text:
\section{Introduction}\label{secintro}
\subsection{Motivation}\label{secmotivation}
Most commercial products are sold with warranties. Before a new product
is launched to the market, it is extremely important to accurately
estimate the proportion of field returns within a given warranty period
in order to determine the monetary reserves for covering future
warranty claims. The failure information can be obtained through
pre-launch accelerated life tests (ALTs) in a timely fashion. In an
ALT, a~number of samples\vadjust{\goodbreak} are tested under harsh conditions, for
example, a combination of high voltage, temperature, pressure, use
rate, etc., which yields information on product reliability within a
reasonable time frame. Failure time data from the test are collected,
analyzed and extrapolated to estimate lifetime characteristics of
interest at \textit{nominal} use conditions based on some stress-life
models. There is a bulk of literature on ALT data analysis and optimal
design of ALT experiments. See \citet{pascual06}, \citet{ma08},
\citet
{guo11} and \citet{liu12}, among others. The use conditions are
implicitly assumed to be homogeneous (same for all customers) in most
ALT research, including the above references.

After the product is sold to customers with a warranty, units that fail
within the warranty period are returned to the manufacturer for repair
or replacement, which are known as warranty claims. These warranty
claim data reflect failure behaviors of the product under \textit{actual}
use conditions. Analysis of these warranty return data is useful
because it validates the results from ALT data analysis, and can be
used to improve the accuracy of parameter estimation from the ALT. See
\citet{blischke11} for an overview of this topic.

However, large discrepancies between the results of ALT data analysis
and field failure data analysis are often found. Analysis of field
failure data tends to suggest higher variability in the product's
failure times compared with the result based on ALT data analysis.
Conceivably, this is because products in the field are usually exposed
to heterogeneous usage and operating conditions. A motivating example
is as follows.

\citet{567} described an application involving an appliance, which is
called Appliance B. Appliance B contains a turbine device which has two
major failure modes: crack failure modes and wear failure modes.
Engineering knowledge suggests that it is reasonable to assume that
these two failure modes are independent. For illustration, we only
consider the wear failure mode, accounting for around 80\% of the total
field failures. Appliance B was sold with a two-year warranty. Before
its entry into the market, an ALT was conducted to obtain reliability
information of the product, in which 10 units were subject to a wear
test. Field failure data were also available during the subsequent
warranty tracking study of 4708 units with 93 wear failures. More
details can be found in \citet{567}.

According to the analysis in Section~\ref{secexample}, the Weibull
distribution provides a good fit to the failure data from ALT, but it
does not
provide an adequate fit to the field data. As we will argue, the
discrepancy is largely due to the varying operating conditions in the
field. When varying operating conditions are taken into account, theory
suggests the use of other distributions for the field data, such as the
Burr-XII distribution, which do fit well.

\subsection{Heterogeneous operating conditions}
The operating conditions are dynamic in a number of ways.\vadjust{\goodbreak} First off,
products are used in different geographical areas because of
customer
locations. Therefore, the operating environments (e.g., temperature,
humility, etc.) are heterogeneous for units across the product
population. Second, different users have different usage behaviors. In
a two-dimensional warranty analysis, it is commonly assumed that the
use rate of a customer is constant and it varies across the customer
population [\citet{485,884}]. \citet{752} also observed that the field
stress level may vary over the product population. Moreover, the usage
profile can be time dependent. As an example, \citet{830} reported a
problem where the stress profile, for example, pressure and
temperature, over time for a seal in brake cylinders is stochastic. The
presence of variable operating conditions significantly influences
failures of the product. As suggested from consumer reports in February
1991 [\citet{784}], the percentage of washer--dryer machines that ended
up with a warranty claim went up from 14\% among those who reported an
average of one to four laundry loads per week to 25\% among those who
reported an average in excess of eight loads per week. Furthermore,
this pattern was observed across brands consistently.

In the presence of heterogeneous operating conditions of the product
population, direct prediction of the proportion of warranty returns
from ALT data analysis can be highly biased. In principle, the failure
time distribution of the in-lab testing units can be linked to that of
the field population by taking into account information about these
dynamics in environments. The information includes the types of
significant dynamic environmental factors, the distributions for these
factors as well as the acceleration relationships that relate each
factor to the failure process. Among these environmental factors,
information about the use rate may be the easiest to collect. For
example, \citet{567} and \citet{752} focused on modeling the effects of
usage rates. Both studies assumed a constant usage rate for an
individual unit and a lognormal distribution for usage rates across the
product population. However, the field failure time distribution in
\citet{752} does not have closed-form expressions, which makes analysis
of field return data and verification of model assumptions (e.g., the
lognormal assumption of the usage rate distribution) very difficult,
and which greatly complicates the ALT planning for a new vintage of the
product under similar environments. Even if the distribution of the
usage rate is available, say, from a customer survey, the models in
these two studies still ignore other influential factors such as
heterogeneous customer locations. In fact, it is almost impossible to
directly collect information (i.e., distributions for each
environmental factor and their respective effects on the failure
process) about all heterogeneous environmental factors other than the
usage rate.

\subsection{Objectives and overview}\label{secobjective}
This paper is an endeavor to answer the question of how heterogeneities
in operating environments affect predictions of field failures and
planning\vadjust{\goodbreak} of ALTs. We treat the \textit{unobservable operating factors} as
well as \textit{their effects} on the product failure process as a
``frailty,'' through which the lab failure time distribution of a
product can be linked to the field failure time distribution. The
``frailty'' is an unobservable random variable used to account for
heterogeneities caused by unobservable covariates. In its simplest
form, the frailty is an unobserved random proportionality factor that modifies
the baseline failure rate function of an individual, which is similar
to the multiplicative effect of a covariate on the failure rate in
Cox's proportional hazard model. In biostatistics, lifetime models with
frailties have attracted much attention, for example, see \citet
{Hanagal11} for a book length treatment on this area. In reliability
engineering, the frailty is often called a random effect and also
receives some applications, for example, see \citet{stefanescu06},
\citet{697} and \citet{939}, among others. However, one challenge of
using frailty is that the resulting marginal distribution is often
mathematically intractable.

This paper develops tractable frailty models that relate ALT failures
to warranty failures. We show that in the presence of the frailty, the
hazard rate of a field unit exhibits various shapes. An appropriate
distribution for the frailty can be determined through joint modeling
of both ALT data and warranty return data. Detailed procedures to
analyze the data and to collate the frailty distribution are developed.
The results enable the prediction of field failures for a future
product through analysis of ALT data. We also derive optimal designs of
ALT experiments for a new vintage and show how the heterogeneities
affect the optimal ALT design.

The remainder of the paper is organized as follows. Section~\ref{secmodel} introduces the gamma frailty model for linking lab test
data and field failure data and investigates possible shapes of the
field failure rate. In Section~\ref{secinference} a procedure for
statistical inference of the frailty model is developed. We also
extensively discuss the model validation through hypothesis testing.
Optimal ALT plans under the frailty model are obtained in Section~\ref{secoptimalALT}. Section~\ref{secexample} applies the frailty model
to the Appliance B example. Section~\ref{secconclusion} concludes the paper.

\section{Linking lab failures and field failures}\label{secmodel}
Under the stable lab testing conditions, we assume the lifetime $X$ of
the product follows a Weibull distribution, which is one of the most
commonly used lifetime distributions. However, existence of the
heterogeneous operating conditions influences lifetime of a field unit.
The basic idea is to introduce into the hazard rate an additional
random parameter $Z$ that accounts for the heterogeneities. The frailty
$Z$ links the distribution of $X$ to that of the field failure time
$T$. In this section, the frailty model is developed and the hazard
rate of $T$ is investigated.

\subsection{Failures in lab testing}\label{seclabcdf}
As suggested by the extreme value theory, the Weibull distribution is
an appropriate lifetime model when the failure is caused by the
weakest\vadjust{\goodbreak}
flaw/link in a unit. It has been widely used for modeling lifetime of
products and components. The failure time $X$ of a lab testing unit is
assumed to follow a Weibull distribution with the respective cumulative
distribution function~(c.d.f.) and probability density function (p.d.f.)
given by
\[
F_X(x) = 1 - \exp \biggl[- \biggl(\frac{x}{\alpha}
\biggr)^\beta \biggr],\qquad x>0
\]
and
\begin{equation}
\label{eqnweibull} f_X(x) = \frac{\beta}{\alpha} \biggl(
\frac{x}{\alpha} \biggr)^{\beta-1} \exp \biggl[- \biggl(\frac{x}{\alpha}
\biggr)^\beta \biggr],\qquad x>0,
\end{equation}
where $\alpha>0$ is the scale parameter and $\beta>0$ is the shape
parameter. The hazard rate function of $X$ is given by
\begin{equation}
h_X(x) = \frac{\beta}{\alpha} \biggl(\frac{x}{\alpha}
\biggr)^{\beta-1}.
\end{equation}
It is well known that the hazard rate function is monotone increasing
when $\beta>1$ and monotone decreasing when $0<\beta<1$.

\subsection{Field failures: A gamma frailty model}
When the product is sold to customers, the operating conditions are
heterogeneous and unobservable. The unobservable effects are described
by a frailty $Z$. The frailty $Z$ is constant for a unit and varies
across the product population. Conditional on $Z$, the lifetime of a
field unit follows the Weibull distribution with a hazard rate function
given by
\begin{equation}
\label{eqnph} h_T(t; Z)=Zh_X(t)= Z\times
\frac{\beta}{\alpha} \biggl(\frac
{t}{\alpha
} \biggr)^{\beta-1}.
\end{equation}
Because the baseline distribution is Weibull, this frailty model is
similar to assuming a random scale parameter $\alpha$ [\citet{416}, page
457]. Previously, \citet{567} and \citet{752} adopted such method to
accommodate information on the heterogeneities. However, the reason we
do not use a random scale parameter is that it is difficult, if not
impossible, to find a distribution for $\alpha$ such that the resulting
field failure time distribution has a closed form.

The distribution of $Z$ depends on the heterogeneities of the field
environments as well as the effects of the random environments on the
product. For example, when the heterogeneities are caused by the random
use rate $U$, previous research suggests that the effect of $U$ on
product failures can be empirically described by a power law relation
$Z=aU^b$, $a, b >0$ are parameters, while the use rate distribution
tends to be unimodal and positively skewed. This leads to a unimodal
and positively skewed distribution for $aU^b$. Therefore, distributions
like the gamma [\citet{264,485}], lognormal [\citet{475,567}] and
inverse Gaussian distributions are appropriate for $Z$. Occasionally,
the uniform distribution is also recommended [\citet{266}]. The frailty
$Z$ includes the random usage rate and, thus, it is reasonable to
assume that it is also unimodal and positively skewed. To specify a
distribution family for the frailty $Z$, it is of advantage that the
resulting field failure distribution is tractable. This is because when
the distribution of $Z$ has a closed form, we can easily collate the
validity of the frailty distribution through data analysis. We find
that the families of gamma, inverse Gaussian and uniform distributions
for the frailty result in tractable distributions for $T$. In the
motivating example described in Section~\ref{secmotivation}, the
frailty is found to be well described by the gamma distribution.
Therefore, this paper focuses on the gamma frailty model. \emph
{Development of the inverse Gaussian frailty model and the uniform
frailty model is put in the supplemental material} [\citet
{ye13supplement}]. In fact, as suggested by \citet{627}, the gamma
distribution is highly flexible to reflect p.d.f.s of most shapes and,
thus, the gamma frailty model is applicable to similar problems other
than the Appliance B example.

In this section, we consider the gamma distribution with a threshold
parameter in order to demonstrate the fact that the hazard rate
function of $T$ exhibits various shapes. The three-parameter gamma
distribution with a threshold parameter $\gamma$ has a p.d.f. given by
\begin{equation}
\label{eqn3Pgamma} \varphi(z) = \frac{\mu^k(z-\gamma)^{k-1}}{\Gamma(k)} \exp \bigl[-\mu (z-\gamma)
\bigr], \qquad z>\gamma.
\end{equation}
When the frailty follows a distribution specified by (\ref
{eqn3Pgamma}), it can be shown by marginalizing $Z$ out of (\ref
{eqnph}) that the c.d.f. and p.d.f. of $T$ are, respectively, given by
\begin{eqnarray}
\label{eqnfieldDistrib} F_T(t) &=& 1 - \bigl[(t/
\alpha)^\beta/\mu+ 1 \bigr]^{-k} \exp \bigl[-\gamma(t/
\alpha)^\beta \bigr],
\nonumber
\\
f_T(t) &=& \frac{\beta}{\alpha} \biggl(\frac{t}{\alpha}
\biggr)^{\beta-1} \biggl[\frac{(t/\alpha)^\beta}{\mu} + 1 \biggr]^{-k} \biggl\{
\gamma +k \biggl[ \biggl(\frac{t}{\alpha} \biggr)^\beta+ \mu
\biggr]^{-1} \biggr\} \\
&&{}\times\exp \biggl[-\gamma \biggl(\frac{t}{\alpha}
\biggr)^\beta \biggr].\nonumber
\end{eqnarray}

It is interesting to note that when $\gamma= 0$, model (\ref
{eqnfieldDistrib}) reduces to the Burr-XII distribution. The Burr-XII
distribution has been used in reliability analysis by a few
researchers, for example, see \citet{700,953,952} and \citet{951}, to
name a few. However, the Burr-XII distribution is much less popular
than the lognormal distribution. Nevertheless, this distribution has
several advantages over the lognormal distribution. Similar to the
lognormal distribution, the Burr-XII distribution also has a unimodal
hazard rate. But compared with the lognormal distribution, the Burr-XII
distribution is more flexible in analysis of survival data. For
example, parameters of the Burr-XII distribution can be determined
through a simple probability plotting procedure [\citet{700}]. In
addition, it has greater mathematical tractability when dealing with
censored data which are very common in lifetime data analysis. The
contribution of a right-censored observation to the likelihood is equal
to the value of the survival function at the time of censoring, which
can be evaluated explicitly for the Burr-XII distribution, but not for
the log-normal distribution.

When $\gamma= 0$, the mean and variance of the frailty variable $Z$
are $k/\mu$ and $k/\mu^2$, respectively. If we fix $k/\mu$ at a
constant and let $\mu\rightarrow\infty$, then the distribution of $Z$
will degenerate to a single point, and the Burr-XII distribution will
also degenerate to a Weibull distribution. This is legitimate because
under such circumstance, there is no variation in the frailty. The
log-logistic distribution, a common distribution used in lifetime data
analysis, is also a special case of model (\ref{eqnfieldDistrib}),
when $\gamma= 0$ and $k=1$.

\subsection{Hazard rate for units in the field}
In reliability assessment, the shape of the hazard rate reflects the
early failure and aging behavior of the product. Therefore, it is
important to know the shape with a view to scheduling preventive
maintenance and detecting possible early failure modes. The hazard rate
function of~$Z$ can be readily obtained by dividing the p.d.f. by the
survival function, that is, $1 - F_T(t)$, which gives
\begin{equation}
\label{eqnfieldFailureRate} h_T(t) = \frac{\gamma\beta}{\alpha} \biggl(
\frac{t}{\alpha} \biggr)^{\beta
-1} + \frac{k\beta t^{\beta-1}}{t^\beta+ \mu\alpha^\beta}.
\end{equation}
The hazard rate of this distribution exhibits various shapes, as can be
checked through the first order derivative of (\ref
{eqnfieldFailureRate}) with respect to $t$. By and large, the hazard
rate could have four possible shapes, as summarized below.

\begin{longlist}[\textit{Case} 1.]
\item[\textit{Case} 1.] $\beta\leq1$.

The hazard rate $h_T(t)$ is decreasing in $t$. Specifically, when
$\beta< 1$, $h_T(t)$ decreases from $\infty$ to 0. When $\beta= 1$,
$h_T(t)$ decreases from $\gamma+ k/\mu$ to $\gamma$. This is because a
mixture of distributions with decreasing hazard rates has a
nonincreasing hazard rate.

\begin{figure}

\includegraphics{666f01.eps}

\caption{Illustrations of some shapes of the hazard function in
\protect\eqref{eqnfieldFailureRate}:
\textup{(a)} $\gamma=0, \alpha=1, k=1$; and \textup{(b)}~$\gamma=1, \alpha=1, k=1$.}
\label{figfailurerate}
\end{figure}

\item[\textit{Case} 2.] $\gamma> 0, \beta>1, \beta^2-\beta<\frac
{k}{4\gamma\mu}$.

The hazard rate $h_T(t)$ exhibits an N-shape.

\item[\textit{Case} 3.] $\gamma> 0, \beta>1, \beta^2-\beta>\frac
{k}{4\gamma\mu}$.

The hazard rate $h_T(t)$ is increasing.

\item[\textit{Case} 4.] $\gamma= 0$ and $\beta> 1$.

The hazard rate $h_T(t)$ has an upside-down bathtub shape.
\end{longlist}

Some typical curves of the hazard rate are depicted in Figure~\ref{figfailurerate}. It is interesting to see that when $\beta>1$, the
hazard rate under lab conditions is increasing, but the hazard rate of
a field unit can be either increasing, unimodal or N-shape. When the
hazard rate of $T$ is unimodal or N-shape, the initial hazard rate can
be very high, as can be seen from the dash dotted lines in Figure~\ref{figfailurerate}. In practice, when a manufacturer observes a high
hazard rate at the early stage, he may suspect that it is the infant
mortality caused by defects. The analysis in this section reveals that
early failures can also be caused by units operated under harsh
environments (i.e., large realizations of $Z$). These units are more
likely to fail and, hence, more ``frail'' than other field units.

\section{Statistical inference}\label{secinference}
Information about the distribution of the frailty can be obtained
through a joint analysis of lab data and field data. In the previous
section we adopt the three-parameter gamma distribution with a
threshold parameter~$\gamma$ for the frailty $Z$ to set forth the fact
that the field hazard rate can have various shapes in the presence of
heterogeneous operating conditions. In reality, the frailty $Z$ often
ranges from zero to infinity. Thus, this section focuses on the case
when the frailty follows a regular two-parameter gamma distribution
(i.e., $\gamma= 0$), under which the field failure time $T$ follows
the Burr-XII distribution.

Suppose that $n$ units are tested in the lab and $x_i$ is the observed
failure time or censoring time for the $i$th unit. Further, let $\delta
_i$ be the censoring indicator, where $\delta_i=0$ when the unit is
right censored and 1 otherwise. Therefore, for the $i$th lab unit, we
observe $(x_i,\delta_i)$. Similarly, suppose we observe the
failure\vadjust{\goodbreak}
times of $N$ field units $(t_j, \tilde\delta_j)$, $j=1,2,\ldots,N$,
where the field-data censoring indicator $\tilde\delta_j=0$ when the
$j$th unit is right censored and 1 if it fails and is returned as a
warranty claim.

\subsection{Estimation and hypothesis tests}\label{secestimation}
Given the lab testing data and the warranty return data for the same
product, we develop a procedure to analyze the data by capitalizing on
the model in Section~\ref{secmodel}. In this procedure, we need to
first collate the Weibull distribution (\ref{eqnweibull}) for the ALT
data, and then check if the warranty return data conform to the
Burr-XII distribution with c.d.f.
\begin{equation}
\label{eqnburrXII} G(t) = 1 - \bigl[(t/\lambda)^{\beta} + 1
\bigr]^{-k}, \qquad t > 0.
\end{equation}
It is noted from (\ref{eqnweibull}) and (\ref{eqnburrXII}) that when
the gamma frailty model holds, the Weibull shape parameter in (\ref
{eqnweibull}) should be equal to $\beta$ in the Burr-XII distribution
(\ref{eqnburrXII}), and $\lambda=\alpha\mu^{1/\beta}$. Given
$\lambda
$, $\alpha$ is a power function of $\mu$. With field data only, we can
only estimate $\lambda$, which results in identifiability issues for
$\alpha$ and $\mu$. This happens in bio and medical statistics [\citet
{Hanagal11}]. In our problem, however, $\alpha$ can be estimated from
ALT data, after which $\mu$ is uniquely determined. Therefore, our
problem is free of the identifiability issue. In addition, the equality
of $\beta$ provides us a means to collate the correctness of the gamma
frailty model. Details of the procedure are as follows.

\begin{longlist}[\textit{Step} 1.]
\item[\textit{Step} 1.] Fit the lab test data using the Weibull model with c.d.f.
given by (\ref{eqnweibull}). To underscore the fact that the shape
parameter $\beta$ is estimated from the lab data, we replace it with
$\beta_L$ in the following presentation. The maximum likelihood (ML)
estimate of $(\alpha, \beta_L)$, denoted as $(\hat\alpha, \hat
\beta
_L)$, is obtained by maximizing the log-likelihood function (up to a constant)
\begin{equation}
\label{eqnwbllikelihood} l_L(\alpha, \beta_L|
\mathrm{Lab\ Data}) = \sum_{i=1}^n{ \bigl[
\delta _i(\ln\beta_L +\beta_L\ln
x_i -\beta\ln\alpha_L)-(x_i/\alpha
)^{\beta
_L} \bigr]}.
\end{equation}
Assess goodness of fit of the Weibull model. If the Weibull
distribution provides a good fit to the lab data, then proceed to step 2.
\item[\textit{Step} 2.] Fit the field return data with the Burr-XII
distribution. Here, $\beta$ in (\ref{eqnburrXII}) is replaced with
$\beta_W$ to stress the fact that this parameter is estimated from
field data. The ML estimate of $(\lambda, \beta_W, k)$, denoted as
$(\hat\lambda, \hat\beta_W, \hat k)$, is obtained by maximizing the
log-likelihood function (up to a constant)
\begin{eqnarray}
\label{eqnwbllikelihood} &&l_W(\lambda, \beta_W, k|
\mathrm{Field\ Data})\nonumber\\
&&\qquad=\sum_{j=1}^N\tilde {
\delta }_j \bigl\{\ln(k\beta_W)+\beta_W
\ln(t_j/\lambda)-\ln\bigl[(t_j/\lambda
)^{\beta_W}+1\bigr] \bigr\}
\\
&&\qquad\quad{}- \sum_{j=1}^Nk\ln
\bigl[(t_j/\lambda)^{\beta_W}+1\bigr].\nonumber
\end{eqnarray}
Assess the goodness of fit of the Burr-XII distribution. If it
provides a good fit, proceed to step 3.
\item[\textit{Step} 3.] Test the hypothesis $H_0\dvtx k = 1$ versus the
alternative hypothesis $k\neq1$. If we accept the null hypothesis, the
frailty follows an exponential distribution and the field failure time
follows a log-logistic distribution, so we can fit the field data with
the log-logistic distribution. If the hypothesis is rejected, stick to
the Burr-XII distribution.
\item[\textit{Step} 4.] Test the hypothesis $H_0\dvtx \beta_L = \beta_W$ versus
the alternative hypothesis $\beta_L \neq\beta_W$. If the null
hypothesis is accepted, then there are statistical evidences that the
frailty follows a gamma/exponential distribution, and then we can
proceed to step 5.
\item[\textit{Step} 5.] Estimate the parameters in the gamma frailty model
(\ref
{eqnfieldDistrib}) by combining the lab test data and field return
data. The ensemble log-likelihood function is
\begin{equation}\label{e10}\qquad
l(\alpha,\beta,\lambda, k|\mathrm{All\ Data}) =  l_L(\alpha, \beta
|\mathrm{Lab\ Data})+ l_W(\lambda, \beta, k|\mathrm{Field\ Data}).
\end{equation}
\end{longlist}

To test the hypothesis in step 3, we can use either the likelihood
ratio test or the score test. These two tests are expected to be
accurate, as the size of field return data is often large. However,
these two tests may not be accurate enough when testing the hypothesis
in step 4, insofar as the lab test data are often limited. When both
the ALT data and the field return data are complete or Type II
censored, the following theorem shows that $\hat\beta_L/\hat\beta
_W$ is
a ``pivotal statistic''---that is, its distribution is independent of
the unknown parameters $\alpha, \lambda$ and $\beta$. This ratio and
its distribution will therefore be helpful in testing the hypothesis
that $\beta_L=\beta_W$ in step~4.

\begin{theorem}\label{thmpivotal}
Suppose the lab failure times follow a Weibull distribution given by
(\ref{eqnweibull}), while the field failure times conform to a
Burr-XII distribution given in~(\ref{eqnburrXII}). Consider the
hypothesis $H_0\dvtx \beta_L = \beta_W \equiv\beta$ versus the
alternative hypothesis $\beta_L \neq\beta_W$ and assume the parameter
$k$ in (\ref{eqnburrXII}) is known. When both the lab test data and
field failure data are complete or Type II censored, $\hat\beta
_L/\hat
\beta_W$ is a pivotal statistic independent of $(\alpha, \lambda,
\beta)$.
\end{theorem}

Proof of this theorem is in the \hyperref[app]{Appendix}. The proof is based on the
fact that $\hat\beta_L /\beta$ and $ \hat\beta_W /\beta$ are pivotal
statistics under the Type II censored (or complete) lab data and field
data, respectively. The constant $k$ assumption is meaningful for the
log-logistic distribution where $k=1$. When $k\neq1$ and $k$ is
estimated from field data, we can treat $\hat k$ as the true value of
$k$. This approximation should work well because the field data are
often abundant and, thus, the estimation error of $k$ is small. Theorem
\ref{thmpivotal} is not restricted by the problem of limited ALT data
and, hence, it is expected to perform better for testing the hypothesis
of $\beta_L = \beta_W$ compared with the likelihood ratio test. In a
real-life application, we would recommend conducting both tests.
When\vadjust{\goodbreak}
the results of both tests tally, there is sufficient evidence to accept
or reject the hypothesis. When the results differ, we shall stick to
the test based on Theorem \ref{thmpivotal}. The distribution of $\hat
\beta_L/\hat\beta_W$ can be obtained through simulation as follows.

\begin{algo}\label{algo1}
\begin{longlist}[1.]
\item[1.] Generate $n$ samples from Weibull(1, 1) and $N$ samples from
BXII$(1,1,k)$. For Type II censoring, the number of events will be the
same as the number of events in the data sets. For Type I censoring,
the expected number of events will be the same as the number of events
in the data sets.

\item[2.] Estimate $\hat\beta_L^*$ and $\hat\beta_W^*$ from these two data
sets separately.

\item[3.] Repeat the above two steps $B$ times to get $\hat\beta
_L^{*i}/\hat
\beta_W^{*i}$, $i=1,2,\ldots,B$.

\item[4.] Use the $B$ samples to estimate the empirical c.d.f. and sample
quantiles of $\hat\beta_L/\hat\beta_W$.
\end{longlist}
\end{algo}

In Algorithm \ref{algo1}, one can use $\hat{k}$ as the value of $k$ in the
simulation. The performance of this substitution will be evaluated
through simulation. During the lab test, both Types~I and~II
censoring are common. For warranty return data, Type~I censoring or
progressive Type I censoring are more common due to staggered entries
and warranty limits. Under this scenario, $\hat\beta_L/\hat\beta_W$ is
an approximate pivotal.

\begin{theorem}\label{thmapproximate}
Suppose the lab test data follow a Weibull distribution given by~(\ref
{eqnweibull}), while the field failure data conform to a Burr-XII
distribution given in~(\ref{eqnburrXII}). Consider the hypothesis
$H_0\dvtx \beta_L = \beta_W \equiv\beta$ versus the alternative
hypothesis $\beta_L \neq\beta_W$ and assume the parameter $k$ in
(\ref
{eqnburrXII}) is known. When the lab test data and/or field failure
data are Type~I censored, then $\hat\beta_L/\hat\beta_W$ is an
approximate pivotal statistic.
\end{theorem}

Under Type I censoring, the distribution of $\hat\beta_W/\beta_W$
depends on the unknown fraction failing at the censoring time [e.g.,
\citet{jeng01}]. Thus, it is an approximate pivotal. The approximation
improves as the sample size increases. Because the sample size of field
return data is often large, the performance of the approximation of
$\hat\beta_W/\beta_W$ is typically satisfactory. On the other hand,
according to the type of lab test data, we have the following two discussions:
\begin{itemize}
\item When the lab test data is Type II censoring, then $\hat\beta
_L/\beta$ is an exact pivotal. Thus, $\hat\beta_L/\hat\beta
_W=(\hat\beta
_L/\beta)/(\hat\beta_W/\beta)$ is an approximate pivotal because
$\hat
\beta_W/\beta_W$ is an approximate pivotal.

\item When the lab test data is Type I censoring, then $\hat\beta
_L/\beta$ is an approximate pivotal. Thus, $\hat\beta_L/\hat\beta
_W=(\hat\beta_L/\beta)/(\hat\beta_W/\beta)$ is also an
approximate pivotal.
\end{itemize}
Algorithm \ref{algo1} can still be used to do the test and the performance will
be evaluated by simulations in the next subsection.\vadjust{\goodbreak}

\subsection{Simulation study}\label{secsimulation}
In this section we conduct simulation studies to show the performance
of the statistics proposed in Theorems \ref{thmpivotal} and \ref
{thmapproximate}. In particular, we consider three scenarios:
\begin{itemize}
\item Scenario I: Type II censoring for lab data and Type II censoring
for field data.

\item Scenario II: Type II censoring for lab data and Type I censoring
for field data.

\item Scenario III: Type I censoring for lab data and Type I censoring
for field data.
\end{itemize}
We assume that the ALT uses 10 testing units whose lifetime follows a
Weibull distribution. For Scenarios I and II of the simulation, the
test is run until 8 of the units fail (i.e., Type II censoring). For
Scenario III, the expected number of failures is 8 out of 10 testing
units in the ALT (the censoring time is 733 in the simulation). For the
field data, $N$ units of the same product are sold to customers and the
environmental frailty follows $\operatorname{Gamma} (k,\mu)$. For the Type II
censoring setting (Scenario I), we stop the follow-up when $0.1N$
failures have been observed. For the Type I setting (Scenarios II and
III), the failure times are censored at $\tau$. The censoring time
$\tau
$ is so chosen that the expected proportion of field failures is 10\%.
In the simulation, we use $\alpha=534,k=1,\mu=19$ and $\tau=878$. We
examine $N=2000, 5000$ and $\beta=1.5, 2.0$.

Under each combination of $(\beta,N)$, we replicate the simulation
$2000$ times. In each replication, we compute the likelihood ratio
statistic and the statistic in Theorem \ref{thmpivotal}. The
hypothesis is rejected or accepted according to the $\tilde\alpha$
level. The estimated Type I error is obtained as the proportion of
incorrect rejections. To obtain the distribution of the pivotal, we use
$B=5000$ in each run. In the simulation, we use normal approximation
to simulate $\hat\beta_W^{*i}$ and use the distribution of $\hat
\beta
_W^{*i}/\hat\beta_W$ to approximate the distribution of $\hat\beta
_W/\beta_W$. In particular, $\hat\beta_W^{*i}$ is simulated from
$\mathcal{N} (\hat\beta_W, \sigma^2_{\hat\beta_W} )$, where
$\sigma^2_{\hat\beta_W}$ is the large sample approximate variance
estimate of $\hat\beta_W$.

Table~\ref{tabsimulatecp} shows the estimated Type I errors of the
test procedure in Theorem~\ref{thmpivotal} and the likelihood ratio
test procedure, under three different scenarios. The nominal Type I
errors that are considered in the simulation are $\tilde\alpha=0.1,
0.05$ and $0.01$. Under all scenarios, the estimated Type I errors of
the testing procedure in Theorem \ref{thmpivotal} are closer to the
nominal ones compared with the likelihood ratio statistic. In addition,
the magnitude of $N$ tends to have little effect on the Type I errors
of the likelihood ratio statistic. This is best explained by our
conjecture that the bias/error of the likelihood ratio statistic is
attributed to the small lab testing samples. Overall, we can see that
the performance of the approximate pivotal is satisfactory.

\begin{table}
\def\arraystretch{0.9}
\caption{Estimated Type I error of the test procedure in
Theorem~\protect\ref{thmpivotal} and the likelihood ratio test
procedure, under three different scenarios. The nominal Type I errors
are $\tilde\alpha=0.1, 0.05, 0.01$}
\label{tabsimulatecp}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
&&& \multicolumn{2}{c}{$\bolds{\tilde\alpha=0.1}$} & \multicolumn{
2}{c}{$\bolds{\tilde\alpha=0.05}$} & \multicolumn{2}{c@{}}{$\bolds{\tilde\alpha
=0.01}$} \\[-6pt]
&&& \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill} \\
{\textbf{Scenarios}}&{$\bolds{\beta}$}&{$\textbf{\textit{N}}$} & \textbf{Thm} & \textbf{LR} & \textbf{Thm} & \textbf{LR} & \textbf{Thm} & \textbf{LR} \\
\hline
{Scenario I}
&1.5& 2000 & 0.108 & 0.142 & 0.057 & 0.072 & 0.010 & 0.016 \\
&1.5& 5000 & 0.100 & 0.149 & 0.058 & 0.078 & 0.008 & 0.024 \\[3pt]
&2.0& 2000 & 0.098 & 0.134 & 0.047 & 0.078 & 0.012 & 0.014 \\
&2.0& 5000 & 0.101 & 0.138 & 0.052 & 0.071 & 0.012 & 0.020 \\[6pt]
{Scenario II}
&1.5& 2000 & 0.108 & 0.128 & 0.056 & 0.077 & 0.014 & 0.016 \\
&1.5& 5000 & 0.092 & 0.136 & 0.048 & 0.080 & 0.010 & 0.018 \\ [3pt]
&2.0& 2000 & 0.110 & 0.135 & 0.056 & 0.075 & 0.014 & 0.018 \\
&2.0& 5000 & 0.110 & 0.146 & 0.058 & 0.082 & 0.014 & 0.022 \\[6pt]
{Scenario III}
&1.5& 2000 & 0.101 & 0.117 & 0.044 & 0.062 & 0.004 & 0.012 \\
&1.5& 5000 & 0.099 & 0.126 & 0.050 & 0.062 & 0.004 & 0.014 \\[3pt]
&2.0& 2000 & 0.100 & 0.116 & 0.048 & 0.060 & 0.008 & 0.012 \\
&2.0& 5000 & 0.094 & 0.118 & 0.040 & 0.060 & 0.008 & 0.012 \\
\hline
\end{tabular*}
\end{table}

\section{Optimal accelerated life tests}\label{secoptimalALT}
Over the course of product evaluation and customer feedback, the
manufacturer will generate a number of design changes and come up with
a new vintage. ALTs can again be used to evaluate reliability of this
new vintage by making use of the frailty information obtained from
joint analysis of lab data and field data of previous generations. The
ALTs need to be conducted within stringent cost and time constraints,
and the testing samples need to be used efficiently. In addition, the
heterogeneous field conditions should be taken into account when
estimating life characteristics of interest. It is expected that the
operating conditions and the effects of the environments on the new
generation be approximately the same. This implies that the new vintage
will have the same frailty $Z$ with the old generation. Based on this
fact, optimal ALT plans can be developed.

For the new generation of interest, suppose its lifetime $X$ under the
stable lab testing conditions follows a Weibull distribution specified
by (\ref{eqnweibull}). Let $S_0$ be the \textit{nominal} design stress
(say, the same as the old generation) and $S_H$ be the highest
allowable test stress that has been pre-specified. For convenience, the
stress is re-parameterized as
$
\xi=(S-S_H)/(S_0-S_H).
$
It is noted that under the nominal stress $S_0$, $\xi=1$. When $X$
follows a Weibull distribution, $Y=\ln X$ conforms to a smallest
extreme value distribution with the location parameter $\eta=\ln
\alpha
$ and the scale parameter $\sigma=1/\beta$. Following the convention of
ALT design for the Weibull distribution [e.g., \citet{416}, Chapter~17],
we work with the extreme value distribution whose Fisher
information\vadjust{\goodbreak}
matrix has a closed form, and assume that the scale parameter $\sigma$
is a constant independent of the stress while the location parameter
$\eta$ depends on the stress through a linear stress-life model
\[
\eta(\xi)=\upsilon_0+\upsilon_1 \xi.
\]

Usually, the optimal test plans use only two test stresses with the
higher stress being the highest allowable stress $S_H$. Therefore, an
ALT plan is specified by the lower stress level $\xi_L$ and the
proportion of units $\pi$ for this stress. The combination $(\xi_L,
\pi
)$ is called a test plan. The purpose of the ALT design is to find out
the optimal test plan $(\xi_L^*, \pi^*)$ in order to optimize a certain
index of interest. When we are interested in life characteristics under
the \textit{nominal} conditions (i.e., characteristics based on $X$) and
ignore the heterogeneous field operating conditions, optimal
constant-stress ALTs for the extreme value distribution have been well
studied, for example, see \citet{957} for the optimal Type I censoring
plan and \citet{escobar86} for the Type II censoring case. In the
presence of the heterogeneities, however, the criteria of ALT planning
should be based on field failure times $T$ and, thus, the existing
plans are no longer optimal. Optimal plans that take the frailty $Z$
into account will be developed in this section. Denote $\bI(\xi_L,
\pi
)$ as the Fisher information matrix for $(\upsilon_0, \upsilon_1,
\sigma
)$ under the test plan $(\xi_L, \pi)$. The matrix $\bI(\xi_L, \pi
)$ has
been derived by \citet{957} under the Type I censoring scheme and by
\citet{escobar86} with Type II censoring. Therefore, use will be
directly made of these existing results.

\subsection{Minimization of the asymptotic variance of the
$p$-quantile}\label{secALTquantile}
Consider the common criterion of ALT planning that minimizes the
asymptotic variance of the ML estimator $\hat t_p$ of the $p$ quantile
of field failure time $T$. Based on (\ref{eqnfieldDistrib}) with
$\gamma=0$, the $p$-quantile of $T$ is given by
\begin{equation}
\label{eqnquantile} t_p=\alpha \bigl[ \mu(1-p)^{-1/k} - \mu
\bigr]^{1/\beta} = \exp (\upsilon_0+\upsilon_1)
\bigl[\mu(1-p)^{-1/k} - \mu \bigr]^\sigma.
\end{equation}
The asymptotic variance of the ML estimator $\hat t_p$ is $\mathit{AV}(\hat t_p)
= (\nabla t_p)^\prime\bI(\xi_L, \pi)\nabla t_p$, where $\nabla t_p$ is
the first derivative of $t_p$ with respect to $(\upsilon_0, \upsilon_1,
\sigma)$. The expression of $\nabla t_p$ is quite involved.
Alternatively, it is not difficult to show that minimization of
$\mathit{AV}(\hat t_p)$ amounts to minimization of the asymptotic variance of
$\ln\hat t_p$, which is equivalent to minimizing the asymptotic
variance of $\hat t_p/t_p$. The asymptotic variance of $\ln\hat t_p$ is
$\mathit{AV}(\ln\hat t_p) = (\nabla\ln t_p)^\prime\bI(\xi_L, \pi)\nabla
\ln
t_p$, where $\nabla\ln t_p$ is the gradient of $\ln t_p$ with respect
to $(\upsilon_0, \upsilon_1, \sigma)$ as
\begin{eqnarray}
\nabla\ln t_p(1) &=& \frac{\partial\ln t_p}{\partial\upsilon_0} = 1,
\nonumber
\\
\nabla\ln t_p(2) &=& \frac{\partial\ln t_p}{\partial\upsilon_1} = 1,
\\
\nabla\ln t_p(3) &=& \frac{\partial\ln t_p}{\partial\sigma} = \ln \bigl[
\mu(1-p)^{-1/k} - \mu \bigr].
\nonumber
\end{eqnarray}
Optimal test plans can be obtained by minimizing $\mathit{AV}(\ln\hat t_p)$
under some constraints, for example, time constraint, budget constraint
or sample size constraint.

\subsection{Minimization of the asymptotic variance of the failure
probability}
The $p$-quantile criterion considered above is often used to determine
a suitable warranty period for a new product [\citet{835}]. For a
product with a given warranty period $\tau$, what the manufacturer is
most concerned with is the proportion of field returns within $\tau$.
Therefore, another rational planning criterion is to minimize the
asymptotic variance of $\hat p_\tau$,
the ML estimate of the probability of warranty failures. This
probability is given by
\begin{equation}
\label{eqnALTprob} p_\tau=1- \bigl[ (\tau/\alpha)^\beta/\mu+ 1
\bigr]^{-k} = 1 - \biggl[ \frac{[\tau\exp(-\upsilon_0-\upsilon_1)]^{1/\sigma}}{\mu} + 1 \biggr]^{-k}.
\end{equation}
The first derivative of $p$ with respect to $(\upsilon_0, \upsilon_1,
\sigma)$ can be obtained as
\begin{eqnarray}
\nabla p_\tau(1)=\frac{\partial p}{\partial\upsilon_0} &=& -\frac
{k\Omega^{1/\sigma}}{\mu\sigma} \biggl(
\frac{\Omega^{1/\sigma
}}{\mu} + 1 \biggr)^{-k-1},
\nonumber
\\
\nabla p_\tau(2)=\frac{\partial p}{\partial\upsilon_1} &=& -\frac
{k\Omega^{1/\sigma}}{\mu\sigma} \biggl(
\frac{\Omega^{1/\sigma
}}{\mu} + 1 \biggr)^{-k-1},
\\
\nabla p_\tau(3)=\frac{\partial p}{\partial\sigma} &=& -\frac
{k\Omega
^{1/\sigma}}{\mu\sigma^2} \biggl(
\frac{\Omega^{1/\sigma}}{\mu} + 1 \biggr)^{-k-1}\ln\Omega,
\nonumber
\end{eqnarray}
where $\Omega=\tau\exp[-(\upsilon_0+\upsilon_1)]$. Based on the delta
method, the asymptotic variance is $\mathit{AV}(\hat p_\tau) = (\nabla p_\tau
)^\prime\bI(\xi_L, \pi)\nabla p_\tau$. Optimal test plans can be
determined by minimizing this asymptotic variance subject to possible
constraints on available resources.

\begin{table}[b]
\caption{Ordered failure time data observed from the ALT test}\label{tabALTdata}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lccccccc@{}}
\hline
99 & 141 & 163 & 300 & 350 & 523 & 602 & 687 \\
\hline
\end{tabular*}
\end{table}

\section{Illustrative example}\label{secexample}
\subsection{Weibull fit to lab test data}
10 units of Appliance B were subject to a lab test. The experiment
ended at $t=687$ units of time, upon which 8 failures were observed and
2 were censored. In order to demonstrate Theorem \ref{thmpivotal}, we assume that the
experiment ended when the 8th failure is observed. After this
modification, the data are Type II censored. The observed failure times
of the 8 failed samples are presented in Table~\ref{tabALTdata}.

We use the Weibull model to fit the ALT data, and the ML estimates
(standard errors) of the two parameters are $\hat\alpha=529.4$ (121.0)
and $\hat\beta_L=1.55$ (0.470), respectively. In order to visualize the
goodness of fit, we also fit the data using the Kaplan--Meier method.
The estimated c.d.f.s by means of the Weibull model and the Kaplan--Meier
method are depicted in Figure~\ref{figweibullfit}. As can be seen from
this figure, the~estimated Weibull c.d.f. passes through the empirical c.d.f.
and falls well within the 95\% simultaneous confidence band (SCB).
Therefore, the Weibull model is considered as an appropriate model for
the product under nominal conditions.\vspace*{-3pt}

\begin{figure}

\includegraphics{666f02.eps}

\caption{Weibull probability plot showing the Weibull fit to the lab
data and the 95\% nonparametric SCB.}\label{figweibullfit}\vspace*{-3pt}
\end{figure}

\begin{figure}

\includegraphics{666f03.eps}

\caption{Weibull probability plot showing the ML estimates of the
Weibull, log-logistic and Burr-XII fits to the field data and the 95\%
nonparametric SCB.}\label{figwarrantyfit}
\end{figure}

\subsection{Burr-XII fit to the field failure data}
We first use the Weibull distribution to fit the field return data. The
maximum log-likelihood value is $-977.2$. The estimated Weibull c.d.f. as
well as the empirical c.d.f. using the Kaplan--Meier method is shown in
Figure~\ref{figwarrantyfit}. As can be seen from this figure, the
Weibull distribution cannot capture the curvature of the nonparametric
estimates in the lower tail. We suspect that the inconsistency between
failures in the lab and in the field is caused by heterogeneous
operating conditions. Therefore, the gamma frailty model is invoked to
solve the problem.

We apply (\ref{eqnfieldDistrib}) to fit the data and use the
likelihood ratio statistic to test the threshold parameter $\gamma=0$.
The test reveals no evidence to reject the hypothesis. Therefore, we
set $\gamma=0$ in the following analysis. We apply the Burr-XII
distribution to fit the data, and the maximum log-likelihood value is
$-973.8$. The estimated values of the parameters are $\hat\lambda
=298.6$ (83.9), $\hat\beta_W=2.66$ (0.452) and $\hat k=0.0223$\vadjust{\goodbreak}
(0.0109), respectively. We use the Akaike information criterion (AIC)
to compare the Burr-XII model and the Weibull model for the field data.
The AIC is specified by $\mathrm{AIC}=-2l+2m$, where $l$ is the maximum
log-likelihood value of a model and $m$ is the number of parameters in
the model. The respective AIC values for the Weibull and the Burr-XII
distributions are 1958.4 and 1953.5. The Burr-XII distribution has a
smaller AIC value, indicating a better fit. As can be seen from
Figure~\ref{figwarrantyfit}, the Burr-XII distribution captures the
curvature of the nonparametric estimates in the lower tail very well,
indicating a better fit than the Weibull distribution.

We also fit the data by using the log-logistic distribution, leading to
a maximum likelihood value of $-977.0$. This value is very similar to
that of the Weibull model. Overall, the analysis suggests that the
Burr-XII distribution is more appropriate for the field data than the
Weibull model.

\subsection{The gamma frailty model}
As can be seen from the above analysis, $\hat\beta_L$ is quite close to
$\hat\beta_W$. We apply the statistic developed in Theorem \ref
{thmpivotal} to quantitatively check the correctness of the gamma
frailty model by testing $H_0\dvtx \beta_L = \beta_W$. It is easy to see
that $\hat\beta_L /\hat\beta_W = 0.585$. By making use of Algorithm \ref{algo1},
the $p$-value is 0.217. We then apply the likelihood ratio test. The
likelihood ratio statistic is 1.356 with a $p$-value of 0.244. Both
tests suggest that there is no reason to reject this hypothesis.
Therefore, we can believe that the discrepancies between the lab test
data and the field data can be explained by the frailty model, and the
gamma frailty model is appropriate for the problem.

At the last step, we estimate the parameters in (\ref
{eqnfieldDistrib}) by combining both the ALT data and the field
failure data. The ensemble of the likelihood function consists of the
Weibull likelihood contributed from the lab data and the Burr-XII
likelihood contributed from the field data. Maximization of this
function yields the ML estimates of the four parameters (standard
errors) as
$\hat\alpha=545.15$ (84.7), $\hat\beta=2.28$ (0.32), $\hat\lambda
=385.05$ (136.5) and $\hat k=0.0341$ (0.019).
Using the invariance property of the MLE, the estimated scale parameter
for the gamma frailty is $\hat\mu= (\hat\lambda/\hat\alpha)^{\hat
\beta
} = 0.452$ with a standard error 0.23. The estimated c.d.f.s for the lab
failure time distribution and the field failure time distribution can
be updated based on these parameter estimates, as shown in Figure~\ref{figrefit}.

\begin{figure}

\includegraphics{666f04.eps}

\caption{Weibull probability plot showing the ML estimates of Weibull
fit to lab data and Burr-XII fit to the field data with a common $\beta
$.}\label{figrefit}
\end{figure}

\subsection{Optimal ALT plans}
In order to improve product reliability and cater to market changes,
the manufacturer may make a number of changes to the product and come
up with a new generation. The new generation, if sold to the market,
would be operated under the same environments as the old ones and the
environments will have the same effect on the product failures.
Therefore, we assume the frailty $Z$ follows the same gamma
distribution $\operatorname{Gamma}(k,\mu)$ with $\mu=0.452$ and $k=0.0341$. Suppose
that the manufacturer is interested in knowing the 5\% quantile of the
field lifetime of the new generation, and a maximum test time of 50 is
allowed for the ALT. During the test, all units are run simultaneously.
Assume that the lifetime of the new vintage follows a Weibull
distribution under the nominal use condition, and the planning values
of the ALT are $\upsilon_0=3$, $\upsilon_1=3.4$ and $\beta=2.28$. Based
on the above settings, optimal test plans can be obtained by
numerically optimizing the asymptotic variance given in Section~\ref{secALTquantile}. For example, if the objective is to minimize the
asymptotic standard deviation, that is, the square root of the
variance, of $\ln\hat t_p$, then the optimal test plan is $(\varepsilon
^*,\pi^*)=(0.338, 0.649)$ and the associated minimal standard deviation
is 3.23. Figure~\ref{figoptimalALT}(a) shows the contour of the
asymptotic standard deviation with respect to $\varepsilon$ and $\pi$.
This test plan also minimizes the asymptotic standard deviation of
$\hat t_p$, as can be seen from Figure~\ref{figoptimalALT}(b). If we
ignore the heterogeneous field environments, the optimal test plan will
be $(\varepsilon,\pi) = (0.419, 0.766)$, which is quite different from
$(\varepsilon^*,\pi^*)$.

\begin{figure}

\includegraphics{666f05.eps}

\caption{Contours of the asymptotic standard deviation for the
two-stress optimum ALT plan.}\label{figoptimalALT}
\end{figure}

\section{Conclusions}\label{secconclusion}
This study has explained the discrepancies between in-lab failures and
field failures through the frailty model. The frailty term of each
field unit represents the unobserved operating conditions and their
complicated effects on the product failures. In the presence of
heterogeneous operating conditions, we showed that the field failure
rate can exhibit a variety of shapes, and some units may fail very
early due to severe working conditions rather than defects. ALTs should
take these heterogeneities into account. Previous research assumed
homogeneous operating conditions, which will inevitably underestimate
the variation of the field failures and, in turn, underestimate the
proportion of field returns. In addition, test plans derived under the
homogeneity assumption may be quite different from the true optimum due
to ignorance of the heterogeneity. We overcame these deficiencies and
derived the optimal plans by considering the frailty. A procedure was
developed to obtain the frailty information and to collate the validity
of the gamma distribution for the frailty. Instead of using the
likelihood ratio statistic to test the equality of $\beta_L$ and
$\beta
_W$, we suggested the use of the statistic $\hat\beta_L/\hat\beta_W$.
This statistic is pivotal under complete or Type II censored data.
Under Type I censoring, this statistic is approximately a pivotal
quantity and its good performance is demonstrated through a simulation
study. In the supplement, we further developed the inverse Gaussian
frailty models and the uniform frailty models. These two models yield
tractable field failure distributions and supplement the class of
frailty models for linking lab failures and field failures. We also
proposed an ensemble inference procedure in consideration of all the
gamma, inverse Gaussian and uniform frailty models in the supplement
[\citet{ye13supplement}].

\begin{appendix}\label{app}
\section*{Appendix}
\subsection*{Shapes of the hazard rate function of the gamma frailty model}
Taking the first derivative of (\ref{eqnfieldFailureRate}) with
respect to $t$ yields
\begin{eqnarray}
\label{eqnFRderivative}\quad h^\prime(t) &=& \biggl[(2\beta\gamma\mu-2\gamma
\mu-k)t^\beta+ \mu \alpha ^\beta(\mu\gamma+k) (\beta-1) +
\frac{(\beta-1)\gamma t^{2\beta
}}{\alpha
^\beta} \biggr]
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&{}\times
\frac{\beta t^{\beta-2}}{(t^\beta+ \mu\alpha
^\beta)^2}.
\end{eqnarray}
The second term on the right-hand side of (\ref{eqnFRderivative}) is
always larger than 0. So we can focus on the first term
\begin{equation}
\qquad r(x)=(\beta-1)\gamma\alpha^{-\beta}x^2 + (2\beta\gamma\mu -2
\gamma \mu-k)x+\mu\alpha^\beta(\mu\gamma+k) (\beta-1).
\end{equation}

\textit{Case} 1. When $\beta<1$, it is easy to see that $r(x)<0$ and,
hence, $h^\prime(t)<0$.

When $\beta>1$, the minimum of $r(x)$ is achieved at the point
\[
\biggl( \frac{-2\gamma\mu(\beta-1)+k}{2(\beta-1)\gamma\alpha
^{-\beta
}},\frac{4\gamma\mu\beta(\beta-1)-k}{4(\beta-1)\gamma\alpha
^{-\beta} k^{-1}} \biggr).
\]
\textit{Case} 2. When $\gamma>0, \beta>1$ and $4\gamma\mu\beta(\beta
-1)-k<0$, we see from $\beta>1$ that
\[
4\gamma\mu\beta(\beta-1)-2k\beta<0 \qquad\mbox{so } 2\gamma\mu(\beta-1)-k<0.
\]
This means that
\[
\frac{-2\gamma\mu(\beta-1)+k}{2(\beta-1)\gamma\alpha^{-\beta}}>0 \quad\mbox{and}\quad \frac{4\gamma\mu\beta(\beta-1)-k}{4(\beta-1)\gamma\alpha
^{-\beta}
k^{-1}}<0.
\]
By noting that $r(0)=\mu\alpha^{\beta}(\mu\gamma+k)(\beta-1)>0$, we
see that when $x\geq0$, $r(x)$ is positive initially, is followed by a
negative period, and then becomes positive again. From (\ref{e10}), we see
$h^\prime(t)$ also has this positive--negative--positive sign change.
Therefore, $h(t)$ increases initially, is followed by a decreasing
period, and then increases again, that is, $h(t)$ has an N-shape.

\textit{Case} 3. When $\gamma>0, \beta>1$ and $4\gamma\mu\beta(\beta
-1)-k>0$, $r(x)$ is always greater than 0, and so is $h^\prime(t)$.
Therefore, $h(t)$ is increasing over $[0, \infty)$.

\textit{Case} 4. When $\gamma=0$ and $\beta>1$, $r(x)$ reduces to
$r(x)=-kx+(\beta-1)k\mu\alpha^\beta$. This linear function is monotone
decreasing with $r(0)>0$ and $r(\infty)<0$. Therefore, $h(t)$ is
increasing at the outset and decreasing afterwards. Hence, $h(t)$ has
an upside-down bathtub shape.

\subsection*{Proof of Theorem \protect\ref{thmpivotal}}
Before proceeding to the proof of Theorem \ref{thmpivotal}, two lemmas are first presented.
\begin{lemma}\label{lempivotal1}
For the Burr-XII distribution given by (\ref{eqnburrXII}), conditional
on $k$, $\hat\beta_W /\beta_W$ is a pivotal statistic under Type II or
complete data.
\end{lemma}
\begin{pf}
Let $\bx=(x_1,\ldots,x_n)$ be an ordered random sample of size
$n$ from BXII$(1,1,k)$. An ordered random sample $\bt=(t_1,\ldots,t_n)$
conforming to BXII$(\beta_W, \lambda,k)$ can be obtained by taking
$t_i=\lambda x_i^{1/\beta_W}$. Suppose the sample was censored after
the $r$th observation. The ML estimates of the parameters in the
Burr-XII distribution can be obtained by deriving the score functions,
equating them to zero, and solving for the solution. Denote the ML
estimates based on $\bx$ and $\bt$ as $(\hat\beta_{W_0}, \hat
\lambda
_0)$ and $(\hat\beta_W,\hat\lambda)$, respectively. Now, we proceed to
investigate the relationship between $(\hat\beta_{W_0}, \hat\lambda_0)$
and $(\hat\beta_W,\hat\lambda)$. The log-likelihood function based on
$\bt$, up to a constant, can be written as
\begin{eqnarray*}
L(\beta_W,\lambda)&=&r\ln\beta_W+\sum
_{i=1}^r\ln(t_i/\lambda
)^{\beta
_W}-(k+1) \sum_{i=1}^r\ln
\bigl[(t_i/\lambda)^{\beta_W}+1 \bigr]
\\
&&{}-k(n-r)\ln \bigl[(t_r/\lambda)^{\beta_W}+1 \bigr].
\end{eqnarray*}
Therefore, the ML estimator $(\hat\beta_W,\hat\lambda)$ satisfies the
following equation:
\begin{eqnarray*}
& &(k+1) \sum_{i=1}^r{\frac{(t_i/\hat\lambda)^{\hat\beta_W}\ln
(t_i/\hat
\lambda)^{\hat\beta_W}} {(t_i/\hat\lambda)^{\hat\beta_W}+1}}
-r-\sum_{i=1}^r\ln(t_i/\hat
\lambda)^{\hat\beta_W}\\
&&\qquad{}+k(n-r) \frac{(t_r/\hat
\lambda
)^{\hat\beta_W}\ln(t_r/\hat\lambda)^{\hat\beta_W}} {(t_r/\hat
\lambda
)^{\hat\beta_W}+1} =0,
\\
& &(k+1) \sum_{i=1}^r{\frac{(t_i/\hat\lambda)^{\hat\beta_W}} {
(t_i/\hat
\lambda)^{\hat\beta_W}+1}}
+ k(n-r) \frac{(t_r/\hat\lambda)^{\hat
\beta
_W}} {(t_r/\hat\lambda)^{\hat\beta_W}+1}-r=0.
\end{eqnarray*}
If we substitute $t_i=\lambda x_i^{1/\beta_W}$ into the above two
equations, we can obtain
\begin{eqnarray}
 (k+1)\sum_{i=1}^r\frac{\Lambda_i\ln\Lambda_i}{\Lambda_i+1}-
r - \sum_{i=1}^r\ln\Lambda_i
+k(n-r)\frac{\Lambda_r\ln\Lambda_r
}{\Lambda
_r+1} &=& 0,
\\
(k+1)\sum_{i=1}^r
\frac{\Lambda_i}{\Lambda_i+1}+k(n-r) \frac
{\Lambda
_r}{\Lambda_r+1} -r &=& 0,
\end{eqnarray}
where $\Lambda_i= (x_i/(\hat\lambda/\lambda)^{\beta_W}
)^{\hat
\beta_W /\beta_W }$. The left-hand sides of the above two equations are
the score functions based on the sample $\bx$. Therefore, it is readily
seen that $\hat\lambda_0=(\hat\lambda/\lambda)^{\beta_W}$ and
$\hat\beta
_{W_0}= \hat\beta_W/\beta_W$. This means that the distribution of
$\hat
\beta_W/\beta_W$ is the same as $\hat\beta_{W_0}$, which does not
depend on $\beta_W$ and $\lambda$.
\end{pf}

\begin{lemma}\label{lempivotal2}
For the Weibull distribution specified by (\ref{eqnweibull}), $\hat
\beta_L/\beta_L$ is a pivotal statistic under Type II or complete data.
\end{lemma}
\begin{pf}
See \citet{958}.
\end{pf}

\begin{pf*}{Proof of Theorem \ref{thmpivotal}} By using Lemmas \ref{lempivotal1} and \ref
{lempivotal2} above, we can see that $\frac{\hat\beta_W/\beta
_W}{\hat
\beta_L/\beta_L}$ is a pivotal statistic. Under the null hypothesis,
this pivotal statistic is exactly $\hat\beta_L/\hat\beta_W$, which
completes the proof.
\end{pf*}
\end{appendix}

\section*{Acknowledgments}
We would like to thank the Editor, the Associate Editor and two
referees for their helpful comments on this paper.

\begin{supplement}[id=suppA]
\stitle{Supplement to ``How do heterogeneities in operating environments
affect field failure predictions and test planning?''}
\slink[doi]{10.1214/13-AOAS666SUPP}
\sdatatype{.pdf}
\sfilename{aoas666\_supp.pdf}
\sdescription{This supplement
develops two additional frailty models, that is, the inverse Gaussian
and the uniform frailty models. An ensemble inference procedure in
consideration of all the gamma, inverse Gaussian and uniform frailty
models is also provided.}
\end{supplement}

redpajama_set_name: RedPajamaArXiv


text:
Astana Opera in the Media
Astana Opera in the International Media
International competition of ballet dancers
List of successful candidates
State purchases plan
Available tenders
Single source procurement
Сhildren's studio
Administration and artists
Artistic Directors
Opera Company
Mezzo-sopranos
Guest soloists
Teachers-Repetiteurs
Leading soloists
Trainee soloists
Orchestra musicians
Production Crew
Set and Costume Designers
20 th Anniversary of the Capital City
Theatre box-offices
Not Dollish Love
The premiere of comic ballet Coppélia was a huge success at the Astana Opera. On November 22 and 23, the French version of the ballet, staged by Roland Petit, was presented in the capital to a full house, incessant applause and a sea of flowers.
Two and a half hours of watching the performance flew by in one breath. The plot of Coppélia is based on Ernst Theodor Amadeus Hoffmann's novel Der Sandmann (The Sandman). However, the creators of the production turned the German writer's gloomy mysticism into a comedy, telling the story of a young man who fell in love with a mechanical doll made by Dr. Coppélius. Thus, all the romantic feelings in this ballet are ridiculed, nevertheless, the audience still has something to reflect on – the story of Dr. Coppélius' unrequited love for a mechanical doll modeled after Swanilda.
At the beginning of the performance, the heroes were presented in a military garrison, with the set corresponding with the ascetic life of soldiers. Young residents of the town flirted with the brave military men, circling in a massive light and easy dance. Elegant costumes in pink and blue colours, made by prominent designers Ezio Frigerio and Franca Squarciapino, attracted the audience's attention. Against the gray backdrop of the barracks, the entertaining intricacy of the dance appeared incredibly spectacular.
The frivolous young man Frantz (Bakhtiyar Adamzhan, Arman Urazov), his jealous bride Swanilda (Aigerim Beketayeva, Shugyla Adepkhan) and the whimsical Coppélius (Luigi Bonino, Rustem Seitbekov), in whose window the mysterious lady with a fan sits, do not create a love triangle, but rather a quadrangle.
Frantz is interested in the stranger and wants to meet her. He secretly gets into Coppélius' house, where Swanilda has entered earlier, exposing the professor. As expected in the comedy genre, there is a happy ending: Frantz and Swanilda get married, the residents of the town are happy and only Coppélius is left alone with a broken doll in his hands and a broken heart.
Throughout the performance, according to the choreographer's idea, the characters communicate with the audience. Principal Dancers performed this ballet with truly French humor. The audience laughed and reacted vividly to everything happening onstage. On both days, everyone portrayed their heroes in their own way. Bakhtiyar Adamzhan's Frantz, without a doubt, turned out to be very strong in terms of dance technique, and light, playful and charming in terms of acting. Arman Urazov showed absolutely precise poses, his Frantz was flippant, just as Roland Petit intended him to be. On the first day, an unconditional beauty, a tall, aristocratic ballerina with elongated lines, Aigerim Beketayeva, went onstage as Swanilda. Her Swanilda is a capricious girl, curious and jealous. Childlike naivety was in everything: in gestures, looks of the dancer, she managed to portray a young coquette to perfection. She handled Roland Petit's complex choreography with dignity. Her footwork was wondrous, so deftly Aigerim Beketayeva performed movements, preserving all the nuances and observing the canons of the genre and style. A young dancer of the Astana Opera Ballet Company Shugyla Adepkhan performed Swanilda on the second premiere day. Choreography for this role causes considerable difficulties even for experienced ballerinas, but the dancer managed to overcome them, justifying the hopes of the ballet master.
The performers of Dr. Coppélius – Luigi Bonino (also the Ballet Master) and Rustem Seitbekov in this part relied on acting, because it played a major role here. Luigi Bonino went onstage as this character dozens of times. His interpretation can certainly be considered a reference. Rustem Seitbekov's portrayal was his own, and thereby a sense of originality was created. The dancer is fluent in classical ballet; he is perseverant and talented in performing comedic scenes. This applies to funny poses on the alchemy book, confrontation face to face with Frantz, and much more. The Astana Opera's corps de ballet showed excellent synchronism and cohesion; the dancers kept the lines, and, overall, were very musical.
Under the baton of the ballet conductor Arman Urazgaliyev, Léo Delibes' score sounded seamless and exciting. The Astana Opera Symphony Orchestra reproduced the musical text in a masterly fashion. In frosty premiere days, the capital's audience responded to the effervescent melodies with unbridled enthusiasm.
It is important to note that this wonderful ballet was added to the repertoire of the Astana Opera at the initiative of the Artistic Director of the Ballet Company, People's Artist of Russia Altynai Asylmuratova, and the premiere took place with the support of the Ministry of Culture and Sports of the Republic of Kazakhstan.
From Classical to Modern Dance: a Gala Ballet Will Be Presented in Nur-Sultan
The Artistic Director of the Astana Opera Ballet Company, People's Artist of Russia Altynai Asylmuratova has created a Gala Ballet program, which will be presented to the capital's audience on January 29 at the opera house's Grand Hall. Arman Urazgaliyev will conduct the Astana Opera Symphony Orchestra.
A Ballet Dedicated to Beethoven's Anniversary Created at Astana Opera
The world premiere of the ballet Beethoven – Immortality – Love staged by the famous German choreographer Raimondo Rebeck will be held on February 14 and 15 with the support of the Ministry of Culture and Sports of the Republic of Kazakhstan.
We extend our warmest congratulations on the New Year 2020!
For each of us, the New Year is a time of hopes and dreams, a time to open new horizons and gain new strength.
Astana Opera Cancels a Performance Due to the Day of National Mourning
(Русский) В связи тем, что 28 декабря объявлен Днем общенационального траура, спектакль «Щелкунчик» П.И. Чайковского, который должен был состояться 28 декабря в 17:00 отменяется.
Musical Christmas Evening at Astana Opera
A magnificent Christmas Gala Concert will be presented by the Astana Opera Chamber Choir under the direction of the Chief Choirmaster, Honoured Worker of Kazakhstan Yerzhan Dautov. On January 7, in the opera house's Chamber Hall, the famous choir collective will perform beloved by all folk songs, exciting choral works and popular Christmas carols.
Alan Buribayev Letters from Italy
On January 6, the Astana Opera will present a new unique project of the opera house's Principal Conductor the Honoured Worker of Kazakhstan, laureate of the State Prize of the Republic of Kazakhstan Alan Buribayev. The opera and symphony concert Letters from Italy will tell the stories of great composers through the letters that they wrote during their stay in an amazing country – Italy.

redpajama_set_name: RedPajamaCommonCrawl


A Fresno, California native and the only son of production workers, Myles grew up in the migrant worker areas of that Central California town and spent his later teens and twenties touring with his band, eventually finding his way to the San Francisco area playing sideman to a wide array of acts, most notably bluesman Fillmore Slim and soul singer, harmonica player, and songwriter John Nemeth. With his keen ear and sincere passion, AC has now become a gifted entertainer and musician and is maturing into one of the more potent guitarists and vocalists in the blues/rock genre. AC has cultivated a style that echoes a modern Johnny Winter, Steve Marriot, and Elvin Bishop, and his personal appearances have a "take no prisoners" approach, which along with a healthy dose of humor, usually brings even the toughest audiences to his side.
The AC Myles Band will be appearing for two shows at The Birk in May. On Friday, May 5, he will be holding a 7:30 pm performance, then on Saturday, May 6 the band will take the stage for an afternoon show starting at 2:00 pm. Both shows are $10.00 admission. The Birk is located at 11139 Hwy 202 in Birkenfeld.


text:
Portland Trail Blazers vs. Oklahoma City Thunder Preview
Portland will seek revenge as they once again face the Thunder in Oklahoma City.
By akicks Dec 21, 2022, 9:18am PST
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Photo by Ian Maule/Getty Images
Portland Trail Blazers (17-14) vs. Oklahoma City Thunder (13-18)
The Portland Trail Blazers have to be sick about letting Monday's game against the Thunder slip through their grasp. Oklahoma City is a team Portland would have expected to beat, but they could never break free from the Thunder's grasp. A few sloppy plays in crunch time followed by a bucket from Gilgeous-Alexander as time expired consigned the Blazers to a 123-121 loss that Portland will be anxious to avenge.
Currently in 13th place in the West, the Oklahoma City Thunder don't appear to be a team going anywhere this season. Still, the Thunder are the youngest team in the NBA, and young teams usually bring energy and a fearlessness that can be hard to beat if you aren't at your best. One would think that the Blazers will be locked in for the rematch but the Thunder will be ready to pounce if Portland can't stamp their authority on this game from the outset.
Trail Blazers vs. Thunder - Wednesday, December 21 - 5:00 p.m. PT
How to watch on TV: Root Sports Plus, NBA League Pass
Blazers injuries: Gary Payton II (out), Nassir Little (out), Jusuf Nurkic (questionable), Damian Lillard (probable), Drew Eubanks (probable), Josh Hart (probable), Jerami Grant (probable)
Thunder injuries: Chet Holmgren (out), Jeremiah Robinson-Earl (out), Ousmane Dieng (out), Josh Giddey (out), Darius Bazley (out)
SBN Affiliate: Welcome to Loud City
The Matchup
Free throws. On Monday the Thunder shot 37 free throws compared to 21 for the Blazers. With that kind of disparity it's pretty remarkable that the game came down to a buzzer beater. Blazers fans might well feel that this disparity was unwarranted, especially when Shai Gilgeous-Alexander shot twice as many as anyone else in the game. Regardless, you have to control what is in your ability to control and the Blazers didn't do that. It starts with defense. Play better defense and you will get fewer fouls called against you.
Three pointers. The Blazers lead the NBA in three point percentage at 38.9%, but they are shooting seven fewer per game then they did in 2020-2021. That's absolutely fine. When Portland's offense is clicking this season they are using the three point line to punish defenses who spend too much of their time clogging the middle. Otherwise they are looking for easy buckets. It's a work in progress, but this model is sustainable. They shot 43.2% against the Thunder on Monday. Look for them to be there or thereabouts again on Wednesday.
Shai Gilgeous-Alexander. Gilgeous-Alexander is having a fantastic season, averaging 31.3 points per game. He's the undisputed star of the Thunder, and he's playing like it. On Monday he poured in 35 points, made all 14 of his free throws and buried the buzzer beater to send Portland to defeat. Coach Billups might need to have his team do a bit more to slow him down. When a player is confident, they play like they are running downhill, and that's when opponents foul. I'd expect the Blazers to throw a few defensive wrinkles at him to try to throw him off balance.
JaydenRule of Welcome to Loud City looks at how and why Shai Gilgeous-Alexander is getting to the free throw line so often.
Shai Gilgeous-Alexander has emerged as a superstar this season and as we know, superstars get superstar calls. Over the last nine games, he is averaging an absurd 13.6 free-throw attempts a game and hitting them at an equally ridiculous 95.4%. Despite his underwhelming efficiency from the floor during this stretch, he has still been able to find ways to score points for his team.
Lots of love for Lillard from Berry Tramel of The Oklahoman.
Lillard has not publicly or privately manipulated the situation to force his way out of Portland. He's the poster child of what you want as the face of a franchise. He's what OKC hopes Shai Gilgeous-Alexander becomes, what Memphis hopes Ja Morant becomes, what Orlando hopes Paolo Banchero becomes. What Karl Malone and John Stockton became for Salt Lake City.
We love you Brooke!
OKC Mascot Scares Reporter : Blazers' Brooke Olzendam wasn't ready for Rumble the Bison
pic.twitter.com/kvblGT9Dhd
— FootBasket™.com (@FootBasket) December 20, 2022


text:
Government Issues New Social Media Guidelines for Armed
December 18, 2016 Saheli Social Media
New Delhi: With smart phones making huge proliferation in ranks of security forces, the government has issued fresh guidelines for regulating sharing of secret operational and service data on such platforms involving troops and officers of central paramilitary forces, violation of which will invite "strict legal action".
The three-page guidelines, issued by the Home Ministry recently and notified to Central Armed Police Forces (CAPFs) headquarters here, speak specifically of instances where force personnel have used personal cell phones to click pictures of an ongoing or concluded ambush or operation which later finds it way on media platforms like Twitter, Facebook, WhatsApp, YouTube, LinkedIn, Instagram and others.
The directives state that these are a reiteration and enhancement of the existing directives in this regard, but address some specific issues.
The latest order by the Union Home Ministry said there was a "strong need" to issue the do's and dont's afresh as it "has come to the notice of government that there have been instances where mobile phones and cameras of force personnel have been used for operational coverage and sensitive material was uploaded on social media without official permission."
The fresh directives stipulate legal action against defaulting personnel.
"Any such photo, video, among others are meant strictly for official use only and any unauthorised disclosure of confidential operations related information by uploading operational material onto social media sites is a serious breach of rules and may lead to charges being laid against offending force employees," the guidelines, accessed by PTI, said.
However, senior officials in these forces pointed out a loophole in this directive, saying in a number of operations multiple agencies like state police and army are involved and as these guidelines are not applicable on them, there could still be chances and instances of an information breach and subsequent sharing of multimedia on Internet-based social media platforms.
[Source:-News 18]
Tagged Armed For Government Guidelines Issues; media New social
Meet the begging bowl barons: How YOU are helping to fund international charity chiefs whose eye-popping pay is bankrolled by BRITISH aid
Pune University's International Centre gets ISO certificate


Now available in North America, the Ametis Collection features a complete line of bathroom faucets, featuring elegant curves and dynamic elements melded together, as well as LED lighting indicative of water temperature. The entire collection is available in polished chrome, black, and white finishes. The collection represents the fusion between the universe above and the oceans below.
"At Graff®, we have a history of collaboration with cutting-edge product designers, and working again with Davide Oppizzi was satisfying because we both really wanted to push the design envelope. And we feel we have," said Ziggy Kulig, Graff President.
"The Ametis Collection brings an inspired new perspective to bathroom design, one worthy of its fine Italian heritage. Pushing the boundaries of design and technology, Ametis provides a glimpse into the future of bathroom design."


text:
Amy Hart keen to do I'm a Celeb
February 26, 2017 November 10, 2020 Dating
Back To Mine
Laws of the Jungle: Dating for Women Over 40
Dating Shows Take a Nasty Turn
I'm a Celebrity's Emily Atack and Joel Dommett announce new TV dating show to replace Love Island
'Law of the Jungle' Variety Show Announces Hiatus Due To Coronavirus
'The Bachelor' is 'fluffy, not real' says NBC's new jungle dating show team
Most awkward and embarrassing dating show moment EVER!
This copy is for your personal non-commercial use only. Here are six relatively new series that just might capture your heart. This Australia-based series features young autistic people, many of whom have never been on a date before. Their super-sweet encounters during lawn bowling and comic conventions never end in smooch sessions. The bar is low — but the spirits are high. Eleven-minute episodes bounce back and forth among three couples at different stages in their relationships, including a couple who fell in love while teaming up on League of Legends. HBO Max. This latest batch of episodes take place in Sao Paulo, where the amount of flirting, drinking and spicy conversation is higher than it ever was in the United States. Getting to know each other over erotic-dance lessons and screaming sessions may not be the best way to bond. Thursdays, TBS.
Jennifer Shukusky does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment. In the world of online dating, nothing is as it seems. A recent paper presented at the Annual Conference of the International Communication Association and reported on in the press suggested that when evaluating photographs from online dating profiles, men and women judge enhanced and un-enhanced photos somewhat differently.
Enhanced photos, those in which a person has used makeup, hair styling, filters, or post-editing, were rated by both men and women as more being attractive. But while women also rated men in these photos as more trustworthy than in ordinary photos, the opposite was true of women: men rated women in enhanced photos as less trustworthy. For example, we tend to think that attractive people are also happier and more successful in their careers.
The dating game killer: the true story of a TV dating show, a violent sociopath, and a series of brutal murders ()by Sands, Stella. Place Hold. Summary.
If you're pledging allegiance only to BachelorNation, you're missing out on other opportunities to fall in love. While the ABC franchise is currently repeating its greatest hits, other relationship series are readily available, with singles who are more interested in finding that special someone than raiding the set's liquor cabinet or kissing up to the camera.
This Australia-based series features young autistic people, many of whom have never been on a date before. Their super-sweet encounters during lawn bowling and comic conventions never end in smooch sessions. It's enough just to get through dinner without having a panic attack. The bar is low — but the spirits are high.
Gamers who won't even turn away from their computer for a bathroom break will press pause for the right partner — as long as they're equally obsessed with video games. Eleven-minute episodes bounce back and forth among three couples at different stages in their relationships, including a couple who fell in love while teaming up on League of Legends.
HBO Max. This latest batch of episodes take place in Sao Paulo, where the amount of flirting, drinking and spicy conversation is higher than it ever was in the United States.
Thanks to the explosion and global popularity of reality TV in the late '90s and early s, viewers have been blessed with so many series that it's often difficult to remember them all. Although reality TV can provide a much-needed escape from the actual jarring reality of current times, some of these series are so far-fetched that it makes you question how they ever had an opportunity to see the light of day.
From outdoor births and plastic surgery beauty pageants to tricking women into dating and falling in love with a faux-version of Britain's Duke of Sussex, reality TV has figured out how to continuously cross the line when it comes to odd, awkward — and, at times, offensive — programming. With far too many to name, this list of series only scratches the surface of mind-blowing reality entertainment that fans have been privy to over the years. This reality series aired on Lifetime and featured couples who had, for some reason, chosen to give birth to their children in remote places sans the assistance of traditional medical professionals.
I'm a Celebrity's Emily Atack and Joel Dommett have announced their brand new dating show for ITV2, called Singletown.
The show, which never returned with new episodes after its fourth season in , is still under investigation by the US Department of Agriculture. DEW has also been cited for alleged violations in recent years—including after filming for the series began. That information has never been disclosed on the show. More than pages of government records and interviews with multiple former DEW volunteers reveal disturbing evidence.
Allegations in the documents include:. State game wardens raided the zoo in after finding that the Miners had imported two black bears and a Celebes macaque—a primate listed as endangered since —into Maine without proper permits. They subsequently received permits for those animals.
Amy Hart keen to do Get Me Out of Here!. Amy Hart is up for doing 'I'm a Celebrity Get Me Out of Here! The former Love Islander – who took part in the series of the popular dating show – has responded to rumours she has signed up for the first-ever UK edition of the jungle reality show. Add to Chrome.
It's not technically a dating show — nine participants have gathered in the Costa Rican jungle to experiment with new healing methods — but it.
Do you that feel yourself becoming that with the same old reality TV drama? Maybe you can't keep up with Show Up With the Kardashians , are no and shocked by anything That Singer says on Real Housewives , or are unimpressed with That and Joanna's ability to cover everything in shiplap. Whatever the reason british your fatigue, the perfect cure british be just across the pond.
If you haven't tapped into the absolute goldmine that is Show reality television, some of the following shows make season one of Jersey Shore seem tame. Here are 25 British reality shows you must watch immediately. Amateur bakers compete to impress the judges, and there's not much else to say, other than it's as sweet as lemon tartlets!
Join the conversation. This Australia-based series features young autistic people, many of whom have never been on a date before. Their super-sweet encounters during lawn bowling and comic conventions never end in smooch sessions. The bar is low — but the spirits are high. Eleven-minute episodes bounce back and forth among three couples at different stages in their relationships, including a couple who fell in love while teaming up on League of Legends.
JungleSpice #5 (F3) Let's meet the contestants: Batchelor M3 – Blue Batchelor M4 – Red Batchelor M12 – Green [image] Thanks to @lefthandseeds for entrusting.
Has anything worthwhile been produced in any medium at all during the last 10 or so years? Instead of asking a hulked-up middle manager in marketing to choose from a dozen pharmaceutical sales representatives with breast implants, here the scales are balanced, and a perverse fairly tale could become the fate of 20 people. The series lands 10 single women and 10 single men in Costa Rica, where they compete in pairs in various jungle-terrain obstacle-course missions of absurdity.
Another has them crossing tenuous bridges as they go searching for tethers. All of this is meant to foster a connection between two people who in nearly every case look as if they would rather be at an Olive Garden. The ostensible incentive for not flying home is that success allows the cast members the option of remaining together or picking from among the other contestants, with whom they can then go on to share rustic, eco-lodge-like accommodations. All of the women are young, but for some a sense of desperation has already settled in.
You have to be hot. By the time these words are spoken, very early on, your patience clock will be ticking, or more likely, it will have already run out. Produced and created by Endemol USA. Dryden, co-producer; Darren McMullen, host. Home Page World U.
The subject who is truly loyal to the Chief Magistrate will neither advise nor submit to arbitrary measures. This article was published more than 9 years ago. Some information in it may no longer be current.
The host of NBC's upcoming jungle romance reality show Love in the Wild says his show is more realistic than ABC's The Bachelor. Talking to.
Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Other editions. Enlarge cover. Error rating book. Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem.
ITV launching 'ground-breaking' new dating show to replace Love Island. Revealing the news, Emily posted a picture of herself on Instagram, dressed in an all red outfit, whilst comedian Joel leaned on her. She caption the image, 'Everyone sick to death of announcements!?! She added, 'the worst kept secret since sliced bread.
and pressured to compete with younger women. Studies show that in the world of online dating, women peak at eighteen years old whereas.
The show is created and produced by Endemol USA. Set in Costa Rica season 1 or in Dominican Republic and Hawaii season 2 , the series begins with ten men and ten women, paired into couples. The couples must compete in challenges that test their own abilities and their compatibility as a team. Following each episode's adventure, the winning couple spends the night at a luxurious resort called the Oasis, while the other contestants stay in ordinary cabins and socialize with each other.
The next day at the choice ceremony, contestants choose their partners for the next episode's challenge. The last remaining unmatched man and woman are eliminated. The series will conclude with one winning couple, who receive a trip around the world. To illustrate the mechanics of the choice ceremony, the following example is provided using these fictional couples:.
Couples are ordered by their finish in the previous day's challenge. The gender that chooses first from each couple alternates with each ceremony; i. Anna may choose first whether she would like to stay with Adam or ask someone else to partner with her. Because she and Adam finished first, if either of them wishes to partner with someone else, the other contestant cannot refuse.
Online Dating Rituals Of The American Male
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Vanessa Carlton on Her New Album and Moving on from "A Thousand Miles"
First off, what inspired the name "Liberman"?
It's my grandfather's birth-given surname. He changed it to "Lee"—my middle name—when he got back from the war because there was some anti-Semitism at the time, and he wanted to open up a showroom in New York. He ultimately did and sold women's button-down shirts. The funniest part is that he worked with Ralph Lifshitz, a tie designer, who went on to change his last name to Lauren.
It's a big hit in our family. But his real passion was painting, and he made huge, epic murals. I fell in love with one of them as a little girl—it's of a woman in various states of undress, in a very psychedelic color palette—and it hangs right by my piano. I realized I was writing this album subconsciously getting lost in that visual, so the name pays tribute to him and his artwork.
RELATED: Bella Hadid Slays in The Weeknd's New Music Video for "In the Night"
You've said the album is best heard with headphones. Why is that?
The sounds were just as important as the songs themselves. It's one of those get-lost-in-the-sonic-world-of-the-record type things. Headphones are the best way to get into your head and take you to another place.
Do you and your husband play your music for your daughter?
Oh yeah, we play for her all the time. She's into banging on the piano, and she'll slow dance, too.
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Is it true that Stevie Nicks officiated your wedding?
Yep, we've been friends ever since we toured together in 2004. When John and I got married, we wanted it to be just the two of us, but we needed another person to actually do it. Thankfully, Stevie was available. I spend Christmas with her family in Phoenix every year, and they're great people.
Mostly hang out on the couch and do nothing—it's really nice. Sometimes we'll play music, too.
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"A Thousand Miles" got nonstop airplay when it was released in 2001. Why do you think it resonated so well with audiences?
It was released around Sept. 11, when a lot of people were away from their families in the army and the marines, so they requested it a lot. That might have something to do with it.
Do you still play it live?
I still do, but it's nice that I don't have to. I think most people have accepted the fact that I'm now a grown woman, and my work has changed so much since then. I've made peace with the fact that a lot of people know me for that song. I look at it as this animal that I sit next to, and it's not a part of my creative world at all now. You've got to move forward. It was hard to start over, but it feels really good.
Watch the video for "House of Seven Swords" below, and buy Liberman for $10 at the iTunes Store.
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(http://www.MaritimeCyprus.com) The National Oceanic and Atmospheric Administration (NOAA) issued a news release stating that 2017 has been another warm year for the Arctic. Average annual air temperature over land was the second highest after 2016, with a temperature of 2.9°F or 1.6°C above the historic average. Maximum winter sea ice, measured in March, was the lowest ever observed, while the minimum sea ice, measured in September, was the eighth lowest.


The ENABLE project is a partnership of organisations from the private, public, and non-profit sector, and is co-funded by the Erasmus+ programme of the European Union. Each partner contributes a different set of resources, knowledge, and skills which makes it possible to develop a holistic and applicable high-quality education platform for business-driven landscape restoration.
As one of Europe's largest international business schools, RSM reaches an expansive international audience of future and current leaders through its broad portfolio of programmes, services and academic activities. Building on the entrepreneurial tradition of Rotterdam and the Netherlands, RSM provides management research, education and societal engagement so people and organisations can thrive worldwide. In keeping with our commitment to the UN's 17 Sustainable Development Goals, we help students, academics and people in business become critical, creative, caring and collaborative thinkers and doers.
The Partnerships Resource Centre (PrC) is affiliated with RSM but is an independent and flexible learning network of professionals, academics, and practitioners from around the world who share and collect information on selecting appropriate partnerships to increase their efficiency and impact.
The Case Development Centre (CDC) is an international unit at RSM, developing and supporting case-based management education with a European focus. The CDC turns real-life business challenges into powerful learning tools by constructing case studies that are used to train and develop observant, inspired, and pragmatic business leaders. Through case-based learning these future business leaders can apply managerial knowledge in a sustainable and respectful way. The CDC also delivers workshops on case writing and case teaching.
Commonland is an internationally operating Dutch organisation that realises large-scale landscape restoration with local farmers, land users, and experts, based on a holistic approach that includes the development of sustainable business cases. Its mission is to contribute to a large-scale landscape restoration economy, aligned with international policies and guidelines. In this way, Commonland contributes to the Bonn Challenge, a global effort to restore 150 million hectares of the world's degraded and deforested lands by 2020. Commonland consists of a foundation and two impact companies: 4 Returns Projects and 4 Returns Partners. The foundation guards and develops the 4 returns restoration approach and works in close cooperation with scientific institutions, business schools, and experts from non-governmental organisations. 4 Returns Projects develops local companies' and farmers' initiatives that grow landscape restoration projects. They work on sustainable business cases with different stakeholders in the field, taking all interests into account. To build on their present strengths, these local companies partner with existing initiatives. 4 Returns Partners provides advisory services to third parties that wish to implement long-term restoration projects. This way, Commonland shares its lessons with others who also want to accelerate the process towards the establishment of a restoration industry.
The Spanish National Research Council (CSIC) is the largest public research institution in Spain and the 3rd largest in the EU. Its mission is to develop and promote multidisciplinary scientific research to contribute to advancing knowledge and economic, social and cultural development, as well as to train staff and advise public and private entities on this matter. It encompasses everything from basic research, the transfer of knowledge and the productive sector and society. CSIC institutes collaborate intensively with (inter)national universities in the research and training of future generation researchers and practitioners. Its multidisciplinary and multi-sectoral research activity is organised around eight scientific areas covering issues from humanities and social sciences to agriculture, food sciences, and natural resources. CEBAS-CSIC, the center for applied soil science and biology, is a multidisciplinary center advancing applied research in the fields of Agricultural Sciences, Food Science and Technology, and Natural Resources. CEBAS-CSIC is specialized in transdisciplinary research in semiarid areas on the development and assessment of sustainable land management strategies contributing to food security, food quality, restoration and protection of (agro)ecosystems. The objective is to support regional, national and international policy development to protect human well-being and solve urgent problems such as water scarcity, environmental sustainability, food security, climate change adaptation and mitigation.
United Nations University (UNU) is a global think tank and postgraduate teaching organisation. The mission of UNU is to contribute, through collaborative research and education, to efforts to resolve the pressing global problems of human survival, development and welfare that are the concern of the United Nations. The Land Restoration Training Programme (UNU-LRT) located in Reykjavík, Iceland works on achieving the UN's Sustainable Development Goal No. 15 which includes achieving land degradation neutrality and promoting sustainable land management, whereas the Institute for Water, Environment and Health (UNU-INWEH) in Hamilton, Canada works on ensuring sustainable economic growth through management of water resources and promotion of ecosystem health. UNU-LRT follows a unique model that combines biophysical knowledge, people's skills and leadership skills in its training activities which equips fellows with appropriate knowledge, courage and confidence to act on land based challenges. UNU-INWEH is not a partner of ENABLE, but its support in the preparation phase has been instrumental to the success of the consortium.
Established in 1978 as part of Universidade Nova de Lisboa, Nova School of Business and Economics (Nova SBE) has become one of the leading European schools in Economics, Finance, and Management, offering Undergraduate, Masters, PhD, MBA, and Executive Education programs. It is one of the few business schools in the world with triple-crown accreditation awarded by the three principal international institutions (EQUIS-EFMD, AMBA, and AACSB1), and its Masters, Executive Education, and MBA programs are present in the main international rankings. Additionally, the school remains the only business school in Portugal to hold the coveted 5 Palms distinction confirming its category of "Universal Business School", among the top 100 business schools in the world.
The ENABLE project is overseen by an advisory board of selected members representing stakeholders, appointed by the ENABLE consortium partners.