty in PSC receive a collective $8 million in extramural grants and contracts each year in the areas of computational biology, structural biology, synthetic chemistry, nanomedicine, drug transportation, pharmacogenomics, and industrial pharmaceutics.
We strive to advance basic, translational, and clinical research across the spectrum of drug discovery and development.
With strengths in structure-based drug design, our expertise in physical, computational, synthetic, and protein chemistry cuts across several centers of excellence in Computer Aided Drug Design, Nuclear Magnetic Resonance (NMR), and Mass Spectrometry. Innovative research in complex biological systems, proteomics, metabolomics, pharmacogenomics, and therapeutic engineering seeks to develop targeted innovative medicines and devices to improve human health. Optimization of therapeutic dosing regimens is the goal of our concentrated efforts in pharmacokinetics, pharmacometrics and drug delivery, including nanotechnology. Interdisciplinary research in collaboration with the University of Maryland School of Engineering is the primary focus of the Bio- and Nano-technology Center.
In this four-minute video, Paul Shapiro, PhD, professor and chair of the Department of Pharmaceutical Sciences at the School of Pharmacy, provides an overview of the wide array of research activities within the department.
Research conducted in the department is interdisciplinary, and researchers have access to state-of-the-art facilities.
Our current research areas include medicinal chemistry, biochemistry, bioinorganic chemistry, cellular and molecular biology, computational chemistry, biophysics, microbiology, neuroscience, pharmacometrics, pharmacokinetics, drug formulation, drug transport and delivery, industrial pharmaceutical resarch, and translational and regulatory sciences.
The Department of Pharmaceutical Sciences publishes a periodic newsletter that contains the latest information and updates about special topics of interest in the department.


719 full-text articles. Page 6 of 21.
Early in April 1917, Lenin crossed the Russian border and returned to his homeland, courtesy of a sealed train arranged by the German government. Almost 100 years ago, the Russian Revolution shook the world – and it still does today. Learn how fresh ideas, websites, group activities, a terrific new book, and a teaching unit from the Choices Program (Brown University) – featuring an exciting role play – can energize your classroom. See how the Russian Revolution can ignite your students' passion for history!


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FRESNO, Calif. (KFSN) -- A Central Valley man who spent nearly a decade in jail after facing murder charges is now home with his family.
Neko Wilson was arrested back in 2009 for the killings of a Kerman couple.
Outside of the Fresno County Jail, screams could be heard from inside the lobby.
This as Wilson walked out after spending 9 years, 2 months and 13 days incarcerated.
"It's just unbelievable it so real it's a miracle," said Wilson.
The 36-year-old was arrested back in 2009 for his involvement in the murder of a Kerman couple during a marijuana home robbery.
Wilson was one of at least six people in on the plot.
However, he never left his car or went into the house when the killings of Gary and Sandra Debartolo took place.
Still, for nearly a decade Wilson faced murder charges under the felony murder rule which allows prosecutors to charge people with murder if they take part in a dangerous felony and someone gets killed in the process.
"They charged me with the death penalty," said Wilson. "I didn't kill nobody you can't fathom understand that and to get the chance to walk out."
That chance made possible because of his brother, a Bay Area attorney, who fought to change the law and won last month in getting a slight tweak.
"The people we are talking about they didn't kill anyone they did not murder anyone and they did not have a willingness to kill," said Neko's brother. "The felony murder rule is unjust it's barbaric."
But Sheriff Margaret Mims disagrees and supports the old law.
"Criminals don't always act alone and in this particular case there were several people involved who all should be held accountable and the unfortunate thing is now they won't be and it's not fair to the Debartolo's and community of crime," said Sheriff Mims.
Nonetheless, Jamilia Rand from Sacramento says this new bill has given her hope because she claims her son is in jail for a crime he also did not commit.
"This case marks the very first case that someone has been freed and I am hopeful that my son's case will come back with a not guilty conviction," said Rand.


Embedded with exotic ingredients and crafted to perfection.
Inspired by the fine art of chocolate making, Cadbury brings you a chocolate that's crafted to perfection. It is embedded with exotic Ingredients and comes in a variety of delectable flavours. Bon Apetit.
Contains added nature identical & artificial (Ethyl Vanilin) flavouring substances.
Allergen Information : Contains Milk, Almonds.
Manufactured on equipment that also processes products containing tree nuts.
Humidity and Temperature may cause product to develop a Whitish Layer, this does not affect its fitness for consumption.


text:
\section{Introduction}
\label{section:introduction}

Historically, randomized experiments were often carried out in medical and agricultural settings. In these settings,  sample sizes were often modest, typically on the order of hundreds or (occasionally) thousands of units. Outcomes commonly studied included average mortality or crop yield, and were characterized by relatively well-behaved data with thin tails.
Standard analyses in those settings typically  involved  estimating the average effect of the treatment by the difference in average outcomes by treatment group, followed by constructing  confidence intervals for the average causal effect using Normal distribution based
approximations. These methods originated in the 1920s, e.g.,  \citet{neyman1923, fisher1937design}, but they continue to be the standard in modern applications. See  \citet{wu2011experiments} for a recent discussion.

More recently many experiments are conducted using digital technology (see, e.g., \cite{kohavi2020trustworthy} for an overview).
 \citet[p.20]{gupta2019top} claim that  ``Together these organizations [Airbnb, Amazon,
Booking.com, Facebook, Google, LinkedIn, Lyft, Microsoft, Netflix, Twitter, Uber, and Yandex] tested more than one hundred thousand experiment treatments last year."
The settings for these online experiments are substantially different from those in biomedical and agricultural settings.
First, the experiments are often on a vastly different scale, with the number of units on the order of hundreds of thousands to tens of  millions. Second, the outcomes of interest, variables such as time spent by a consumer, sales per consumer or payments per service provider, are characterized by distributions with extremely thick tails. Third, the average treatment effects are often small relative to the standard deviation of the outcomes, even if their magnitude is substantively important.

For example, \citet{lewis2015unfavorable} analyze challenges with statistical power in experiments designed to measure digital advertising effectiveness. They discuss a hypothetical experiment where the average expenditure per potential customer is \$7, with a standard deviation of \$75, and where an average treatment effect of \$0.35 (0.005 of a standard deviation) would be substantial in the sense of being highly profitable for the company. In that example, an experiment with power for a treatment effect of \$0.35  equal to 80\%, and  a significance level for the test of means of 0.05, would require a sample size of 1.4 million customers.
As a result confidence intervals for the average effect of an advertisement are likely to include zero even if the average effect were substantively important, even with large sample sizes, {\it e.g.,} over a hundred thousand units.
However, the fact that a confidence interval for the average treatment effect includes zero does not mean that the data are not informative about the presence of causal effects of the treatment. Using Fisher exact p-value calculations \citep{fisher1937design} with well-chosen  statistics ({\it  e.g.}, the Hodges-Lehman difference in average ranks, \citet{rosenbaum1993hodges}), one may well be able to establish conclusively that treatment effects are present. However, the magnitude of the treatment effect is important for decisions about advertising. In this type of example, point estimates of average effects are of more importance for decision makers than rejections of the sharp Fisher null hypothesis of no effects whatsoever.

This sets the stage for the problem we address in this paper. In the absence of additional information, there is no estimator for the average treatment effect that is more efficient than the difference in means. To obtain more precise estimates, we either need to change focus away from the average treatment effect, or we need to make additional assumptions.
One approach to changing the question slightly is to transform the outcome
({\it e.g.} taking logarithms or winsorizing the outcome) followed by a standard analysis estimating the average effect of the treatment on the transformed outcome.
In this paper we choose a different approach, making additional assumptions on the distribution of the data. The key conceptual contribution is that we postulate   low-dimensional parametric models for the unit-level treatment effects.
Note that we do not use a parametric  models for the full outcome distributions, which might involve interaction between potential treatment and control values because specifying a parametric model that well approximates the full outcome distribution is more challenging than doing so for treatment effects. Unlike outcomes, treatment effects tend to be small and often have little variation.
Instead, we propose a semiparametric model of the outcome distribution that incorporates a parametric model only for the treatment effects but not for the joint outcome distributions (\citet{bickel1993efficient}, \citet{bickel2015mathematical}). For this semiparametric set-up, we derive the influence function, the semiparametric efficiency bound, and propose semiparametrically efficient estimators. It turns out that the parametrization of the treatment effect can be very informative, potentially making the asymptotic variance for these semiparametric estimators much smaller than the corresponding asymptotic variance for the difference-in-means estimator. In addition, even if the model for the treatment effect is misspecified, the estimand corresponding to the estimator may have an interesting, causal, interpretation.

The remainder of the paper is organized as follows.
First, in Section \ref{section:example} we consider a special case where we assume that the treatment effect is additive and constant. This implies that the two potential outcome distributions differ only by a shift.
This case is closely related to the classical two-sample problem, as discussed in \citet{hodges1963estimates}, and to problems considered in the literature on descriptive statistics for nonparametric models as in \citet{bickel1975descriptiveI, bickel1975descriptiveII}, \citet{doksum1974empirical}, \citet{doksum1976plotting}.
Particularly relevant is the work of \citet{jaeckel1971robust}, \citet{jaeckel1971some}. One way to interpret Section \ref{section:example} is that it presents, in a causal inference framework, extensions of Jaeckel's work to the two sample context in the setting of semiparametric theory. In the process of doing so, we generalize Huber's model \citep{huber1964} and the estimator based on trimmed means to include asymmetry, and present a simplified version of the results of \citet{chernoff1967asymptotic} (see also \citet{bickel1967some}, \citet{govindarajulu1967generalizations} and \citet{stigler1974linear} on linear combinations of order statistics). In particular, we exhibit efficient M and L estimates for outcome distributions that are known up to a shift. We then analyze partially adaptive estimates, in particular flexible trimmed means (as in \citet{jaeckel1971robust}) and fully adaptive estimates of both types, as discussed, for example, in \citet{bickel1993efficient}.
In this setting the problem is closely related to the literature on robust estimation of locations ({\it e.g.,} \citet{bickel1975descriptiveI, bickel1975descriptiveII, jaeckel1971robust, hampel2011robust, huber2011robust,
 bickel1976descriptive, bickel2012descriptiveIV}).
In Section \ref{estimation_model} we  consider the
case where we have more flexible parametric models for the treatment effect.
In Section \ref{section:simulations} we provide some simulation evidence regarding the finite sample properties of the proposed methods in controlled settings.
In this section
we also provide two applications of the methods, one to data on  housing prices and one to data on medical expenses. In both cases the outcome distributions are thick-tailed.
Section \ref{section:conclusion} concludes.
A software implementation for R is available at \url{https://github.com/michaelpollmann/parTreat}.

\section{Constant Additive Treatment Effects}\label{section:example}

In this section we focus on a special
case of the general problem where the treatment effect is additive and  constant across units. In Section
\ref{estimation_model} we discuss the general case. We start by setting up the problem more formally and introducing some notation. Then we discuss robust estimation in the one-sample case to motivate a class of weighted quantile treatment effect estimators. We then discuss the formal semiparametric problem and show the adaptivity of the proposed estimators.
Finally we consider partial adaptivity and robustness.

\subsection{Set Up}

We consider a set up with a randomized experiment with $n$ observations drawn randomly from a large population. With probability $p\in(0,1)$ a unit is assigned to the treatment group. Let  $n_1$ and $n_0=n-n_1$ denote the number of units assigned to the treatment and control group.
Using the potential outcome set up \citep{neyman1923, rubin1974estimating, imbens2015causal},  let $Y_i(0)$ and $Y_i(1)$ denote the two potential outcomes for unit $i$, and let the binary treatment be denoted by $Z_i\in\{0,1\}$.
The realized outcome is $Y_i=Y_i(Z_i)$.
We assume that there is no interference between units, so that the treatment assignment for one unit does not affect the outcomes for any other unit. For all units in the sample we observe the pair $(Z_i,Y_i)$.

The cumulative distribution functions for the two potential outcomes are $F_0(\cdot)$ and $F_1(\cdot)$
with  inverses $F^{-1}_{0}(s)$ and   $F^{-1}_{1}(s)$ and means and variances $\mu_0$, $\mu_1$, $\sigma_0^2$ and $\sigma_1^2$.
Note that by the random assignment assumption the distribution of the potential outcome $Y_i(z)$ is identical to the conditional distribution of the realized outcome $Y_i$ conditional on $Z_i=z$.

We are interested in the average treatment effect in the population,
\begin{equation} \tau=\mathbb{E}[Y_i(1)-Y_i(0)].\end{equation}
The  natural estimator for  this average treatment effect is the difference in sample averages
\begin{equation} \hat\tau=\overline{Y}_1-\overline{Y}_0,\hskip1cm
{\rm where}\ \
\overline{Y}_1=\frac{1}{n_1}\sum_{i=1}^{n} Z_i Y_i,\hskip0.4cm  \overline{Y}_0=\frac{1}{n_0}\sum_{i=1}^{n} (1-Z_i)Y_i,\end{equation}
are the averages of the observed outcomes by treatment group. Under standard conditions
\begin{align*}
\sqrt{n}(\hat{\tau}-\tau) & \stackrel{d}{\Rightarrow} \mathcal{N}\left(0,\frac{\sigma^2_0}{1-p}+\frac{\sigma^2_1}{p}\right).
\end{align*}
$\frac{n_{1}}{n}\stackrel{P}{\rightarrow}p$
and $\frac{n_{0}}{n}\stackrel{P}{\rightarrow}(1-p)$, with $p\in(0,1)$.

The concern with the conventional estimator $\hat\tau$ is that it may be imprecise.
In particular in settings where the outcome distribution is thick tailed, sometimes extremely so, confidence intervals may be wide.
At the same time, it is possible that there is substantial evidence in the data that the treatment does have some effect.
Randomization based tests on ranks or other transformations of the outcome ({\it e.g.,} \citet{rosenbaum_book}) may offer such evidence. However, the magnitude of the causal effect is often of primary interest, and it is not sufficient to simply reject the null of no effect.
We address this by incorporating ideas from the robust  estimation literature
\citep{hampel2011robust, huber2011robust} and
by imposing some restrictions on the joint distribution of the potential outcomes motivated by
the semiparametric literature
\citep{bickel1975descriptiveI, bickel1975descriptiveII, bickel1976descriptive, bickel2012descriptiveIV}
 to develop new estimators.

The quantile treatment effect  \citep{lehmann1975nonparametrics} plays an important role in our setup. For quantile $u\in(0,1)$, define
\begin{equation}\label{response}
{\Delta}(u)  =F_{1}^{-1}(u)-F_{0}^{-1}(u),\ \ 0\leq u\leq 1,
\end{equation}
which is closely related to what
\citet{doksum1974empirical} and \citet{doksum1976plotting}
  label the {\it response} function:
$\tilde \Delta(y)=
F^{-1}_{1}\left( F_{0}(y)\right)-y=\Delta(F_0(y))
.$
The natural estimate for the quantile treatment effect
${\Delta}(u)$
is the empirical plug-in,
\begin{align*}
\hat{{\Delta}}(u)=\hat{F}_{1}^{-1}(u)-\hat{F}_{0}^{-1}(u)
\end{align*}
where $\hat{F}_{1}^{-1}(u)$ equals $Y_{([n_1 u])}^{(1)}$, defined as the $[n_1u]^{\rm th}$ order
statistic of $Y_{i}|Z_{i}=1$, $i=1,\dots,n_{1}$, where $n_{1}=\sum_{i=1}^{n}Z_{i}$
and similarly for $\hat{F}_{0}^{-1}(u)$.

A natural class of parameters summarizing the difference
between the $Y_i(1)$ and $Y_i(0)$ distributions consists of weighted averages of the quantile treatment effects:
\begin{align*}
\tau(F_{0},F_{1};W) & =\int_{0}^{1}{\Delta}(u)dW(u)
\end{align*}
where the weights integrate to one, $W(0)=0, W(1)=1$.
Different choices for the weight function correspond to different estimands.
The constant weight case, $W'(u)\equiv 1$, corresponds to
the population average treatment effect $\tau=\mathbb{E}(Y(1)-Y(0))$.
The median corresponds to the case where  $W$ puts all its mass at
$1/2$.
We thus allow $W$ to permit point masses.

For a given weight function $W$ we can estimate the parameter $\tau(F_{0},F_{1};W)$
using a weighted average quantile estimator:
\begin{equation}
\label{waqe}
\hat{\tau}_{W}  \equiv\tau(\hat F_{0},\hat F_{1};W)=\int_{0}^{1}\Bigl(\hat{F}_{1}^{-1}(u)-\hat{F}_{0}^{-1}(u)\Bigr)
dW(u)\end{equation}
\[\hskip1cm=\frac{1}{n_{1}}\sum_{i=1}^{n_{1}}w_{i}^{(1)}Y_{(i)}^{(1)}-\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}w_{i}^{(0)}Y_{(i)}^{(0)}+o_{P}(1)
\]
where
\begin{align*}
w_{i}^{(z)} & = W\left(\frac{i}{n_{z}}\right)-W\left(\frac{i-1}{n_{z}}\right),
\end{align*}
and $Y_{(i)}^{(z)}$ again are  the order statistics in treatment group $z$.

\subsection{Efficient estimation}

We begin by considering the nonparametric model for a single sample,
with cumulative distribution $F$ where the interest is in the weighted
quantile $\int_{-\infty}^{\infty}F^{-1}(u)dW(u)$ for a given weight funtion $W()$. For this one-sample
case, \cite{jaeckel1971robust,jaeckel1971some} builds on \cite{chernoff1967asymptotic} to
show that under simple conditions on $W$ and $F$, for a sample
size of $n$,
\begin{align}
\label{een}
\int_{0}^{1}\Bigl(\hat{F}^{-1}(u)-F^{-1}(u)\Bigr)dW(u)=\int_{-\infty}^{\infty}\psi(x,F)d(\hat{F}(x)-F(x))+o_{P}(n^{-1/2}),
\end{align}
where the influence function $\psi$ is related to the weight function
$W$ by
\begin{align}
\psi(x,F) & = -\int_{x}^{\infty}\frac{1}{f(y)}dW(F(y)) + \int_{-\infty}^{\infty}\frac{F(y)}{f(y)}dW(F(y))\label{eq:2.5}.
\end{align}
the last term ensures that $\int_{-\infty}^{\infty}\psi(x,F)dF(x)=0$.

Note that if the derivatives $\psi'$ and $W'=w$ exist, by \eqref{eq:2.5},
\begin{align*}
\psi(x,F) & =-\int_{x}^{\infty}w(F(y))dy+\int_{-\infty}^{\infty}F(y)w(F(y))dy
\end{align*}
so that $\psi'(x,F)=w(F(x))$.
Note that for the
median, \eqref{eq:2.5} yields, $\psi(x,F)=\frac{sign(x-F^{-1}(\frac{1}{2}))}{2f(F^{-1}(\frac{1}{2}))}$.
Our formula \eqref{eq:2.5} is slightly more general than Jaeckel's, and in the appendix we establish sufficient conditions on $F$ and $W$ for our version of his result to hold.

Expression (\ref{een}) in turn implies that
\begin{align*}
\int_{0}^{1}\Bigl(\hat{F}^{-1}(u)-F^{-1}(u)\Bigr)dW(u) & \stackrel{d}{\Rightarrow}\mathcal{N}(0,\sigma^{2}(F,W)).
\end{align*}
where the variance equals
\[
\sigma^{2}(F,\omega)=\int_{-\infty}^{\infty}\psi(x,F)^{2}dF(x).
\]

\cite{jaeckel1971robust,jaeckel1971some}'s results
extend in the following way to the two-sample setting we are interested in.
If $\tau(F_{0},F_{1},W)$ is estimated by
\begin{align}
\hat{\tau}(F_{0},F_{1},W) & =\int_{0}^{1}(\hat{F}_{1}^{-1}(u)-\hat{F}_{0}^{-1}(u))dW(u) \label{eq:tau-hat-F0-F1-W}
\end{align} then, under regularity conditions
given in the appendix (Theorem A.1),
\begin{align}
\hat{\tau}(F_{0},F_{1},W) & =\tau(F_{0},F_{1},W)+\frac{1}{n}\sum_{i=1}^{n}\left(Z_{i}\frac{\psi(Y_{i},F_{1})}{p}-(1-Z_{i})\frac{\psi(Y_{i},F_{0})}{1-p}\right)+o_{P}(n^{-1/2})\label{eq:2.6}
\end{align}
where $\psi(y,F)$ is given by \eqref{eq:2.5} for given $W$.

Now let us return to the primary focus of this section, the estimation of the average treatment effect under the constant treatment effect assumption.
In that case
the quantile treatment effects are all equal:
\begin{align*}
{\tau}(u) & \equiv\tau.
\end{align*}
Then, for all weight functions $W$,
\begin{align}
\tau(F_{0},F_{1},W) & =\tau.
\end{align}

Suppose that in addition to knowing that the treatment effect is constant,
we know $F_{0}$ up to a shift. That is, $F_{0}(x)=F(x-\eta)$ where $F(\cdot)$ is known with derivative $f$,
$\eta$ unknown, and, because of the constant treatment effect, $F_{1}(x)=F(x-\eta-\tau)$.
For this fully parametric model (with unknown parameters $\eta$ and
$\tau$), if the Fisher information $I(f)=\int\left(\frac{f'}{f}\right)^{2}(x)f(x)dx=\int \left(-\frac{f'}{f}\right)'(x)f(x)dx$ satisfies $0<I(f)<\infty$, the maximum likelihood estimator  of $\tau$, suitably
regularized ({\it e.g.,}  \citet{lecam-yang-1988}), has
influence function,
\begin{align}
\psi_{f,\eta}(Z,Y; \tau) & =-\frac{1}{I(f)}\cdot\left(\frac{Z}{p}\cdot\frac{f'}{f}(Y-\eta-\tau)-\frac{1-Z}{1-p}\cdot\frac{f'}{f}(Y-\eta)\right)\label{eq:psi_f(Z,Y)}.
\end{align}

If $f'/f$ is absolutely continuous, then
\begin{equation}
w_{f}(F(x))  =\frac{1}{I(f)}\left(-\frac{f'}{f}\right)'(x)  \text{ or } w_f(u) = \frac{1}{I(f)}\left(-\frac{f'}{f}\right)'(F^{-1}(u)) \label{eq:w_f(F(x))}
\end{equation}
provides an efficient $L$ estimate when substituted appropriately
in \eqref{eq:tau-hat-F0-F1-W},
leading to \begin{align} \hat{\tau}(F_{0},F_{1},W) & =\int_{0}^{1}(\hat{F}_{1}^{-1}(u)-\hat{F}_{0}^{-1}(u)) w_f(u) du.
\label{eq:tau-hat-F0-F1-ww}
\end{align}

\subsection{Fully adaptive estimation}

It follows from semiparametric theory \citep{bickel1993efficient} that even if the density $f$
is unknown, substituting a suitable estimate of $f$ (and $\eta$) in \eqref{eq:psi_f(Z,Y)}
or (\ref{eq:tau-hat-F0-F1-ww}), will yield estimators with influence functions
given by \eqref{eq:psi_f(Z,Y)}, or equivalently by
\begin{align}
\psi_{f_0}(Z,Y; \tau) & =-\frac{1}{I(f_0)}\cdot\left(\frac{Z}{p}\cdot\frac{f_0'}{f_0}(Y-\tau)-\frac{1-Z}{1-p}\cdot\frac{f_0'}{f_0}(Y)\right),
\end{align}
where $f_0(\cdot) \equiv F_0'(\cdot) \equiv f(\cdot - \eta)$.

This is a consequence of the
orthogonality of the tangent space with respect to $f$.

Formally we have for the unknown $f(\cdot)$ case:
\begin{theorem}
\label{thm:efficiency}For all $f$ such that $f'$ exists and $0<I(f)<\infty$:
\begin{enumerate}
\item[$(i)$] There exist $\sqrt{n}$ consistent estimators $\hat{\tau}$ and $\hat{\mu}$.
\item[$(ii)$] Under mild conditions (see \eqref{eq:fhat-score-convergence} and \eqref{eq:fhat-info-convergence} in the appendix), we can construct an $M$ estimate $\hat{\tau}^{if}$ such that
\begin{align}
\sqrt{n}(\hat{\tau}^{if}-\tau) & \stackrel{d}{\Rightarrow}\mathcal{N}\left(0,\frac{1}{p(1-p)I(f)}\right).\label{eq:(A)}
\end{align}
\item[$(iii)$] Under mild conditions (see Lemma \ref{lem:linear-order-statistics} in the appendix), we can construct an $L$ estimate $\hat{\tau}^{waq}$ (weighted average quantile)
such that $\hat{\tau}^{waq}$ satisfies \eqref{eq:(A)}.
\end{enumerate}
\end{theorem}

Our conditions are not optimal (see \citet{stone1975adaptive} for minimal ones in
the one sample case).

The $M$ estimate is of the form,
\begin{align}
\hat{\tau}^{if} & \equiv\tilde{\tau}+\frac{1}{n}\left\{\sum_{i=1}^{n/2}\psi_{\hat{f}_{0(2)}}(Z_{i},Y_{i};\tilde{\tau})+\sum_{i=1+n/2}^{n}\psi_{\hat{f}_{0(1)}}(Z_{i},Y_{i}; \tilde{\tau})\right\} \label{eq:tau-if}
\end{align}
where $\hat{f}_{0(1)}$ is an estimate of $f_0$ using
$\{(Z_{i},Y_{i}): 1\leq i \leq \frac{n}{2}\}$, and $\hat{f}_{0(2)}$ using
$\{(Z_{i},Y_{i}): \frac{n}{2} < i\leq  n\}$, a one step estimate
using the sample splitting technique \citep{klaassen1987consistent}.

The $L$ estimate is obtained by direct plug-in of weight function
estimation using sample splitting as above. Details
are in the appendix.

Thus, both the estimators $\hat{\tau}^{waq}$ and $\hat{\tau}^{if}$
are adaptive to models in which the distribution of control and treated
potential outcomes is known up to location. Theorem \ref{thm:efficiency}
implies that the influence function based estimator is as efficient
as the maximum likelihood estimator based on the true distribution
function. For instance, if potential outcomes are normally distributed,
the maximum likelihood estimator is the difference in means, and the
influence function estimator has the same limiting distribution. If,
however, the potential outcomes follow a double exponential distribution,
the difference in means estimator is not efficient; the difference
in medians is the efficient estimator. Under this true distribution,
the influence function based estimator adapts and has the same limiting
distribution as the difference in medians. For the Cauchy distribution the optimal weights are more complicated,
\(w_f(u) \propto - \cos(2 \pi u) \sin(\pi u)^2\),
but the influence function based estimator has the same limiting distribution,
without requiring a priori knowledge about the distribution.

This influence function based estimator $\hat{\tau}^{if}$ is a special
case of the estimator developed by \citet{cuzick1992efficient,cuzick1992semiparametric}
for the partial linear regression model setting.

Although these estimators are efficient under the constant additive
treatment model, as we shall see in Section 3, if the model is violated,
estimates of the types derived from $f$ known continue to estimate
at rate $n^{-1/2}$, meaningful measures of the treatment
effect as discussed in Section 2.1. This is unfortunately not the
case for $\hat{\tau}^{if}$ and $\hat{\tau}^{waq}$ because estimation
of $f'$ and $f''$ introduces components of variance of order larger
than $n^{-1/2}$. However, there is a partial remedy.

\subsection{Partial Adaptation}

An interesting alternative to full adaptation in this problem was
studied by \citet{jaeckel1971some} in the one-sample symmetric case.  See also
\citet{yu2017robust}. The estimator proposed by \citet{jaeckel1971some} is
\begin{align*}
\hat{\mu}_{\alpha} & \equiv\frac{1}{1-2\alpha}\int_{\alpha}^{1-\alpha}\hat{F}^{-1}(u)du
\end{align*}
for a sample from $F(x-\mu)$ with $f$ symmetric. We can interpret this as restricting the class of weight functions to one indexed by a scalar parameter $\alpha$:
\[
w_{\alpha}(u)  =
\left\{\begin{array}{ll}
\frac{1}{1-2\alpha}\hskip1cm& {\rm if}\ \alpha\leq u\leq1-\alpha\\
  0& \text{ otherwise}.
  \end{array}\right.
\]
This weight function
yields the $\alpha$-trimmed mean. We can then choose the value of $\alpha$
that minimizes the asymptotic
variance of $\hat{\mu}_{\alpha}$. This asymptotic variance is equal to
\begin{align*}
\sigma_{\alpha}^{2}\equiv\frac{1}{n(1-2\alpha)^{2}} & (\mathbb{E}(X-\overline{\mu})^{2}1(F^{-1}(\alpha)\leq X\leq F^{-1}(1-\alpha))\\
 & +\alpha\cdot(F^{-1}(1-\alpha)-\overline{\mu})^{2}+\alpha(F^{-1}(\alpha)-\overline{\mu})^{2}),
\end{align*}
with
\begin{align*}
    \overline{\mu} = \int_{\alpha}^{1-\alpha} F^{-1}(u)du + \alpha (F^{-1}(\alpha) + F^{-1}(1-\alpha)),
\end{align*}
which can be estimated by replacing $F$ with the empirical distribution, denoted as $\hat{\sigma}^2_{\alpha}$.
\iffalse
\begin{align*}
\hat\sigma_{\alpha}^{2}=\frac{1}{n(1-2\alpha)^{2}}( & \sum_{i=1}^{n}(X_{i}-\hat{\mu}_{\alpha})^{2}1(\hat{F}^{-1}(\alpha)\leq X_{i}\leq\hat{F}^{-1}(1-\alpha))\\
 & +\alpha\cdot(\hat{F}^{-1}(1-\alpha)-\hat{\mu}_{\alpha})^{2}+\alpha\cdot(\hat{F}^{-1}(\alpha)-\hat{\mu}_{\alpha})^{2}).
\end{align*}
\fi
Let $\hat\alpha=\arg\min_\alpha\hat\sigma^2_\alpha$, and let $\hat{\mu}_{\hat\alpha}$
be the corresponding estimator for $\mu$.
\citet{jaeckel1971some} shows that $\hat{\mu}_{\hat\alpha}$ is adaptive for estimating $\mu$ over a Huber family of densities. In the Huber family  $(X-\mu)/\sigma$ has density $f$ for varying $\mu$, $\sigma>0$:
\[
\log f(x)= \left\{
\begin{array}{ll}
-\frac{x^{2}}{2}-c(k),\hskip1cm & {\rm if}\ |x|\leq k\\ \\
-\frac{k|x|}{2}-c(k) & {\rm if}\ |x|>k.
\end{array}\right.
\]
where $c(k)$ makes $\int f(x)dx=1$, and
$k=-F^{-1}(\alpha)$. Adaptivity here means that
using the trimming proportion optimizing the variance estimate,
in fact yields an estimate which is efficient for the member of the Huber family generating the data. He optimizes $0<\alpha_0 \leq \alpha \leq \alpha_1 <\frac{1}{2}$.

Because this family includes among others the
 Gaussian ($k\rightarrow \infty$) and  double
exponential ($k=0$), this family
is very flexible. For more properties, see \cite{huber2011robust}.

In the two-sample case, it is reasonable to consider asymmetric weight
functions leading to the natural generalization,
\begin{align*}
\hat{\tau}_{\alpha,\beta} & =\frac{1}{1-(\alpha+\beta)}\int_{\alpha}^{1-\beta}(\hat{F}_{1}^{-1}(u)-\hat{F}_{0}^{-1}(u))du.
\end{align*}

We show in the online appendix that Jaeckel's result on partial adaptation
can, for our two sample problem, be extended to a generalization of the  Huber
family whose members are symmetric iff $k_{1}=k_{2}$, defined by
$(X-\mu)/\sigma \sim f$:
\begin{align}
\log f(x)=\begin{cases}
-\frac{x^{2}}{2}-c(k_{1},k_{2}), & \text{ if }-k_{1}\leq x\leq k_{2},\\
\frac{k_{1}x}{2}-c(k_{1},k_{2}), & \text{ if }x<-k_{1},\\
-\frac{k_{2}x}{2}-c(k_{1},k_{2}), & \text{ if }x>k_{2},
\end{cases} \label{eq:extended-huber}
\end{align}
where $c(k_{1},k_{2}) \equiv \log \Bigl(2\left(\frac{e^{-k_{1}/2}}{k_{1}} + \frac{e^{-k_{2}/2}}{k_{2}}\right) + \sqrt{2\pi}(\Phi(k_{2}) - \Phi(-k_{1}))\Bigr)$ and $\Phi$ is the CDF of $\mathcal{N}(0, 1)$.

This family can be equivalently parametrized by $F(-k_{1})$ and $F(k_{2})$.

\section{The General Parametric Treatment Effect Case}
\label{estimation_model}

In some settings, the assumption of an additive model may be too restrictive.
In this section, we develop estimators given a general parametric model for this difference.

The starting point is a model governing the relation between the two potential outcomes:
\begin{assumption}[Parametric Model for Unit-level Treatment Effects]\label{as:par-outcomes}
The unit-level potential outcomes satisfy
\begin{equation*}
Y_i(1) = h(Y_i(0), \theta), \qquad \mathrm{with}\ \ \frac{\partial}{\partial y}h(y, \theta)>0,
\end{equation*}
for some known function $h$ and an unknown parameter, $\theta \in \mathcal{R}$. Equivalently,
\begin{equation*}
    F_1(h(y,\theta)) = F_0(y).
\end{equation*}
\end{assumption}
As before we initially assume $F_0$ known.
In terms of the quantile treatment effects $\tau(u)$  Assumption \ref{as:par-outcomes} implies the restriction
\[ \tau(u)=F_1^{-1}(u)-F^{-1}_0(u)=h(F_0^{-1}(u),\theta)-F^{-1}_0(u)
.\]
Given Assumption \ref{as:par-outcomes}, the population average treatment effect can be characterized as
\begin{equation*}
\tau^{\rm pop}=\int_0^1 \bigl( h(F_0^{-1}(u),\theta)-F_0^{-1}(u)\bigr) du
\end{equation*}
In practice, however, estimating $\tau^{\rm pop}$ may still be subject to substantial sampling variance.
As an alternative, we suggest to estimate the in-sample, as opposed to population, average treatment effect. An insight is that estimators of this object can  have a much lower variance.
We define the in-sample average treatment effect as
\begin{equation}\label{tau_is}
\begin{aligned}
\tau^{\rm is} & = \frac{1}{N} \sum_{i=1}^{N}\Bigl(Y_i(1) - Y_i(0)\Bigr) \\
& = \frac{1}{N}\sum_{i=1}^{N} \Bigl\{ Z_i (Y_i - h^{-1}(Y_i,\theta)) + (1-Z_i)(h(Y_i,\theta) - Y_i)\Bigr\}.
\end{aligned}
\end{equation}
When \(h\) is not just an additive function, \(\tau^{\rm is}\) is sample-dependent, and thus stochastic.
In particular when the variance of \(Y_i\) is large because of thick tails for the potential outcome distributions, the variance of \(\tau^{\rm is}\) will be large, too.
We therefore focus on the variance of estimators $\hat\tau$ relative to \(\tau^{\rm is}\) for the particular sample at hand, rather than relative to the population average \(\tau^{\rm pop}\).

Because $F_0$ is known, we can estimate $\theta$ efficiently by some version of maximum likelihood to get an estimate $\hat{\theta}$ and
\begin{equation*}
    \hat{\tau} = \int_0^1 \left( h(F_0^{-1}(y), \hat{\theta}) - F_0 ^{-1}(y) \right) dy
\end{equation*}
as an estimate of $\tau$. The density of $Y$ given $Z$ is
\begin{equation*}
   f(y|z)= \left(f_0(y)\right)^{1-z} \left(f_1(y, \theta) \right)^z.
\end{equation*}
By Assumption~\ref{as:par-outcomes}
\begin{align}
    f_1(y, \theta) = f_0(h^{-1}(y,\theta)) \frac{\partial h^{-1}(y,\theta)}{\partial y} ,\label{eq:f1(y,theta)}
\end{align}
and the score function is
\begin{equation*}
    \dot{\ell}(y,z,\theta) \equiv z \cdot \frac{\partial}{\partial \theta} \log f_1(y, \theta)
\end{equation*}
yielding,
\begin{equation*}
    \hat{\theta} = \theta + \frac {1}{n} \sum_{i=1}^n I^{-1} \dot{\ell}(Y_i,Z_i,\theta) + o_P(n^{-1/2})
\end{equation*}
where $I=E\left(\dot{\ell}(Y,Z,\theta)\right)^2$.

If $F_0$ is assumed unknown, to obtain an efficient influence function we must
\begin{itemize}
\item[1)] Compute the tangent plane as $f_0$ varies with $\theta$ fixed. The tangent plane is given by
\begin{equation*}
    \dot{P}_f = \{u(Y,Z) = (1-Z) v(Y,\theta) + Z v(h^{-1}(Y,\theta),\theta): \int v^2(y,\theta)f_0(y)dy <\infty, \int v(y,\theta) f_0(y)dy=0\}.
\end{equation*}
(Note that both factors of the likelihood must be varied treating $\theta$ as fixed.)

\item[2)] Project $\dot{\ell}$ on the orthocomplement of the tangent plane to get
\begin{equation*}
    \dot{\ell}^*(Y,Z,\theta) = \dot{\ell}(Y,Z,\theta) - (Z Q(h^{-1}(Y,\theta), \theta) + (1-Z) Q(Y, \theta))
\end{equation*}
where $Z Q(h^{-1}(Y,\theta), \theta) + (1-Z) Q(Y, \theta)$ is the projection of $\dot{\ell}$ on $ \dot{P}_f$.

\item[3)] The efficient influence function is given by
\begin{equation*}
    \psi(Y,Z,\theta) = \frac{\dot{\ell}^*(Y,Z,\theta}{ E\left(\dot{\ell}^*(Y,Z,\theta)\right)^2.}
\end{equation*}
\end{itemize}

\begin{lemma}\label{thm:eif-general}
The efficient influence function for $\theta$ is
\begin{equation*}
\psi_{f_0}(y, z, \theta )=\mathcal{I}^{-1}
\left\{
\frac{z}{p}\cdot g(y,\theta ) - \frac{1-z}{1-p} \cdot g(h(y,\theta ), \theta )
\right\},
\end{equation*}
where
\begin{align*}
g(y, \theta )  &= \frac{\partial}{\partial \theta } \log\left(f_0(h^{-1}(y,\theta))\cdot \frac{\partial h^{-1}(y,\theta)}{\partial y}\right)
\end{align*}
and
\begin{align*}
    \mathcal{I} = \int g^2(h(y,\theta ),\theta )f_0(y)dy.
\end{align*}
\end{lemma}

To use the influence function approach we need a $\sqrt n$-consistent initial estimator $\tilde\theta$. We can do so by a fixed number of quantiles, $u_1,\ldots,u_d$, where $d$ is the dimension of $\theta$, and find the $\theta$ that solves
 \[ \hat F^{-1}_1(u)=h(\hat F_0^{-1}(u),\theta),
\]
for $u=u_1,\ldots,u_d$. Let $\tilde\theta$ denote the solution to this system of equations.
Then:
\begin{theorem}\label{thm:eif-est-general}
Under mild conditions on the estimation of density and its derivative, the estimator $\hat{\theta}^{if}$ below
is efficient for $\theta$, i.e. $\sqrt{n}(\hat{\theta} - \theta) \stackrel{d}{\Rightarrow} \mathcal{N}(0,\frac{1}{p(1-p)} \mathcal{I}^{-1})$:
\begin{align*}
\hat{\theta}^{if} & \equiv\tilde{\theta}+\frac{1}{n}\left\{\sum_{i=1}^{n/2}\psi_{\hat{f}_{0(2)}}(Y_i,Z_{i};\tilde{\theta})+\sum_{i=1+n/2}^{n}\psi_{\hat{f}_{0(1)}}(Y_i,Z_{i}; \tilde{\theta})\right\}
\end{align*}
where $\hat{f}_{0(1)}$ is the estimate of $f_0$ using $\{{(Y_i,Z_i): 1\leq i \leq \frac{n}{2}}\}$, and $\hat{f}_{0(2)}$ is the estimate of $f_0$  using $\{(Y_i,Z_i): \frac{n}{2} < i \leq  n\}$, again a one step estimate using the sample splitting technique \citep{klaassen1987consistent}, similar to \eqref{eq:tau-if}.
\end{theorem}

Note that, just as in the constant treatment effect setting, the efficient influence function is not the one corresponding to $F_0$ known.

\section{Simulations}\label{section:simulations}

We evaluate the performance of the proposed estimators and conventional estimators in a Monte Carlo study.
Throughout most of these simulations, the true unit-level treatment effects are all zero.
We estimate the treatment effect using the proposed efficient estimators based on an additive model.
We consider seven estimators:
The (standard) difference in means, the difference in medians, the Hodges-Lehman (\citet{hodges1963estimates}) estimator,\footnote{The Hodges-Lehmann estimator is equal to the median of all pairwise differences between treated and control observations.}
the adaptively trimmed mean, the adaptively winsorized mean,\footnote{For both trimming and winsorizing, we choose the trimming percentage that minimizes an estimate of the asymptotic variance, allowing anywhere between no trimming (difference in means) and the extreme of trimming all but the medians (difference in medians). While including the extremes is not covered by the theory, this approach appears to work well in our simulations.} the estimator based on the efficient influence function (eif), and the weighted average quantiles (waq) estimator. For the latter two we report only results without sample splitting. Results for the case where we do use sample splitting are very similar and are available from the authors.

We present results for three sets of simulations. First, we report results for settings with  a wide range of known distributions for the potential outcomes, so we can directly assess the ability of the proposed methods to adapt to different distributions. Second, we report results for simulations based on real data, one set based on housing prices and one based on medical expenditures.
Both data sets are examples of relatively heavy tailed distributions as commonly encountered in empirical work.

\subsection{Simulations with Known Distributions}

We simulate samples of \(n=\)20,000 observations, exactly half of which are treated, and report summary statistics based on 10,001 simulated samples (using an odd number so that the median is unique).
We repeat the simulation study for standardized Normal, Double Exponential (Laplace), and Cauchy distributions for the potential outcomes.
The difference in means is the maximum likelihood estimator Normally distributed data, and so will do well there, but may perform poorly for thicker tailed distributions such as the Double Exponential distribution and in particular
 the Cauchy distribution.
The difference in medians is the maximum likelihood estimator for the Double Exponential distribution, and relatively robust to thick tails and outliers, and so  is  expected to perform reasonably well across all specifications, but not as well as the efficient estimators for the Normal.

For the simulations with known distributions we can infer the functional form for the optimal weights for the quantile-based estimator.
The optimal weights for the waq estimator are proportional to the (estimated) second derivative of the log density.
For the Normal distribution, \(\pder[2]{\ln f}{y}(y) = - \frac{1}{\sigma^2}\), implying the optimal weights are constant, so that the semiparametrically efficient estimator approximates the simple difference in means (which is the parametric maximum likelihood estimator in this case).
The density of the double exponential distribution is such that the optimal
 weights asymptotically place all weight close to the median.
The maximum likelihood estimator for the location of a Double Exponential distribution is the sample median.
For the standard Cauchy distribution the efficient weights \(w\) on the difference in \(u\in(0,1)\) quantiles of treated and control distributions are \(w_f(u) \propto - \cos(2 \pi u) \sin(\pi u)^2\), shown in Figure~\ref{fig:weight-cauchy}.
Most of the weight is concentrated around the median, with strictly {negative} weights  outside the $[0.25,0.75]$  quantile range.

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{simulations-output/fig_weight_cauchy_q.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{simulations-output/fig_weight_cauchy_x.pdf}
\caption{\label{fig:weight-cauchy}The efficient weight function for the Cauchy distribution.
The weights, normalized to have mean 1, are plotted against the quantile (left) and against the value of the observations (right).
For the figure on the right, we only plot the range from \(-10\) to \(10\), which corresponds to approximately the \(0.03\) to \(0.97\) quantile.
At this point, the weight is approximately \(-0.04\), and weights for more extreme quantiles are closer to 0.}
\end{figure}

The efficient estimators perform well across distributions, and confidence intervals based either on estimates of the analytic variance formulas or on the bootstrap achieve their nominal coverage levels, as shown in Table~\ref{tab:simulations-parametric}.
Their standard deviations are close to the theoretical efficiency bound, as shown in column 4, labelled relative efficiency, where values larger than one imply standard deviations of the estimator in excess of the efficiency bound.
The efficient influence function and weighted average quantiles estimators are close to the most efficient estimator for the normal, double exponential, and Cauchy distribution.
The last columns show that the confidence intervals are close to their nominal coverage for each distribution.
For computational convenience in the simulations, the confidence intervals based on the bootstrap variance use the \(m\)-out-of-\(n\) bootstrap (\citet{bickel2012resampling}), with \(m= \)2,000 (half treated, half control), to estimate the variance of the estimators.
Even with these smaller sample sizes, the density estimates calculated in the bootstrap samples appear to be sufficiently good to yield reasonable confidence intervals for the estimators.

\begin{table}
    \caption{\label{tab:simulations-parametric}Summary statistics
    for simulations with different distributions:
    Normal, Double Exponential, Cauchy. Statistics
    shown are based on 10,001 simulated samples.
    Each sample has 10,000 treated observations
    and 10,000 control observations. Relative
    efficiency is the ratio of standard deviation
    to the square root of the efficiency bound.
    Columns labeled length show the median length
    of the confidence intervals.}
    \centering
\resizebox{1\linewidth}{!}{
\begin{tabular}{l rrrrr rr rr}
\toprule
& & & & & & \multicolumn{2}{c}{95\% C.I.} & \multicolumn{2}{c}{boot. var. C.I.} \\
\cmidrule(lr){7-8} \cmidrule(lr){9-10}
estimator & bias & \makecell{standard \\ deviation} & \makecell{relative \\ efficiency} & RMSE & MAD & coverage & \makecell{median \\ length} & coverage & \makecell{median \\ length}  \\
\midrule
\multicolumn{10}{l}{Normal Distribution:} \\
\input{simulations-output/tab-normal-N10000-2021-09-01}
\\
\multicolumn{10}{l}{Double Exponential Distribution:} \\
\input{simulations-output/tab-laplace-N10000-2021-09-01}
\\
\multicolumn{10}{l}{Cauchy Distribution:} \\
\input{simulations-output/tab-cauchy-N10000-2021-09-01}
\bottomrule
\end{tabular}
}
\end{table}

\subsection{Simulations with House Price Data}
\label{section:applications}

In the second set of simulations
we use house price data from the replication files of \citet{linden2008estimates,linden2019replication}.
They obtained property sales data for Mecklenburg County, North Carolina, between January 1994 and December 2004.
They dropped sales below \$5,000 and above \$1,000,000, such that 170,239 observations remain, which we take as our population of interest.
Despite the trimming  the distribution is noticeably skewed (skewness \(2.2\)) and heavy-tailed (kurtosis \(9.5\)).
Even after taking logs, the distribution is heavy-tailed with kurtosis equal to \(5.1\).
Figure~\ref{fig:house-prices-weights} plots a histogram for house prices, both in levels and in logs, along with the estimated optimal weights (second derivative of the log density) based on all 170,239 observations.

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{houses-output/fig-weights-house-prices-level.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{houses-output/fig-weights-house-prices-log.pdf}
\caption{\label{fig:house-prices-weights}Histogram of house prices, in levels and logs, as well as estimated optimal weights (second derivative of the log density) based on all 170,239 observations.
The weights are normalized to be mean 1 (black horizontal line). Some weights are below 0 (blue horizontal line).
Vertical lines indicate the \(0.0001\), \(0.001\), \(0.01\), \(0.1\), \(0.9\), \(0.99\), \(0.999\), \(0.9999\) quantiles.}
\end{figure}

We base simulations on this data by drawing samples of size \(n=\)20,000, and randomly assigning exactly half of each sample to the treatment group and the remaining half to the control group, with a zero treatment effect.
Each sample is drawn without replacement, but observations may appear in multiple samples.
Because the true distribution, however, is not available analytically, we cannot calculate the efficiency bound \(\mathcal{I}\) directly.
Instead, we estimate the efficiency bound using density estimates based on all observations.
We compute the same estimators as in the simulations of the previous section, with a small adjustment to the adaptively trimmed and winsorized means.
In settings with positively-valued outcomes, researchers are typically more concerned about the effect of heavy right, rather than left, tails on the variance of estimators.
We therefore fix the trimming and winsorizing percentiles on the left to 0\% (no trimming), and only adaptively choose the threshold on the right.
Without such an adjustment, the estimators tend to trim (too) little in order to avoid losing observations from the left tail, which are unproblematic for the precision of the estimator.

Table~\ref{tab:house-prices-simulations} summarizes the simulation results based on 10,001 simulated samples.
When the house prices are in levels, the standard deviation of the difference in means estimator is twice as large as that of efficient estimators.
For the difference in medians and the Hodges-Lehman estimators, which are less affected by outliers in the data, the standard deviation is larger by approximately 30\% and 20\%, respectively.
Confidence intervals, based on estimated variances and asymptotic normal approximations, have close to nominal coverage throughout, and are meaningfully shorter for the efficient estimators we propose.

We also estimate a proportional treatment effect (multiplicative) model and translate the estimated coefficients into level effects.
Under the multiplicative model, \(Y_{i}(1) = \theta Y_{i}(0)\).
When outcomes are strictly positive, this is identical to an additive model for log outcomes; \(\log(Y_{i}(1)) = \log(\theta) + \log(Y_{i}(0))\).
Given an estimate \(\hat{\tau}_{\rm log}\) of the additive model with the outcome in logs, the estimate of the level treatment effect is then
\begin{equation*}
\hat{\tau} = \frac{1}{n} \Bigl(
\sum_{i=1}^{n}
(1-Z_{i}) \bigl((\exp(\hat{\tau}_{\rm log}) - 1) Y_{i} \bigr)
+
Z_{i} \bigl((1 - \exp(-\hat{\tau}_{\rm log})) Y_{i} \bigr)
\Bigr).
\end{equation*}
For estimates of the in-sample treatment effect, we therefore apply estimates \(\hat{\tau}_{\rm log}\) to a population of interest with known means \(\mu_{Y_{0}}\) and \(\mu_{Y_{1}}\) and fixed treatment probability \(p\) as
\begin{equation*}
\hat{\tau} =
(1-p) \bigl((\exp(\hat{\tau}_{\rm log}) - 1) \mu_{Y_{0}} \bigr)
+
p \bigl((1 - \exp(-\hat{\tau}_{\rm log})) \mu_{Y_{1}} \bigr)
.
\end{equation*}
Using the Delta method, if \(V\) is the asymptotic variance of \(\hat{\tau}_{\rm log}\), then the asymptotic variance of \(\hat{\tau}\), holding \(\mu_{Y_{0}}\), \(\mu_{Y_{1}}\), and \(p\) fixed, is
\begin{equation*}
\Bigl((1-p) \exp(\tau_{\rm log}) \mu_{Y_{0}}
+
p \exp(-\tau_{\rm log}) \mu_{Y_{1}}
\Bigr)^{2}
V
\end{equation*}
which we estimate by replacing \(\tau_{\rm log}\) with \(\hat{\tau}_{\rm log}\) and \(V\) by the estimate of the variance, \(\hat{V}\).
For the purpose of these simulations, we set \(\mu_{Y_{0}} = \mu_{Y_{1}}\) equal to the population mean of house prices, and \(p = 1/2\).

When treatment effects are assumed to be proportional to potential outcomes, the proposed estimators for this multiplicative model are still more efficient than alternative estimators, but the gains are smaller.
The middle panel of Table~\ref{tab:house-prices-simulations} shows the quality of estimates of the multiplicative parameter obtained by transforming outcomes into logs, \(\log(\theta)\).
As can be seen in panel~(b) of Figure~\ref{fig:house-prices-weights}, the distribution of log house prices appears closer to the normal distribution with fewer ``outliers.''
Consequently, the difference in means estimator, which is efficient for the normal distribution, comes noticeably closer to the efficiency bound than when outcomes are in levels.
Nevertheless, the efficient estimators further reduce the variance.
The bottom panel of Table~\ref{tab:house-prices-simulations} shows the same summary statistics when the multiplicative parameter is translated back into a level effect.
The efficient estimators of the multiplicative parameter lead to treatment effects with smaller variance and shorter confidence intervals than the difference in means, irrespective of whether the latter is estimated in levels (top panel) or in logs and then translated into levels (bottom panel).

\begin{table}
    \caption{\label{tab:house-prices-simulations}Summary statistics for simulations based on house price data from \citet{linden2008estimates}.
    Statistics shown are based on 10,001 simulated samples.
    Each sample has 10,000 treated observations and 10,000 control observations.
    Relative efficiency is the ratio of standard deviation to the square root of the efficiency bound, which we calculate from density estimates based on the full sample.
    Columns labeled length show the median length of the confidence intervals.
    The adaptively trimmed mean and adaptively winsorized mean only apply trimming at the top, but not at the bottom.
    }
    \centering
\resizebox{1\linewidth}{!}{
    \begin{tabular}{l rrrrr rr rr}
    \toprule
    & & & & & & \multicolumn{2}{c}{95\% C.I.} & \multicolumn{2}{c}{boot. var. C.I.} \\
    \cmidrule(lr){7-8} \cmidrule(lr){9-10}
    estimator & bias & \makecell{standard \\ deviation} & \makecell{relative \\ efficiency} & RMSE & MAD & coverage & length & coverage & length \\
    \midrule
    \multicolumn{10}{l}{effect in levels based on additive model in levels} \\
    \input{houses-output/tab-house-prices-N10000-level-2021-09-01}
    \\
    \multicolumn{10}{l}{multiplicative parameter: additive model in logs} \\
    \input{houses-output/tab-house-prices-N10000-log-2021-09-01}
    \\
    \multicolumn{10}{l}{effect in levels based on additive model in logs} \\
    \input{houses-output/tab-house-prices-N10000-log-level-2021-09-01}
    \bottomrule
    \end{tabular}
}
\end{table}

\subsection{Medical Expenditures Data}

Next, we present simulation results based on medical expenditure data from the IBM MarketScan Research Database, following the sample construction of \citet[Figure~2]{koenecke2021alpha}.
We restrict the sample to males, age 45--64, with pneumonia inpatient diagnosis and at least 1 year of continuous medical enrollment.
For each patient, we consider the first inpatient admission only to abstract away from any dynamics.
We focus on medical expenditure as the outcome variable.
For each patient, we sum the payments recorded by MarketScan for this admission.
In total, we use data on 103,662 admissions.\footnote{The sample size differs slightly from that reported by \citet{koenecke2021alpha} due to missing expenditure data for a small number of admissions.}
Figure~\ref{fig:medical-expenditures-weights} plots a histogram for medical expenditure in levels and in logs along with the estimated optimal weights (second derivative of the log density).
We also observe a treatment variable in this data set,
the (prior) use of alpha blockers, which \citet{koenecke2021alpha} find may improve health outcomes during respiratory distress by preventing hyperinflammation.

\begin{figure}
\includegraphics[trim={0 0 0 2.3in},clip,width=0.49\linewidth]{health-output/fig_weight_level_health-2021-06-01.pdf}
\includegraphics[trim={0 2.05in 0 0.4in},clip,width=0.49\linewidth]{health-output/fig_weight_level_health-2021-06-01.pdf}
\hfill
\includegraphics[trim={0 0 0 2.3in},clip,width=0.49\linewidth]{health-output/fig_weight_log_health-2021-06-01.pdf}
\includegraphics[trim={0 2.05in 0 0.4in},clip,width=0.49\linewidth]{health-output/fig_weight_log_health-2021-06-01.pdf}
\caption{\label{fig:medical-expenditures-weights}Histogram of medical expenditures per admission, in levels and logs, as well as estimated optimal weights (second derivative of the log density), based on the 98,155 observations in the control group.
For the figure in levels, vertical lines indicate the \(0.001\), \(0.01\), \(0.1\), and \(0.9\) quantiles; the figure is limited to below \$200,000, such that the \(0.99\) and higher quantiles do not appear.
For the figure in logs, vertical lines indicate the \(0.001\), \(0.01\), \(0.1\), \(0.99\), \(0.999\), \(0.9999\) quantiles.
The weights are normalized to be mean 1 (black horizontal line).
Some weights are below 0 (blue horizontal line).}
\end{figure}

We design a simulation study similar to those of the previous sections, treating the receipt of alpha blockers as randomly assigned in our population.
This allows us to study (coverage) properties of the estimators and inference procedures in settings where the parametric treatment effect model is not (necessarily) correctly specified.
For these simulations, each sample is a draw, without replacement, of 200 of the 5,507 observations in the treatment group and 3,565 of the 98,155 observations in the control group.
While the control group is smaller than in our other simulations, it remains sufficiently large to estimate the densities and its derivatives required for our estimators.
In this simulation design, \(F_{0}\) is given by the empirical distribution of the control group, and \(F_{1}\) is given by the empirical distribution of the treatment group, of the full sample.
Although it is not necessarily correct, the additive treatment effect model may offer a reasonable approximation, and we are interested in the performance of inference methods when the conditions for our theoretical results are not (quite) met in this particular application.

The simulation results reported in Table~\ref{tab:health-simulations-log} suggest that the proposed estimators perform reasonably well in this setting despite mis-specification.
 The bias, RMSE, MAD, and coverage are relative to the population difference in means.
The  third panel of the table
translates the multiplicative parameter into levels effects as in the previous section for simulations based on house prices.
Here, we take \(\mu_{Y_{0}} = \)38,745 and \(\mu_{Y_{1}} = \)34,872, as well as \(p = 0.0531\), as in the distribution these simulations draw from.
As for the house price data, the adaptively trimmed mean and adaptively winsorized mean are asymmetric, only trimming the right tail of the data.

\begin{table}
\caption{\label{tab:health-simulations-log} Summary statistics for simulations based on the medical expenditures data in logs, across 1,000 simulated samples.
Each sample has 200 treated observations and 3,565 control observations.
Coverage and median length refer to 95\% confidence intervals.}
\centering
    \begin{tabular}{l rrrrr rr rr}
    \toprule
    & & & & & \multicolumn{2}{c}{95\% C.I.} & \multicolumn{2}{c}{boot. var. C.I.} \\
    \cmidrule(lr){6-7} \cmidrule(lr){8-9}
    estimator & bias & \makecell{standard \\ deviation} & RMSE & MAD & coverage & length & coverage & length \\
    \midrule

\multicolumn{9}{l}{effect in levels based on additive model in levels} \\
    \multicolumn{9}{l}{error and coverage relative to population difference in means} \\
    \input{health-output/tab-health-sim-ln-0-cover-self-0} \\

\multicolumn{9}{l}{multiplicative parameter: additive model in logs} \\
    \multicolumn{9}{l}{multiplicative parameter: error and coverage relative to population difference in means} \\
    \input{health-output/tab-health-sim-ln-1-cover-self-0}
    \\

\multicolumn{9}{l}{effect in levels based on additive model in logs} \\
    \multicolumn{9}{l}{effect in levels: error and coverage relative to population difference in means} \\
    \input{health-output/tab-health-sim-log-level-cover-self-0}
    \bottomrule
    \end{tabular}
\end{table}

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

In many modern settings where randomized experiments are used to estimate treatment effects the presence of heavy-tailed distributions can lead to larger standard errors.
Often researchers use Winsorizing with {\it ad hoc} thresholds to address this.
Here we develop systematic methods for obtaining more precise inferences using parametric models for the treatment effects, while avoiding the specification of models for the potential outcomes. We present results for semiparametric effiency bounds, suggest efficient estimators, and show in simulations that these methods can be effective in realistic settings.

\newpage

\bibliographystyle{chicago}

\section{Proof of partial adaptation with \texorpdfstring{$(\alpha,\beta)$}{(alpha,beta)}-trimmed mean}

\subsection{Main results}
\begin{theorem}
Suppose that $f_0$ falls into the extended Huber family as defined
in Section 2.4. If $\alpha$ and $\beta$ are chosen properly, then for the constant additive treatment effect model,
$\hat{\tau}_{\alpha,\beta}$ is an efficient estimate of $\tau$.
\end{theorem}

\begin{proof}
According to Theorem 1, the information bound is given by $\frac{1}{p(1-p)}I^{-1}$.
According to Theorem \ref{thm:quantile-to-mean}, we have
\begin{align*}
    \sqrt{n}(\hat{\tau}_{\alpha,\beta}-\tau) \stackrel{d}{\Rightarrow} \mathcal{N}\Bigl(0, p^{-1}\sigma_{f_1}^2 + (1-p)^{-1}\sigma_{f_0}^2\Bigr), \label{eq:var-atrimmedmean-diff}
\end{align*}
where $\sigma_{f_j}^2$ is defined by \eqref{eq:sigma_f^2} for $j\in \{0, 1\}$, i.e.
\begin{align}
    \sigma_{f_j}^2 = \frac{1}{(1-\alpha-\beta)^2}\int_{\alpha}^{\beta}\int_{\alpha}^{\beta} \frac{\min(s,t) - st}{f_j(F_j^{-1}(s))f_j(F_j^{-1}(t))}dsdt.
\end{align}
Since $F_{1}^{-1}(t)-F_{0}^{-1}(t)\equiv\tau$, then $\sigma_{f_0}^2 = \sigma_{f_1}^2$. Therefore it is sufficient to prove that $\sigma_{f_0}^2 = I^{-1}$.

For notational simplicity, below we restrict the proof to $\mu=0$ and $\sigma=1$, that is $f_{0}$
is parameterized by $(k_{1},k_{2})$ as \eqref{eq:extended-huber} in Section 2.4 and $f_{1}(x)\equiv f_{0}(x-\tau)$.

Let $\alpha=F_{0}(-k_{1})$ and $1-\beta=F_{0}(k_{2})$.

Then $I=-\int(\frac{f_0'}{f_0})^{'}(x)dF_{0}(x)=\int_{-k_{1}}^{k_{2}}dF_{0}(x)=1-(\alpha+\beta).$

First of all, with integral change of variable,
\begin{align*}
 \sigma_{f_0}^2 = & \frac{1}{(1-\alpha-\beta)^{2}}\int_{-k_{1}}^{k_{2}}\int_{-k_{1}}^{k_{2}}(F_{0}(\min(x,y))-F_{0}(x)F_{0}(y))dxdy.
\end{align*}

Furthermore, it is not hard to verify that $F_{0}(-k_{1})=f_{0}(-k_{1})/k_{1}$
and $1-F_{0}(k_{2})=f_{0}(k_{2})/k_{2}$. Using the fact that $f_{0}'(x)=-xf_{0}(x)$
for $x\in(-k_{1},k_{2})$, then
\begin{align*}
\int_{-k_{1}}^{k_{2}}F_{0}(x)dx & =xF_{0}(x)\big|_{-k_{1}}^{k_{2}}-\int_{-k_{1}}^{k_{2}}xf_{0}(x)dx\\
 & =k_{2}\cdot(1-\beta)+k_{1}\alpha+\int_{-k_{1}}^{k_{2}}df_{0}(x)\\
 & =k_{2}\cdot(1-\beta)+k_{1}\alpha+f_{0}(k_{2})-f_{0}(-k_{1})\\
 & =k_{2}\cdot(1-\beta)+k_{1}\alpha+k_{2}\beta-k_{1}\alpha\\
 & =k_{2}
\end{align*}
and
\begin{align*}
 & \int_{-k_{1}}^{k_{2}}\int_{-k_{1}}^{k_{2}}(F_{0}(\min(x,y))dxdy\\
= & 2\int_{-k_{1}}^{k_{2}}\int_{-k_{1}}^{k_{2}}F_{0}(x)\cdot1(x\leq y)dxdy\\
= & 2\int_{-k_{1}}^{k_{2}}F_{0}(x)\cdot(k_{2}-x)dx\\
= & -\int_{-k_{1}}^{k_{2}}F(x)d(x-k_{2})^{2}\\
= & \int_{-k_{1}}^{k_{2}}(x-k_{2})^{2}f(x)dx+(k_{2}+k_{1})^{2}\alpha\\
= & \int_{-k_{1}}^{k_{2}}\left(k_{2}^{2}f(x)+2k_{2}f'(x)-xf'(x)\right)dx+(k_{2}+k_{1})^{2}\alpha\\
= & k_{2}^{2}(1-\alpha-\beta)+2k_{2}(k_{2}\beta-k_{1}\alpha)+(k_{1}+k_{2})^{2}\alpha-\int_{-k_{1}}^{k_{2}}xf'(x)dx\\
= & k_{2}^{2}+\alpha k_{1}^{2}+\beta k_{2}^{2}-\int_{-k_{1}}^{k_{2}}xdf(x)\\
= & k_{2}^{2}+\int_{-k_{1}}^{k_{2}}f(x)dx\\
= & k_{2}^{2}+1-(\alpha+\beta).
\end{align*}
Thus
\begin{align*}
\sigma_{f_0}^2 & =\frac{1}{(1-(\alpha+\beta))^{2}}\left((k_{2}^{2}+1-(\alpha+\beta))-k_{2}^{2}\right)\\
               & = I^{-1}.
\end{align*}
which completes the proof.
\end{proof}

\begin{theorem}
Let
\begin{align*}
(\hat{\alpha},\hat{\beta}) & =\arg\min_{\alpha_{0}\leq\alpha,\beta\leq\alpha_{1}}(p^{-1}\hat{\sigma}_{1}^{2}(\alpha,\beta)+(1-p)^{-1}\hat{\sigma}_{0}^{2}(\alpha,\beta))
\end{align*}
where $p^{-1}\hat{\sigma}_{1}^{2}(\alpha,\beta)+(1-p)^{-1}\hat{\sigma}_{0}^{2}(\alpha,\beta)$ is the estimate of the asymptotic variance as derived in \eqref{eq:var-atrimmedmean-diff} and $\hat{\sigma}_j^2$ is the estimate of $\sigma_{f_j}^2$ as given in Proposition \ref{prop:alpha-beta-trim-rate}.
If $0<\alpha_{0},\alpha_{1}<1$ and $\alpha_{0}+\alpha_{1}<1$, and
$\sigma_{1}^{2}(\alpha,\beta)$ has a unique solution w.r.t. $(\alpha,\beta)$,
which falls into $(\alpha_{0},\alpha_{1})\times(\alpha_{0},\alpha_{1})$,
then $(\hat{\alpha},\hat{\beta})$ is a consistent estimate of the
optimal $(\alpha,\beta)$.
\end{theorem}

\begin{proof}
From Proposition \ref{prop:alpha-beta-trim-rate}, uniform convergence below holds
for $j\in\{0,1\}$:
\begin{align*}
\sup_{\alpha_{0}\leq(\alpha,\beta)\leq\alpha_{1}}|\hat{\sigma}_{j}^{2}(\alpha,\beta)-\sigma_{j}^{2}(\alpha,\beta)| & =o_{P}(1).
\end{align*}
The conclusion holds.
\end{proof}

\begin{remark}
By Proposition \ref{lem:convexity}, the limiting objective function $p^{-1}\sigma_{1}^{2}(\alpha,\beta)+(1-p)^{-1}\sigma_{0}^{2}(\alpha,\beta)$
is strictly convex at the local neighborhood of the optimal $(\alpha,\beta)$.
\end{remark}

\subsection{Supporting results}
\begin{lemma} \label{lem:var-atrimmedmean}
The influence function \eqref{eq:W-to-psi}, specialized for the $(\alpha,\beta)$-trimmed mean, can be simplified to
\begin{align*}
    \psi(x) = & \frac{1}{1-\alpha-\beta}(x - \mu(\alpha,\beta)), & \text{ if } x \in (F^{-1}(\alpha), F^{-1}(1-\beta)) \\
            = & \frac{1}{1-\alpha-\beta}(F^{-1}(1-\beta) - \mu(\alpha,\beta)), & \text{ if } x \geq F^{-1}(1-\beta) \\
            = & \frac{1}{1-\alpha-\beta}(F^{-1}(\alpha) - \mu(\alpha,\beta)), & \text{ if } x \leq F^{-1}(\alpha)
\end{align*}
where
\begin{align*}
    \mu(\alpha,\beta) & =\int_{\alpha}^{1-\beta}F^{-1}(u)du + \alpha F^{-1}(\alpha) + \beta F^{-1}(1-\beta).
\end{align*}
The asymptotic variance $\sigma_f^2$, denoted as $\sigma^2(\alpha,\beta)$ for the $(\alpha,\beta)$-trimmed mean, can be written as
\begin{align*}
\sigma^{2}(\alpha,\beta) & = \int_0^1 \psi^2(F^{-1}(u))du.
\end{align*}
\end{lemma}
\begin{proof}
The proof follows from direct calculation and is omitted.
\end{proof}

\begin{proposition}
\label{prop:alpha-beta-trim-rate}Assume that $f$ is continuous and
that $f(x)>0$ for any $x\in \mathcal{R}$. Let $\hat{\mu}(\alpha, \beta)$ and $\hat{\sigma}^2(\alpha,\beta)$ be the estimate of $\mu(\alpha,\beta)$ and $\hat{\sigma}^2(\alpha,\beta)$ respectively as defined in Lemma \ref{lem:var-atrimmedmean}, by replacing $F$ with $\hat{F}$.
Assume that $\sigma^{2}(\alpha,\beta)$ has a unique minimum w.r.t
$(\alpha,\beta)$ which falls into $(\alpha_{0},\alpha_{1})\times(\alpha_{0},\alpha_{1})$,
where $(\alpha_{0},\alpha_{1})\in(0,1)\times(0,1)$ with $\alpha_{0}+\alpha_{1}<1$.
Let
\begin{align*}
(\hat{\alpha},\hat{\beta}) & =\arg\min_{0<\alpha_{0}\leq\alpha,\beta\leq\alpha_{1}<1}\hat{\sigma}^{2}(\alpha,\beta)
\end{align*}
be the estimate of the optimal $(\alpha,\beta)$. Then $(\hat{\alpha},\hat{\beta})$
is a consistent estimate of the optimal $(\alpha,\beta)$.
\end{proposition}

\begin{proof}
Let $\psi(x)$ be the same as in Lemma \ref{lem:var-atrimmedmean}, and let $\hat{\psi}$ be an estimate of $\psi$  by replacing $F$ with $\hat{F}$. For $u\in (0,1)$, define
\begin{align*}
    e(u) = \psi(F^{-1}(u)) \text{ and }
    \hat{e}(u) =  \hat{\psi}(\hat{F}^{-1}(u)).
\end{align*}

Then
\begin{align*}
|\hat{\sigma}^{2}(\alpha,\beta)-\sigma^{2}| & =|\int_{0}^{1}\hat{e}^{2}(u)du-\int_{0}^{1}e^{2}(u)du|\\
 & =|\int_{0}^{1}(\hat{e}-e)^{2}(u)du+2\int_{0}^{1}e(u)(\hat{e}(u)-e(u))du|\\
 & \leq|\hat{e}-e|_{\infty}^{2}+2|e|_{\infty}\cdot|\hat{e}-e|_{\infty},
\end{align*}
where

\begin{align*}
|e|_{\infty} & =\sup_{\alpha_{0}\leq u\leq1-\alpha_{1}}|e(u)|<\infty\\
|\hat{e}-e|_{\infty} & =\sup_{\alpha_{0}\leq u\leq1-\alpha_{1}}|\hat{e}(u)-e(u)|\\
 & \leq \frac{2}{1-\alpha_0-\alpha_1}\sup_{\alpha_{0}\leq u\leq1-\alpha_{1}}|\hat{F}^{-1}(u)-F^{-1}(u)|\\
 & =O_{P}(n^{-1/2}).
\end{align*}
Therefore the uniform convergence holds:
\begin{align*}
\sup_{0<\alpha_{0}\leq\alpha,\beta\leq\alpha_{1}<1}|\hat{\sigma}^{2}(\alpha,\beta)-\sigma^{2}(\alpha,\beta)| & =o_{P}(1).
\end{align*}
The conclusion follows since the optimal $(\alpha,\beta)$ is unique.
\end{proof}

\begin{proposition}
\label{lem:convexity}Suppose that $f$ is from the extended Huber family as given by \eqref{eq:extended-huber} in Section 2.4. Then $\sigma^2(\alpha,\beta)$ is strictly convex
at the local neighborhood of $(\alpha,\beta)=(F(-k_{1}),1-F(k_{2}))$.
\end{proposition}

\begin{proof}
Note that
\begin{align*}
(1-\alpha-\beta)^{2}\sigma^2(\alpha, \beta) & =\int_{\alpha}^{1-\beta}\int_{\alpha}^{1-\beta}\frac{\min(s,t)-st}{f(F^{-1}(s))f(F^{-1}(t))}dsdt\\
 & \equiv RHS
\end{align*}
then
\begin{align*}
\frac{\partial RHS}{\partial\alpha} & =-2\int_{\alpha}^{1-\beta}\frac{\alpha(1-s)}{f(F^{-1}(s))f(F^{-1}(\alpha))}ds\\
 & =-\frac{2\alpha}{f(F^{-1}(\alpha))}\int_{\alpha}^{1-\beta}\frac{1-s}{f(F^{-1}(s))}ds\\
\end{align*}
and
\begin{align*}
\frac{\partial RHS}{\partial\beta} & =-\frac{2\beta}{f(F^{-1}(1-\beta))}\int_{\alpha}^{1-\beta}\frac{s}{f(F^{-1}(s))}ds.\\
\end{align*}
Hence
\begin{align*}
\frac{\partial^{2}RHS}{\partial\alpha^{2}} & =\frac{2\alpha}{f(F^{-1}(\alpha))}\frac{1-\alpha}{f(F^{-1}(\alpha))}-2\left(\frac{1-\alpha\frac{f'}{f^{2}}(F^{-1}(\alpha))}{f(F^{-1}(\alpha))}\right)\int_{\alpha}^{1-\beta}\frac{1-s}{f(F^{-1}(s))}ds\\
 & =\frac{2\alpha(1-\alpha)}{f^{2}(F^{-1}(\alpha))}-\frac{2(f-\alpha f'/f)}{f^{2}}(F^{-1}(\alpha))\cdot\int_{\alpha}^{1-\beta}\frac{1-s}{f(F^{-1}(s))}ds
\end{align*}
\begin{align*}
\frac{\partial^{2}RHS}{\partial\beta^{2}} & =\frac{2\beta(1-\beta)}{f^{2}(F^{-1}(1-\beta))}-2\left(\frac{f+\beta f'/f}{f^{2}}\right)(F^{-1}(1-\beta))\cdot\int_{\alpha}^{1-\beta}\frac{s}{f(F^{-1}(s))}ds\\
\frac{\partial RHS}{\partial\alpha\partial\beta} & =\frac{2\alpha\beta}{f(F^{-1}(\alpha))f(F^{-1}(1-\beta))}.
\end{align*}

Now to evaluate the values at
\begin{align*}
\alpha & =F(-k_{1})=f(-k_{1})k_{1}^{-1}\\
\beta & =1-F(k_{2})=f(k_{2})k_{2}^{-1},
\end{align*}
we have
\begin{align*}
\frac{f'}{f}(F^{-1}(\alpha)) & =k_{1}\\
\frac{f'}{f}(F^{-1}(1-\beta)) & =-k_{2}\\
\alpha k_{1} & =f(-k_{1})\\
\beta k_{2} & =f(k_{2}).
\end{align*}
Thus
\begin{align*}
\int_{\alpha}^{1-\beta}\frac{s}{f(F^{-1}(s))}ds & =\int_{-k_{1}}^{k_{2}}F(x)dx=xF(x)|_{-k_{1}}^{k_{2}}-\int_{-k_{1}}^{k_{2}}xf(x)dx=k_{2}\\
\int_{\alpha}^{1-\beta}\frac{1-s}{f(F^{-1}(s))}ds & =(k_{2}+k_{1})-k_{2}=k_{1}
\end{align*}
and
\begin{align*}
RHS & =1-\alpha-\beta\\
\frac{\partial RHS}{\partial\alpha} & =-2\\
\frac{\partial RHS}{\partial\beta} & =-2
\end{align*}
and
\begin{align*}
\frac{\partial^{2}RHS}{\partial\alpha^{2}} & =\frac{2(1-\alpha)}{k_{1}^{2}\alpha}\\
\frac{\partial^{2}RHS}{\partial\beta^{2}} & =\frac{2(1-\beta)}{k_{2}^{2}\beta}\\
\frac{\partial RHS}{\partial\alpha\partial\beta} & =\frac{2}{k_{1}k_{2}}.
\end{align*}
Hence
\begin{align*}
\frac{\partial\sigma^2(\alpha, \beta)}{\partial\alpha} & =\frac{\partial}{\partial\alpha}\frac{RHS}{(1-\alpha-\beta)^{2}}\\
 & =\frac{-2}{(1-\alpha-\beta)^{2}}+RHS\cdot\frac{2}{(1-\alpha-\beta)^{3}}=0\\
\frac{\partial\sigma^2(\alpha, \beta)}{\partial\beta} & =\frac{\partial}{\partial\beta}\frac{RHS}{(1-\alpha-\beta)^{2}}=\frac{-2}{(1-\alpha-\beta)^{2}}+RHS\cdot\frac{2}{(1-\alpha-\beta)^{3}}=0
\end{align*}
and the Hessian matrix can be calculated as below:
\begin{align*}
2\cdot(1,1)^{T}(1,1)\cdot\sigma^2(\alpha,\beta)+(1-\alpha-\beta)^{2}\frac{\partial^{2}\sigma^2(\alpha, \beta)}{\partial(\alpha,\beta)\partial(\alpha,\beta)} & =\left(\begin{array}{cc}
\frac{2(1-\alpha)}{k_{1}^{2}\alpha} & \frac{2}{k_{1}k_{2}}\\
\frac{2}{k_{1}k_{2}} & \frac{2(1-\beta)}{k_{2}^{2}\beta}
\end{array}\right)\\
\end{align*}
so
\begin{align*}
\frac{\partial^{2}\sigma^2(\alpha, \beta)}{\partial(\alpha,\beta)\partial(\alpha,\beta)} & =\frac{2}{(1-\alpha-\beta)^{2}}\left(\begin{array}{cc}
\frac{(1-\alpha)}{k_{1}^{2}\alpha}-\frac{1}{1-\alpha-\beta} & \frac{1}{k_{1}k_{2}}-\frac{1}{1-\alpha-\beta}\\
\frac{1}{k_{1}k_{2}}-\frac{1}{1-\alpha-\beta} & \frac{(1-\beta)}{k_{2}^{2}\beta}-\frac{1}{1-\alpha-\beta}
\end{array}\right).\\
\end{align*}
Note that
\begin{align*}
\int_{-k_{1}}^{k_{2}}x^{2}f(x)dx & =\int_{-k_{1}}^{k_{2}}-xf'(x)dx\\
 & =\int_{-k_{1}}^{k_{2}}f(x)dx-xf(x)|_{-k_{1}}^{k_{2}}\\
 & =1-\alpha-\beta-(k_{2}^{2}\beta+k_{1}^{2}\alpha)
\end{align*}
that is,
\begin{align*}
1-\alpha-\beta & =k_{1}^{2}\alpha+k_{2}^{2}\beta+\int_{-k_{1}}^{k_{2}}x^{2}f(x)dx\\
 & =k_{1}^{2}\alpha+k_{2}^{2}\beta+(1-\alpha-\beta)ES^{2}
\end{align*}
where $S$ is defined on $x\in(-k_{1},k_{2})$ with density $\frac{f(x)}{1-\alpha-\beta}$.
Then
\begin{align*}
ES & =(1-\alpha-\beta)^{-1}\int_{-k_{1}}^{k_{2}}xf(x)dx\\
 & =(1-\alpha-\beta)^{-1}\int_{-k_{1}}^{k_{2}}-f'(x)dx\\
 & =\frac{k_{1}\alpha-k_{2}\beta}{1-\alpha-\beta}
\end{align*}

So the determinant of $\frac{\partial^{2}\sigma^2(\alpha,\beta)}{\partial(\alpha,\beta)\partial(\alpha,\beta)}\times\frac{(1-\alpha-\beta)^{2}}{2}$
is
\begin{align*}
 & (\frac{(1-\alpha)}{k_{1}^{2}\alpha}-\frac{1}{1-\alpha-\beta})(\frac{(1-\beta)}{k_{2}^{2}\beta}-\frac{1}{1-\alpha-\beta})-(\frac{1}{k_{1}k_{2}}-\frac{1}{1-\alpha-\beta})^{2}\\
= & \frac{(1-\alpha)}{k_{1}^{2}\alpha}\frac{(1-\beta)}{k_{2}^{2}\beta}-\frac{1}{k_{1}^{2}k_{2}^{2}}+\frac{1}{1-\alpha-\beta}(\frac{2}{k_{1}k_{2}}-\frac{(1-\alpha)}{k_{1}^{2}\alpha}-\frac{(1-\beta)}{k_{2}^{2}\beta})\\
= & \frac{k_{1}^{2}\alpha+k_{2}^{2}\beta+(1-\alpha-\beta)ES^{2}}{k_{1}^{2}k_{2}^{2}\alpha\beta}-\frac{1}{1-\alpha-\beta}\cdot\frac{k_{1}^{2}\alpha(1-\beta)+k_{2}^{2}\beta(1-\alpha)-2k_{1}k_{2}\alpha\beta}{k_{1}^{2}k_{2}^{2}\alpha\beta}\\
= & \frac{(1-\alpha-\beta)^{2}ES^{2}-(k_{1}^{2}\alpha^{2}+k_{2}^{2}\beta^{2}-2k_{1}k_{2}\alpha\beta)}{(1-\alpha-\beta)k_{1}^{2}k_{2}^{2}\alpha\beta}\\
= & \frac{1-\alpha-\beta}{k_{1}^{2}k_{2}^{2}\alpha\beta}\cdot var(S)
\end{align*}
which is greater than 0 since $\alpha+\beta<1$. This completes the
proof.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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