text:
\section{Introduction}
\label{sec-introduction}
The growing research focus in recent years on the analysis of degradation
data has played a major role in the field of reliability inference and management. In particular, reliability data and characteristics
for highly reliable products can be collected through analyzing the obtained degradation
data. Consequently, statistical modeling and inference approaches have been introduced on the basis of
different degradation measuring techniques.

\cite{limon2017literature}, for instance, provided a comprehensive reviews on several approaches in the literature that are utilized to evaluate reliable information from acclerated degradation data.
\cite{RePEc:eee:reensy:v:142:y:2015:i:c:p:369-377} introduced an
analytical optimal ADT design prcedure for more efficient reliability demonstration by minimizing the
asymptotic variance of decision variable in reliability demonstration under the constraints of sample size, test duration, test cost, and predetermined decision risks.  Considering linear mixed effects model (LMEM), \citep{doi:10.1002/asmb.2061} considered the minimum asymptotic variance criterion to characterize optimal design as well as compromise design plans for accelerated degradation tests. Further, \cite{7114339} developed an algorithm-based optimal
ADT approach by minimizing the asymptotic variance
of the MLE of the mean failure time of a system, where
the sample size and termination time of each run of the ADT at a
constant measurement frequency were specified.
\citep{shat2021experimental} charachterized optimal designs for repeated measures accelerated degradation tests with multiple stress variables, where the degradation paths are assumed to follow a linear mixed effects. \citep{7160792} introduced a modeling precedure to
simultaneously analyze linear degradation data and traumatic
failures with competing risks in step stress ADT experiment.

The general theory of optimal design of experiments is efficiently charachterized in terms of the mathematical context of approximate designs (see e.\,g.\ \cite {silvey1980optimal}). For example, \cite{schwabe1996optimum} suggested optimal designs for multi-factor models which can be used to treat more than one stress variable and the choice of time plans simultaneously under various interaction structures.

\cite{dette2010optimal} considered the problem of constructing $D$-optimal designs for linear and  nonlinear random effect models with applications in population
pharmacokinetics.
\cite{grasshoff2012optimal} presented $D$-optimal designs for random coefficient regression models in the basis of geometrical arguments when only one observation is available per unit, i.e. a situation which occurs in destructive testing.

It can be noted that the vast majority of the existing literature focuses on desinging the degradation stress variable(s). As opposed to this approach the time variable will be considered in the present work in order to obtain optimal experimental designs under various measurement plans.

The present paper is organized as follows.
In Sections~\ref{sec-model-formulation} we state the general model formulation and in Section~\ref{sec-information} we exhibit the corresponding information matrix. Section~\ref{sec-failuretime} is devoted to derive the design optimality criterion for estimating a quantile of the failure time distribution under normal use conditions.
In Section~\ref{sec-opt-time-plan} also the measurement times are optimized under the constraint that measurements are taken according to the same time plan for all units, and in
Section~\ref{sec-single-measurements} optimal measurement times are determined in the setting of destructive testing.
The paper closes with a short discussion in Section~\ref{sec-discussion}.

\section{Formulation of the model}
\label{sec-model-formulation}

In this section, we introduce a mixed effects regression model incorporating a product-type structure as in \citep{shat2021experimental}.
In accordance with \citep{shat2021experimental} we assume that there are $n$ testing units $i=1,...,n$, for which degradation $y_{i j}$ will be measured according to a time plan $\mathbf{t}=(t_1,...,t_k)^T$ with $k$ time points $t_j$, $j = 1, ..., k$, $0 \leq t_1 < ... < t_k \leq t_{\max}$.
Each unit $i$ is measutred under a setting $\mathbf{x}_i$ of the stress variable(s), which is kept fixed for each unit throughout the testing, but varies between units.
The number $k$ of measurements and the measurent times $t_1,...,t_k$ are the same for all units.
The measurements $y_{i j}$ are considered as realizations of random variables $Y_{i j}$ which can be charachterized by a hierarchical model.
For each unit $i$ the observation $Y_{i j}$ at $t_j$ is expressed as
\begin{equation}
\label{modelindividualresponse}
Y_{i j} = \mu_{i}(\mathbf{x}_i, t_j) + \varepsilon_{i j} ,
\end{equation}
where $\mu_{i}(\mathbf{x}, t)$ is the mean degradation of unit $i$ at time $t$, when stress $\mathbf{x}$ is applied, and $\varepsilon_{i j}$ is a measurement error.
The mean degradation $\mu_{i}(\mathbf{x}, t)$ is given by a linear model equation in the stress variable $\mathbf{x}$ and the time $t$,
\begin{equation}
	\label{mean-degradation-unit}
	\mu_{i}(\mathbf{x}, t) = \sum_{q = 1}^{p} \beta_{i q} f_{q}(\mathbf{x}, t_j)
	= \mathbf{f}(\mathbf{x}, t_j)^{T} \boldsymbol{\beta}_i
\end{equation}
where $\mathbf{f}(\mathbf{x}, t) = (f_{1}(\mathbf{x}, t), ..., f_{p}(\mathbf{x}, t))^T$ is a $p$-dimensional vector of known regression functions $f_{q}(\mathbf{x}, t)$ in both the stress variable(s) $\mathbf{x}$ and the time $t$, and $\boldsymbol{\beta}_i = (\beta_{i, 1}, ..., \beta_{i, p})^T$ is a $p$-dimensional vector of unit specific parameters $\beta_{i, q}$.
Thus, the response is given by
\begin{equation}
	\label{modelindividuallevel}
	Y_{i j} = \mathbf{f}(\mathbf{x}_{i}, t)^{T}\boldsymbol{\beta}_i + \varepsilon_{i j}.
\end{equation}
Further, the measurement error $\varepsilon_{i j}$ is assumed to be normally distributed with zero mean and some potentially time dependent error variance $\sigma_{\epsilon j}^{2}$.
The vector $\boldsymbol{\varepsilon}_{i} = (\varepsilon_{i 1}, ..., \varepsilon_{i k})^T$ of errors within one unit $i$ is $k$-dimensional multivariate normal with zero mean and positive definite variance covariance matrix $\boldsymbol{\Sigma}_\varepsilon$.
For the regression functions $\mathbf{f}(\mathbf{x},t)$ we suppose a product-type structure with complete interactions between the stress variable $\mathbf{x}$ and the time $t$, i.\,e.\ there are marginal regression functions $\mathbf{f}_1(\mathbf{x})=(f_{1 1}(\mathbf{x}), ..., f_{1 p_1}(\mathbf{x}))^T$ and $\mathbf{f}_2(t)=(f_{2 1}(t), ..., f_{2 p_2}(t))^T$ of dimension $p_1$ and $p_2$ which only depend on the stress variable $\mathbf{x}$ and the time $t$, respectively, such that the vector $\mathbf{f}(\mathbf{x}, t) = \mathbf{f}_1(\mathbf{x}) \otimes \mathbf{f}_2(t)$ of regression functions factorizes into its marginal counterparts ($p = p_1 p_2$).
Here ``$\otimes$'' denotes the Kronecker product.
Then the observation $Y_{i j}$ can be written as
\begin{equation}
	\label{modelindividuallevel-product}
	Y_{i j} = \sum_{r = 1}^{p_1} \sum_{s = 1}^{p_2} \beta_{i rs} f_{1 r}(\mathbf{x}_{i}) f_{2 s}(t_j) + \varepsilon_{i j}
	= (\mathbf{f}_1(\mathbf{x}_i) \otimes \mathbf{f}_2(t_j))^T \boldsymbol{\beta}_i + \varepsilon_{i j},
\end{equation}
where the entries of the vector $\boldsymbol{\beta}_i = (\beta_{i 1 1}, ..., \beta_{i 1 p_2}, ..., \beta_{i p_1 p_2})$ of parameters are relabeled lexicographically according to the marginal regression functions.

Moreover,
we will assume throughout that the marginal regression function $\mathbf{f}_1(\mathbf{x}) = (f_{11}(\mathbf{x}), ..., f_{1 p_1}(\mathbf{x}))^T$ of the stress variable $\mathbf{x}$ contains a constant term, $f_{1 1}(\mathbf{x}) \equiv 1$ say,
and that only the leading $p_2$ parameters $\beta_{i 1 1}, ..., \beta_{i 1 p_2}$ of $\boldsymbol{\beta}_i$ associated with this constant term are unit specific.
The remaining parameters in $\boldsymbol{\beta}_i$ are assumed to have the same value $\beta_{irs}=\beta_{rs}$, $r = 2, ..., p_1$, $s = 1, ..., p_2$, for all units $i = 1, ..., n$.
Hence, for unit $i$ the model \eqref{modelindividuallevel-product} can be rewritten as
\begin{equation}
\label{eq-general-model}
 Y_{i j} = \left(\mathbf{f}_1(\mathbf{x}_{i}) \otimes \mathbf{f}_2(t_j)\right)^{T}\boldsymbol{\beta} + \mathbf{f}_2(t_j)^T\boldsymbol{\gamma}_i + \varepsilon_{i j},
\end{equation}
where $\boldsymbol{\beta} = (\beta_{1 1}, ..., \beta_{p_1 p_2})^T$ is the vector of (aggregate) fixed effect parameters (averaged over the units)
 and $\boldsymbol{\gamma}_i = (\gamma_{i 1}, ..., \gamma_{i p_2})^T$ is the $p_2$-dimensional vector of unit specific deviations $\gamma_{i s} = \beta_{i, 1 s} - \beta_{1 s}$, $s = 1, ..., p_2$, from the corresponding aggregate parameters where $\boldsymbol\gamma_i$ are independent and identically distributed multivariate normal with mean 0 and covariance matrix $\boldsymbol\Sigma_\gamma$.

For the analysis of degradation under normal use, we further assume that the general model~\ref{eq-general-model} is also valid at the normal use condition $\mathbf{x}_u$, i.\,e.\
\begin{equation}
	\label{eq-degr-path-use-cond}
	\mu_u(\mathbf{x}_{u},t) = (\mathbf{f}_1(\mathbf{x}_{u}) \otimes \mathbf{f}_2(t))^{T} \boldsymbol{\beta} + \mathbf{f}_2(t)^T \boldsymbol{\gamma}_u
\end{equation}
describes the mean degradation of a future unit $u$ at normal use condition $\mathbf{x}_u$ and time $t$, and the random effects $\boldsymbol{\gamma}_u$ are multivariate normal with mean zero and variance covariance matrix $\boldsymbol\Sigma_\gamma$.

\section{Information and design}
\label{sec-information}
Denote by $\boldsymbol\theta = (\boldsymbol{\beta}^T, \boldsymbol{\varsigma}^T)^T$ the vector of all model parameters, where $\boldsymbol\varsigma$ collects all variance covariance parameters from $\boldsymbol\Sigma_\gamma$ and $\boldsymbol\Sigma_\varepsilon$. Let $ \mathbf{F}_1 = (\mathbf{f}_1(\mathbf{x}_1), ..., \mathbf{f}_1(\mathbf{x}_n))^{T}$ be the $n \times p_1$ marginal design matrix for the stress variables across units, and denote by $\mathbf{F}_2 = ( \mathbf{f}_2(t_1), ..., \mathbf{f}_2(t_k))^T$ the $k \times p_2$ marginal design matrix for the time variable. The corresponding per unit variance covariance matrix is $\mathbf{V}=\mathbf{F}_2{\boldsymbol{\Sigma}}_\gamma\mathbf{F}_2^T + {\boldsymbol{\Sigma}}_{\varepsilon}$

where $\mathbf{V}$ depends only on $\boldsymbol\varsigma$, see \citep{shat2021experimental} for further details about estimation in the model parameters.
The Fisher information matrix is block diagonal,
\begin{equation}
	\label{info-block}
	\mathbf{M}_{\boldsymbol{\theta}}
	= \left(
		\begin{array}{cc}
			\mathbf{M}_{\boldsymbol\beta} & \mathbf{0}
			\\
			\mathbf{0} & \mathbf{M}_{\boldsymbol\varsigma}
		\end{array}
	\right)
\end{equation}
where $\mathbf{M}_{\boldsymbol\beta}$ and $\mathbf{M}_{\boldsymbol\varsigma}$ are the information matrices for $\boldsymbol\beta$ and $\boldsymbol\varsigma$, respectively, for short.
The particular form of $\mathbf{M}_{\boldsymbol\varsigma}$ will be not of interest.
 By (\ref{eq-degr-path-use-cond}), the information matrix $\mathbf{M}_{\boldsymbol\beta}$ for the parameters $\boldsymbol\beta$ factorizes according to
\begin{equation}
	\label{eq-info-product}
	\mathbf{M}_{\boldsymbol\beta} = \mathbf{M}_1 \otimes \mathbf{M}_2
\end{equation}
into the information matrix $\mathbf{M}_1 = \mathbf{F}_1^T \mathbf{F}_1$ in the marginal model
in the stress variable and the information matrix $\mathbf{M}_2 = \mathbf{F}_2^T\mathbf{V}^{-1}\mathbf{F}_2$ in the marginal mixed effects model in the time variable $t$ where the time plan $\boldsymbol t =(t_1,...,t_k)^T$ plays the role of a (exact) design here.
\section{Optimality criterion based on the failure time under normal use condition}
\label{sec-failuretime}
In accordance with \cite{doi:10.1002/asmb.2061} we are focusing on some characteristics of the failure time distribution of soft failure due to degradation.
As already mentioned, we assume that the model equation (\ref{eq-degr-path-use-cond}) $\mu_{u}(t) = \mu(\mathbf{x}_{u}, t) = (\mathbf{f}_1(\mathbf{x}_{u}) \otimes \mathbf{f}_2(t))^{T} \boldsymbol{\beta} + \mathbf{f}_2(t)^T \boldsymbol{\gamma}_u$ for the mean degradation paths is also valid under normal use condition $\mathbf{x}_u$.
A soft failure due to degradation is expressed as the exceedance of the mean degradation $\mu_u$ over a  failure threshold $y_0$. The failure time $T$ under normal use condition is defined as the first time $t$ the mean degradation path $\mu_u(t)$ reaches or exceeds the threshold $y_0$, i.\,e.\ $T = \min \{t \geq 0;\, \mu_u(t) \geq y_0\}$.
As random effects $\boldsymbol\gamma_u$ are involved in the mean degradation path, the failure time $T$ is random. As in \citep{shat2021experimental} we denote by $\mu(t) = \mathrm{E}(\mu_u(t)) = (\mathbf{f}_1(\mathbf{x}_{u}) \otimes \mathbf{f}_2(t))^{T}\boldsymbol{\beta}$ the aggregate degradation path under normal use condition and by $\boldsymbol\delta  = (\delta_1, ..., \delta_{p_2})^T$ the vector of its coefficients $\delta_{s} = \sum_{r = 1}^{p_1} f_{1 r}(\mathbf{x}_u)\beta_{r s}$, in the regression functions $f_{2 s}$ in $t$, i.\,e.\ $\mu(t) = \mathbf{f}_2(t)^T\boldsymbol\delta = \sum_{s = 1}^{p_2}\delta_s f_{2 s}(t)$.\\

For the purpose of describing certain characteristics of the distribution of the failure time, we will determine the distribution function $F_T(t) = \mathrm{P}(T \leq t)$.
First note that $T \leq t$ if and only if $\mu_u(t) \geq y_0$.
Hence
\begin{eqnarray}
   F_{T}(t) &  = & \mathrm{P}(\mu_u(t) \geq y_0)
   \nonumber
	\\
	& = & \mathrm{P} (- \mathbf{f}_2(t)^T \boldsymbol{\gamma}_u \leq \mu(t) - y_0)
   \nonumber
   \\
	& = & \Phi(h(t)) ,
    \label{eq-failure-time-distribution}
\end{eqnarray}
where
\begin{equation}
	\label{eq-h-tau}
	h(t) = \frac{\mu(t) - y_0}{\sigma_u(t)} ,
\end{equation}
$\sigma_u^2(t) = \mathbf{f}_2(t)^T \boldsymbol\Sigma_\gamma \mathbf{f}_2(t)$ is the variance of the mean degradation path $\mu_u(t)$ at time $t$, and $\Phi$ denotes the standard normal distribution function.

We will be interested in quantiles $t_\alpha$ of the failure time distribution, $\mathrm{P}(T \leq t_\alpha) = \alpha$, up to which under normal use conditions $\alpha \cdot 100$ percent of the units fail and $(1 - \alpha) \cdot 100$ percent of the units persist.
Of particular importance is the median $t_{0.5}$.
By (\ref{eq-failure-time-distribution}) these quantiles can be determined as the solutions of the equation $h(t_\alpha) = z_\alpha$, where $z_\alpha = \Phi^{-1}(\alpha)$ is the $\alpha$-quantile of the standard normal distribution.
For the median ($\alpha = 0.5$) we have $z_{0.5} = 0$ and, hence the median failure time $t_{0.5}$ is the solution of $\mu(t) = y_0$.
In the particular case of straight lines for the mean degradation paths, i.\,e.\ $\mathbf{f}_2(t) = (1,t)^T$ and $\boldsymbol\Sigma_\gamma =\left(\begin{array}{cc}
	\sigma_1^2 & \rho \sigma_1 \sigma_2
	\\
	\rho \sigma_1 \sigma_2 & \sigma_2^2
\end{array}\right)$
where $\sigma_1^2, \sigma_2^2$ and $\rho$ are defined,
the function $h(t)$ specifies to
\begin{equation}
	\label{eq-h-tau-linear}
	h(t) = \frac{\delta_2 t + \delta_1 - y_0}{\sqrt{\sigma_1^2 + 2\rho \sigma_1 \sigma_2 t + \sigma_2^2 t^2}} ,
\end{equation}
where $\delta_{1} = \sum_{r = 1}^{p_1} f_{1 r}(\mathbf{x}_u) \beta_{r 1}$ and $\delta_2 = \sum_{r = 1}^{p_1} f_{1 r}(\mathbf{x}_u) \beta_{r 2}$ are the intercept and the slope of the aggregate degradation path $\mu(t) = \delta_1 + \delta_2 t$ under normal use condition, respectively.
The median failure time is then given by $t_{0.5} = (y_0 - \delta_1) / \delta_2$ which gives a solution $t_{0.5} > 0$ under the natural assumptions that the aggregate degradation path is increasing, $\delta_2 > 0$, and that the aggregate degradation at time $t=0$ is less than the failure threshold, $\delta_1 < y_0$.

In general, under the assumption that the correlation of the random effects is non-negative for the intercept and the slope of the mean degradation path,  $\rho \geq 0$, the function $h(t)$ can be seen to be strictly increasing, $h^\prime(t) > 0$, in $t>0$.
Note that $\Phi(h(0)) = \Phi(-(y_0-\delta_1)/\sigma_1)$ is the probability that under normal use condition the mean degradation path exceeds the threshold $y_0$ already at the initial time $t=0$ where $\lim_{t\to\infty}h(t) = \delta_2/\sigma_2$
and $t_\alpha$ is nondegenerate for
$\Phi(-(y_0-\delta_1)/\sigma_1) < \alpha < \Phi(\delta_2/\sigma_2)$.

In the special case of only a random intercept in the random effects, i.\,e.\ $\sigma_2^2 = 0$, all $\alpha$-quantiles $t_\alpha$ finitely exist for $\alpha \geq \Phi( - (y_0-\delta_1) / \sigma_1)$ and can be determined as $t_\alpha = (y_0 - \delta_1 + z_\alpha\sigma_1) / \delta_2$.

The quantile $t_\alpha = t_\alpha(\boldsymbol\theta)$ is a function of both the  fixed effect parameters $\boldsymbol\beta$ and the variance parameters $\boldsymbol\varsigma$, in general.

By the delta-method the maximum likelihood estimator $\widehat{t}_\alpha$ of ${t}_\alpha$ is asymptotically normal with  asymptotic variance
\begin{equation}
	\label{eq-avar-tau-alpha}
	\mathrm{aVar}(\widehat{t}_{\alpha}) = \mathbf{c}^T \mathbf{M}_{\boldsymbol{\theta}}^{-1} \mathbf{c} ,
\end{equation}
where $\mathbf{c} = \frac{\partial}{\partial\boldsymbol\theta}t_{\alpha}$ is the gradient vector of $t_{\alpha}$ with respect to the parameter vector ${\boldsymbol\theta}$.
The asymptotic variance depends on the design of the experiment through the information matrix $ \mathbf{M}_{\boldsymbol{\theta}}$ and will be chosen as the optimality criterion for the design.
The gradient $\mathbf{c}$ is given by
\begin{equation}
	\label{eq-gradient-tau-alpha}
	\mathbf{c} =  c_0 \left({\textstyle{\frac{\partial}{\partial\boldsymbol\theta}}}\mu(t)\vert_{t=t_\alpha} - z_\alpha {\textstyle{\frac{\partial}{\partial\boldsymbol\theta}}}\sigma_u(t)\vert_{t=t_\alpha}\right) ,
\end{equation}
in view of (\ref{eq-h-tau}) by the implicit function theorem (see e.\,g.\ \cite{krantz2012implicit}), where   $c_0 \neq 0$ is a constant not dprending on the design.
As the aggregate mean degradation $\mu(t)$ only depends on the aggregate location parameters $\boldsymbol\beta$ and the variance $\sigma_u^2(t)$ only depends on the variance parameters $\boldsymbol\varsigma$ the gradient simplifies to $\mathbf{c} = - c_0 (\mathbf{c}_{\boldsymbol\beta}^T, \mathbf{c}_{\boldsymbol\varsigma}^T)^T$, where
\begin{equation*}
	\label{eq-gradient-tau-alpha-beta}
	\mathbf{c}_{\boldsymbol\beta} = {\textstyle{\frac{\partial}{\partial\boldsymbol\beta}}}\mu(t)\vert_{t = t_\alpha} = \mathbf{f}(\mathbf{x}_u, t_\alpha)
\end{equation*}
is the gradient of $\mu(t)$ with respect to $\boldsymbol\beta$ and
\begin{equation*}
	\label{eq-gradient-tau-alpha-sigma}
	\mathbf{c}_{\boldsymbol\varsigma} = - z_\alpha {\textstyle{\frac{\partial}{\partial\boldsymbol\varsigma}}}\sigma_u(t)\vert_{t = t_\alpha}
\end{equation*}
is, up to a multiplicative constant, the gradient of $\sigma_u(t)$ with respect to $\boldsymbol\varsigma$.
The particular shape of $\mathbf{c}_{\boldsymbol\varsigma}$ does not play a role here, in general.
But it is important that $\mathbf{c}_{\boldsymbol\varsigma} = \mathbf{0}$ in the case of the median ($\alpha=0.5$).
By the block diagonal form (\ref{info-block}) of the information matrix the asymptotic variance (\ref{eq-avar-tau-alpha}) of $\widehat{t}_\alpha$ becomes
\begin{equation}
	\label{eq-avar-tau-alpha-sum}
	\mathrm{aVar}(\widehat{t}_{\alpha}) = c_0^2 \left(\mathbf{f}(\mathbf{x}_u, t_\alpha)^T \mathbf{M}_{\boldsymbol{\beta}}^{-1}\mathbf{f}(\mathbf{x}_u, t_\alpha) +z_\alpha^2 \frac{\partial}{\partial\boldsymbol\varsigma}\sigma_u(t_\alpha)^T \mathbf M_{\boldsymbol \varsigma}^{-1} \frac{\partial}{\partial\boldsymbol\varsigma}\sigma_u(t_\alpha)\right)
\end{equation}
which simplifies to $\mathrm{aVar}(\widehat{t}_{0.5}) = c_0^2 \mathbf{f}(\mathbf{x}_u, t_\alpha)^T \mathbf{M}_{\boldsymbol{\beta}}^{-1} \mathbf{f}(\mathbf{x}_u, t_\alpha)$
in the case of the median. For the case of a product-type model the expression related to the fixed effect parameters $\boldsymbol\beta$ decomposes into
\begin{equation}
	\label{eq-avar-tau-alpha-decompose}
	\mathbf{c}_{\boldsymbol\beta}^T \mathbf{M}_{\boldsymbol{\beta}}^{-1} \mathbf{c}_{\boldsymbol\beta} = \mathbf{f}_1(\mathbf{x}_u)^T \mathbf{M}_{1}^{-1} \mathbf{f}_1(\mathbf{x}_u) \cdot \mathbf{f}_2(t_\alpha)^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_\alpha).
\end{equation}

\section{Optimization of the time plan}
\label{sec-opt-time-plan}

In  \citep{shat2021experimental} we treated the situation when there is a fixed time plan $\mathbf{t} = (t_1, ..., t_k)^T$, and only the stress variables $\mathbf{x} \in \mathcal X$ are to be optimized across units. There we noticed that the optimal design for the stress variable does not depend on the time plan used.
We consider now the situation when also the settings $t_1, ..., t_k$ for the standardized time variable $t \in [0, 1]$ may be optimized within units.
As in Section~\ref{sec-model-formulation} we still assume that the same time plan  $\mathbf{t}=(t_1, ..., t_k)^T$ is used for all units. By equations (\ref{eq-avar-tau-alpha-sum}) and (\ref{eq-avar-tau-alpha-decompose}), the asymptotic variance is given by
\begin{equation}
	\label{eq-avar-tau-alpha-sum-decompose}
	\mathrm{aVar}(\widehat{t}_{\alpha}) = c_0^2 \left(\mathbf{f}_1(\mathbf{x}_u)^T \mathbf{M}_{1}^{-1} \mathbf{f}_1(\mathbf{x}_u) \cdot \mathbf{f}_2(t_\alpha)^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_\alpha) + \mathbf{c}_{\boldsymbol\varsigma}^T {\mathbf{M}}_{\boldsymbol{\varsigma}}^{-1} \mathbf{c}_{\boldsymbol\varsigma}\right)
\end{equation}

Here the dependence of the asymptotic variance on the settings at the time plan $\mathbf{t}$ for the time variable comes in through the information matrix $\mathbf{M}_{2}$ in the second marginal model as well as through the information matrix ${\mathbf{M}}_{\boldsymbol{\varsigma}}$ for the variance parameters.
In general, the asymptotic variance is a compound criterion in $\mathbf{f}$ which aims at minimizing a linear combination of a variance term $\mathbf f_2(t_\alpha)^T \mathbf M_2 ^{-1} \mathbf f_2(t_\alpha)$ in the location parameters $\boldsymbol\beta$ and a variance term $\mathbf c_{\boldsymbol\varsigma}^T \mathbf M_{\boldsymbol\varsigma}^{-1} \mathbf c_{\boldsymbol\varsigma}$ in the variance parameters. In this compound criterion the first coefficient involves the marginal information matrix $\mathbf M_1$ for the stress variable. As a consequence the criterion and, hence, the optimal time plan depend on the design for the stress variable. Thus first the marginal desiogn for the stress variable has to be optimized which can be done simultaneously in all time plans. Then the corresponding value has to be inserted into the compound criterion for optimizing the time plan. To facilitate further calculations we restrict to the case of estimating the median failure time $t_{0.5}$.
Consequently, the asymptotic variance simplifies to
\begin{equation}
	\label{eq-avar-tau-median-decompose}
	\mathrm{aVar}(\widehat{t}_{0.5}) = c_0^2 \mathbf{f}_1(\mathbf{x}_u)^T \mathbf{M}_{1}^{-1} \mathbf{f}_1(\mathbf{x}_u) \cdot \mathbf{f}_2(t_{0.5})^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_{0.5}) \, .
\end{equation}
Hence, for the median failure time also the optimization of the measurement times $t_1,..,t_k$ can be performed independently of the marginal model of the stress variables and their settings.
Only the marginal $c$-criterion $\mathbf{f}_2(t_{0.5})^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_{0.5})$ has to be minimized. Remember that the marginal information matrix $\mathbf{M}_{2}$ is given by $\mathbf{M}_{2} = \mathbf{F}_{2}^T \mathbf{V}^{-1} \mathbf{F}_{2}$ where the variance covariance matrix $\mathbf{V}$  of the vector $\mathbf{Y}_i$ of measurements for each unit $i$ is given by $\mathbf{V} = \mathbf{F}_2 {\boldsymbol{\Sigma}}_\gamma \mathbf{F}_2^T + {\boldsymbol{\Sigma}}_{\varepsilon}$.
In the situation of uncorrelated and homoscedastic measurment erros ($\boldsymbol{\Sigma}_\varepsilon = \sigma^2_\varepsilon\mathbf{I}_k$), \citep{Schmelter2007} provided a representation of the invere information matrix in random effects models.
This can be readily extended through the following result to a general, non-singular variance covariance structure $\boldsymbol{\Sigma}_\varepsilon$ for the measurement errors  $\boldsymbol\varepsilon$.
\begin{lemma}
	\label{lem-repr-minv}
	Let $\mathbf{F}$ be a $k \times p$ matrix of rank $p$, $\boldsymbol\Sigma_\gamma$  a non-negative definite $p \times p$ matrix, $\boldsymbol\Sigma_\varepsilon$ a positive definite $k \times k$ matrix, $\mathbf{V} = \mathbf{F} \boldsymbol\Sigma_\gamma \mathbf{F}^T + \boldsymbol\Sigma_\varepsilon$, and $\mathbf{M} = \mathbf{F}^T \mathbf{V}^{-1} \mathbf{F} $.
	Then
	\begin{equation}
		\mathbf{M}^{-1} = (\mathbf{F}^T \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F})^{-1} + \boldsymbol\Sigma_\gamma .
	\end{equation}
\end{lemma}
where the proof of Lemma~\ref{lem-repr-minv} is obtained in the Appendix.
From this we obtain for the marginal information matrix $\mathbf{M}_{2}$ that its inverse can be decomposed to $\mathbf{M}_{2}^{-1} = \mathbf{F}_2^T \boldsymbol{\Sigma}_\varepsilon^{-1} \mathbf{F}_2+ \boldsymbol{\Sigma}_\gamma$. As a consequence, the marginal $c$-criterion $\mathbf{f}_2(t_{0.5})^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_{0.5})$ can be split up into
\begin{equation}
	\label{eq-var-t-extra-decompose}
	\mathbf{f}_2(t_{0.5})^T \mathbf{M}_{2}^{-1} \mathbf{f}_2(t_{0.5}) = \mathbf{f}_2(t_{0.5})^T {\mathbf{M}_{2}^{(0)}}^{-1} \mathbf{f}_2(t_{0.5}) + \mathbf{f}_2(t_{0.5})^T \boldsymbol{\Sigma}_\gamma \mathbf{f}_2(t_{0.5}) ,
\end{equation}
where $\mathbf{M}_{2}^{(0)} = \mathbf{F}_2^T \boldsymbol{\Sigma}_\varepsilon^{-1} \mathbf{F}_2$ is the information matrix in the marginal fixed effect model in the time variable $t$.
Hence, as for any linear criterion, the optimization of the asymptotic variance~(\ref{eq-avar-tau-median-decompose}) of the median failure time ${t}_{0.5}$ with respect to the time plan $\mathbf t$ does not depend on the variance covariance matrix $\boldsymbol{\Sigma}_\gamma$ of the random effects.
For optimization only the term $\mathbf{f}_2(t_{0.5})^T {\mathbf{M}_{2}^{(0)}}^{-1} \mathbf{f}_2(t_{0.5})$ has to be minimized which is the $c$-criterion for extrapolation of the mean response at $t_{0.5}$ in the marginal fixed effect model in $t$.
This leads to the following result for optimization with respect to the time variable.

\begin{proposition}
	\label{prop-extrapolation-t}
	If the time plan $\mathbf t$ is $c$-optimal for extrapolation of the mean response at  $t_{0.5}$ in the marginal fixed effect model with covariance $\boldsymbol\Sigma_\varepsilon$ for the time variable $t$, then $\mathbf t$ minimize the asymptotic variance for the estimator $\widehat{t}_{0.5}$ of the median failure time $t_{0.5}$ under normal use condition.
\end{proposition}

Note that in degradation experiments the median failure time $t_{0.5}$ under normal use condition is typically much larger than the time horizon of the experiment, i.\,e.\ $t_{0.5} > 1$ on the standardized time scale.
However, the above proposition also holds for interpolation, $t_{0.5} \in [0,1]$. The optimal time plan depends on the location parameters $\boldsymbol\beta$ through $t_{0.5}$, but also on the variance covariance structure $\boldsymbol\Sigma_{\varepsilon}$ of the measurement errors within units, and is, hence, local.
If the number $k$ of measurement times $t_1, ..., t_k$ is large, one might be tempted to use the concept of approximate designs also here.

In that case, the approximate design $\tau$, as introduced by\citep{kiefer1959optimum}, will be defined by mutually distinct time points $t_1, ..., t_\ell$ from the standardized experimental time interval $\mathcal{T} = [0,1]$ with corresponding proportions $\pi_1, ..., \pi_\ell > 0$ satisfying $\sum_{j=1}^\ell \pi_j = 1$ with the interpretation that (approximately) $\pi_j k$ measurements are performed at time point $t_j$, $j = 1, ..., \ell$, for each unit.
In the situation of a linear time trend, $\mathbf{f}_2(t) = (1,t)^T$ this leads to essentially the same extrapolation problem as for the stress variable in \citep{shat2021experimental}.  Under the additional assumptions of uncorrelated and homoscedastic measurement errors
and $t_{0.5}>=1$, the $c$-optimal design for extrapolation at $t_{0.5} > 1$ is concentrated on the $\ell = 2$ endpoints $t_1 = 0$ and $t_2 = 1$ of the standardized experimental time interval $\mathcal{T}=[0,1]$ with corresponding proportions $\pi_1 = \frac{t_{0.5} - 1}{2 t_{0.5} - 1}$ and  $\pi_2 = \frac{t_{0.5}}{2 t_{0.5} - 1}$ (see \citep{schwabe1996optimum}, Example~2.1).
Similarly, as for extrapolation of the stress variable in \citep{shat2021experimental}, the proportion $\pi_2$ at the endpoint $t_2 = 1$ in the direction of the extrapolation time $t_{0.5}$ decreases from $1$, when $t_{0.5}$ is close to $1$, to $0.5$, when the distance gets large ($t_{0.5} \to \infty$).

However, from a practical point of view, it often does not seem meaningful to have replications, i.\,e.\ to have more than one measurement at a time at the same unit.
Moreover, even if this would be possible, these measurements would be expected to be strongly correlated with a correlation beyond that caused by the random effects.
For instance, the different measurement times $t_1, ..., t_k$ may be assumed to be at least some sufficiently large $\Delta t > 0$ apart, or more specifically, they are restricted to a grid on the standardized experimental time interval with grid size $\Delta t$.
Nevertheless approximate design theory can be used to determine optimal time plans.
To do so the standardized experimental time interval is discretized to a sufficiently coarse grid, $\mathcal{T} = \{j \Delta t;\, j = 0,1, ..., J\}$, where $\Delta t = 1/J$, i.\,e.\ $\mathcal{T} = \{0, \Delta t, 2 \Delta t, ..., 1\}$.
Additionally constraints are imposed on the proportions $\pi_j$ that none of these proportions is larger than $1/k$, i.\,e.\ the number of measurements at a time is bounded by one and, hence, there are at least $k$ different time points.
Optimal approximate time plans can then be obtained under these constraints by using standard algorithms for design optimization.
Actually, the so obtained proportions may be smaller than one for some of the time points.
But a theoretical result based on an equivalence theorem under constraints (convex optimization with Kuhn-Tucker conditions) guarantees that this only occurs for a small number of time points.  For the $D$-criterion
\citep{sahm2001note} showed that the experimental region splits up in subregions where the weight of the optimal design attains the upper bound $1/k$ and where it is $0$.
Only at the boundary of these subregions weights strictly between $0$ and $1/k$ may occur.
In particular, for straight line regression on $[0,1]$, the subregions with maximal weight $1/k$ are adjacent to the endpoints $0$ and $1$ of the interval while in the interior the optimal design has zero weight.
This result carries over directly to other criteria like the $c$-criterion under consideration.
From this approach efficient exact designs can be obtained by adjusting the weights to the admissible values $0$ and $1 / k$ under the constraint of total weight $1$. For brevity we consider in the following Example the model of Example~{2} in \citep{shat2021experimental} for numerical results.
\begin{table}
	\begin{center}
		\caption{Nominal values for Example~\ref{ex-2}}
		\label{tab-ex-intro-nominal-values}
		\vspace{2mm}
		\begin{tabular}{c|c|c|c|c||c|c|c|c||c||c}
			& $\beta_{1}$ & $\beta_2$ & $\beta_3$ & $\beta_4$ & $\sigma_{1}$ & $\sigma_{2}$ & $\rho$ & $\sigma_\varepsilon$ & ${x}_{u}$ & $y_0$
			\\
			$f_j(x,t)$ & $1$ & $x$ & $t$ & $x t$ & & & & & &
			\\
			\hline
			& $2.397$ & $1.629$ & $1.018$ & $0.0696$ & $0.114$ & $0.105$ & $-0.143$ & $0.048$ & $-0.056$ & $3.912$
		\end{tabular}
	\end{center}
\end{table}

\begin{example}
\label{ex-2} In this model we have straight line regression in both a single stress variable $x$ and the time variable $t$ with an interaction term  $x t$. The response $y_{i j}$ of testing unit $i$ at time $t_j$ is given by
\begin{equation}
\label{example2_111}
 y_{i j} = \beta_{i1} + \beta_2 x_{i } + \beta_{i 3} t_j + \beta_4 x_it_j + \varepsilon_{i j}
\end{equation}
under the nominal values of Table~\ref{tab-ex-intro-nominal-values} which are reproduced from Example~7.2 in \cite{doi:10.1002/asmb.2061} after standardization. The measurement errors are supposed to be uncorrelated and homoscedastic ($\boldsymbol\Sigma_{\varepsilon} = \sigma^2_{\varepsilon} \mathbf{I}_k$), and the measurement errors $\varepsilon_{i j}$ are realizations of a normally distributed error variable with mean zero and error variance $\sigma^2_\varepsilon$. Further, the unit specific parameters $(\beta_{i 1}, \beta_{i 3})^T$ of the units are assumed to be realizations of a bivariate normal distribution with mean $(\beta_{1}, \beta_{3})^T$ and a variance covariance matrix $\boldsymbol{\Sigma}_\gamma =
\left(\begin{array}{cc}
	\sigma_1^2 & \rho \sigma_1 \sigma_2
	\\
	\rho \sigma_1 \sigma_2 & \sigma_2^2
\end{array}\right)$.
	We are searching for a locally $c$-optimal design $\tau^*$ for extrapolation at the median failure time $t_{0.5}=1.583$.
	As constraints for the design we assume that $k = 6$ observation can be taken on a grid with increment $\Delta t = 0.05$ of the standardized experimental time interval $[0,1]$, i.\,e.\ $J = 20$ and $\mathcal{T} = \{0, 0.05, 0.10, ..., 1\}$. By the multiplicative algorithm (see e.g. \citep{silvey1978algorithm} ) adapted to the present constraint situation, the following numerical solution is obtained for the locally $c$-optimal design
	\[
	\tau^* =
		\left(
			\begin{array}{ccccccc}
				0.00 & 0.05 & 0.10 & 0.85 & 0.90 & 0.95 & 1.00
				\\
				0.166 & 0.166 & 0.130 & 0.055 & 0.166 & 0.166 & 0.166
			\end{array}
		\right) \, .
	\]
	The optimal approximate design $\tau^*$ is supported on seven time points which are concentrated to the ends of the standardized experimental time interval $[0,1]$ with maximal admitted weight $1/k$ for all but the two boundary points $t = 0.10$ and $t  = 0.85$ separating the grid points $t = 0.00, 0.05, 0.90,  0.95$, and $ 1.00$ with full weight $1/k$ from those with zero weight $ (t=0.15,...,0.80).$
	This shape of the optimal design is in accordance with the findings of \citep{sahm2001note} for $D$-optimality.
	In view of Proposition~\ref{prop-extrapolation-t} the design $\tau^*$ is also optimal for the estimation of the median failure time.

	For practical use, the optimal approximate design $\tau^*$ may be adjusted to
	\[
		\tau_0 =
			\left(
				\begin{array}{cccccc}
					0.00 & 0.05 & 0.10 & 0.90 & 0.95 & 1.00
					\\
					0.166 & 0.166 & 0.166 & 0.166 & 0.166 & 0.166
				\end{array}
			\right)
	\]
	which is supported on exactly $k = 6$ time points with equal weights $1/k=1/6$ by transferring the weight from the boundary point $t= 0.85$ to the boundary point $t = 0.10$  (see \citep{dorfleitner1999rounding}). The design $\tau_0$ can be realized as an exact design by taking one measurement at each of the six time points $0.00$, $0.05$, $0.10$, $0.90$, $0.95$, and $1.00$.
	To quantify what might have got lost, the quality of the adjusted design $\tau_0$ can be measured in terms of the local $c$-efficiency
	\[
		\mathrm{eff}_c(\tau_0) =\frac{ \mathrm{aVar}(\hat{t}_{0.5};\tau^*) }{ \mathrm{aVar}(\hat{t}_{0.5};\tau_0)} = \frac{\mathbf{f}_2(\hat{t}_{0.5})^T \mathbf{M}_2(\tau^*)^{-1} \mathbf{f}_2(\hat{t}_{0.5})}{\mathbf{f}_2(\hat{t}_{0.5})^T \mathbf{M}_2(\tau_0)^{-1} \mathbf{f}_2(\hat{t}_{0.5})} = 98.70\%
	\]
	 for extrapolation at $t_{0.5}$ in the marginal mixed effects model and, hence, for estimation of the median failure time.
	This indicates that the adjusted design $\tau_0$ is highly efficient and can be favorably used as the time plan for conducting an accelerated degradation testing experiment.
\end{example}

Note that by Proposition~\ref{prop-extrapolation-t} the optimal design $\tau^*$ does not depend on the variance covariance structure $\boldsymbol\Sigma_{\gamma}$ of the random effects, but the efficiency of the adjusted design $\tau_0$ may be affected by the random effects, see equation (\ref{eq-var-t-extra-decompose}).
Nevertheless, the $c$-efficiency $$\mathbf{f}_2(t_{0.5})^T \mathbf{M}_2^{(0)}(\tau^*)^{-1} \mathbf{f}_2(t_{0.5}) / \mathbf{f}_2(t_{0.5})^T \mathbf{M}_2^{(0)}(\tau_0)^{-1} \mathbf{f}_2(t_{0.5})$$ for extrapolation at $t_{0.5}$ in the fixed effect model provides a lower bound for the efficiency in estimating the median failure time.
For further approaches allowing for correlations depending on the distance of the measurement times, e.\,g.\ when measurements follow a random process, we refer to \citep{naether1985randomfields} and \citep{mueller2007spatialdesigns}.

\section{Cross-sectional time plans for destructive testing}
\label{sec-single-measurements}
In
the case of destructive testing only $k = 1$ measurement will be available per unit. Hence, measurements have to be taken at different time points at different units in order to guarantee estimability of the parameters,
\begin{equation}
	\label{eq-single-observation}
	Y_{i} = (\mathbf{f}_1(\mathbf{x}_{i}) \otimes \mathbf{f}_2(t_{i}))^T \boldsymbol{\beta} + \mathbf{f}_2(t_{i})^T\boldsymbol{\gamma}_i + \varepsilon_{i} \, ,
\end{equation}
$i = 1, ..., n$, where all other expressions have the same meaning as in the model equation~\ref{eq-general-model}.
We will restrict to the case of $k = 1$ measurements per unit in the remainder of this section.
In that case all measurements $Y_i$ are assumed to be independent.
Denote by $\sigma^2(t) = \mathbf{f}_2(t)^T \boldsymbol{\Sigma}_\gamma \mathbf{f}_2(t) + \sigma^2_\varepsilon > 0$
the variance function for measurements at time $t$, $\mathrm{Var}(Y_i) = \sigma^2(t_i)$.
Also here an identifiability condition has to be imposed on the variance parameters $\boldsymbol\varsigma$ to distinguish between the variance $\sigma^2_\varepsilon$ of the measurement error and the variance $\sigma_1^2$ of the random intercept, see \cite{grasshoff2012optimal}.
The information matrix $\mathbf{M}_{\boldsymbol\beta}$ for the location parameter $\boldsymbol\beta$ can be written as
\begin{equation*}
	\label{eq-info-single-obs-exact}
		\mathbf{M}_{\boldsymbol\beta} = \sum_{i = 1}^n (\mathbf{f}_1(\mathbf{x}_i) \mathbf{f}_1(\mathbf{x}_i)^T) \otimes \left({\textstyle{\frac{1}{\sigma^2(t_i)}}} \mathbf{f}_2(t_i) \mathbf{f}_2(t_i)^T\right)
\end{equation*}
according to the product-type structure of the model~(\ref{eq-single-observation}).
For estimating the median failure time $t_{0.5}$ the asymptotic variance is given by $\mathrm{aVar}(\widehat{t}_{0.5}) = c_0^2 \mathbf{c}^T \mathbf{M}_{\boldsymbol{\beta}}^{-1} \mathbf{c}$ as in Section~\ref{sec-failuretime}, where the vector $\mathbf{c}$ factorizes according to $\mathbf{c} = \mathbf{c}_1 \otimes \mathbf{c}_2$ into components $\mathbf{c}_1 = \mathbf{f}_1(\mathbf{x}_u)$ and $\mathbf{c}_2 = \mathbf{f}_2(t_{0.5})$ associated with the marginal models for the stress and the time variables, see (\ref{eq-avar-tau-alpha-decompose}).
The experimental settings $\mathbf{x}_i$ for the stress variable and $t_i$ for the time of measurement may be chosen independently from the standardized experimental regions $\mathcal{X}$ and $\mathcal{T}$, respectively. Here the approximate designs $\zeta$ are specified by $m$ mutually distinct combinations $(\mathbf{x}_i, t_i) \in \mathcal{X} \times \mathcal{T}$ with corresponding weights $\eta_i > 0$, $i = 1, ..., \nu$, $\sum_{i = 1}^\nu \eta_i = 1$.
The corresponding normalized information matrix is defined as
\begin{equation*}
	\label{eq-info-single-obs-approx}
	\mathbf{M}(\zeta) = \sum_{i = 1}^\nu \eta_i (\mathbf{f}_1(\mathbf{x}_i) \mathbf{f}_1(\mathbf{x}_i)^T) \otimes (\tilde{\mathbf{f}}_2(t_i) \tilde{\mathbf{f}}_2(t_i)^T) \, ,
\end{equation*}
where $\tilde{\mathbf{f}}_2(t) = \mathbf{f}_2(t) / \sigma(t)$ is the weighted version of the marginal regression function for the time variable (standardized by the standard deviation for measurement at time $t$).
Then, in accordance with \citep{shat2021experimental}, a $c$-optimal design $\zeta^*$  for $\mathbf{c} = \mathbf{c}_1 \otimes \mathbf{c}_2$ can be obtained as the product-type design generated from the $c$-optimal designs $\xi^*$ for $\mathbf{c}_1$ and $\tau^*$ for $\mathbf{c}_2$ in the associated marginal models, respectively.

\begin{theorem}
	\label{th-single-obs-prod-design}
	If $\xi^*$ is $c$-optimal for extrapolation at normal use condition $\mathbf{x}_u$ in the marginal model for the stress variables $\mathbf{x}$ and $\tau^*$  is $c$-optimal for extrapolation at the median failure time $t_{0.5}$ in the weighted marginal model with regression functions $\tilde{\mathbf f}_2(t)=\mathbf f_2(t)/\sigma(t)$ for the time variable $t$, then the design $\zeta^* = \xi^* \otimes \tau^*$ is optimal for estimating the median failure time $t_{0.5}$ in the combined model~(\ref{eq-single-observation}).
\end{theorem}

Note that for the second marginal model we utilize that $\tilde{\mathbf{f}}_2(t_{0.5})$ and $\mathbf{c}_2 = \mathbf{f}_2(t_{0.5})$ only differ by the factor $1 / \sigma(t_{0.5}) > 0$ which does not affect $c$-optimality. The $c$-optimal designs $\xi^*$ for extrapolation at $\mathbf{x}_u$ are the same as obtained in \citep{shat2021experimental}, and the $c$-optimal designs for extrapolation at $t_{0.5}$ can be obtained similarly by Elfving's theorem applied to the weighted regression functions $\tilde{\mathbf{f}}_2(t)$.

The locally optimal designs for estimating the median failure time are influenced by both the location and the variance parameters $\boldsymbol\beta$ and $\boldsymbol\varsigma$.
Therefore we make a sensitivity analysis, how the optimal designs change with the parameters and how well the locally optimal designs perform under parameter misspecification.
For this we note first that the optimal marginal design $\xi^*$ for extrapolation at $\mathbf{x}_u$ is globally optimal and does not depend on the parameters.
In the particular case of straight line degradation paths, as in Example~\ref{ex-2}, the optimal marginal design $\tau^*$ is concentrated on the endpoints 0 and 1 of the standardized time interval, as long as the median $t_{0.5}$ is larger than the upper bound 1 of the interval. Then only the optimal weight $\pi^* = t_{0.5} \sigma(1) / (t_{0.5} \sigma(1) + (t_{0.5} - 1) \sigma(0))$ at $t=1$ depends on the location parameters $\boldsymbol\beta$ through $t_{0.5}$ and on the variance parameters $\boldsymbol\varsigma$ through the ratio $\sigma(1) / \sigma(0)$ of the standard deviations at the support points.

As in Example~\ref{ex-2}, the asymptotic eficiency
\[
	\mathrm{eff}_{\mathrm{aVar}}(\zeta; \boldsymbol\theta) = \frac{\mathrm{aVar}_{\boldsymbol\theta}(\hat{t}_{0.5}; \zeta^*_{\boldsymbol\theta})}{\mathrm{aVar}_{\boldsymbol\theta}(\hat{t}_{0.5}; \zeta)}
\]
of the design $\zeta$ for estimating $t_{0.5}$, factorizes,
\[
	\mathrm{eff}_{\mathrm{aVar}}(\xi \otimes \tau; \boldsymbol\theta) = \mathrm{eff}_{c 1}(\xi) \cdot \mathrm{eff}_{c 2}(\tau; \boldsymbol\theta) \, ,
\]
in the case of product-type designs $\zeta = \xi \otimes \tau$ into the marginal efficiency $\mathrm{eff}_{c 1}(\xi)$ of $\xi$ for extrapolation at $\mathbf{x}_u$ and the marginal efficiency $\mathrm{eff}_{c 2}(\tau; \boldsymbol\theta)$ of $\tau$ for extrapolation at $t_{0.5}$.
For a product-type design $\xi^*\otimes\tau$ with optimal first marginal design $\xi^*$, the asymptotic efficiency reduces to the $c$-efficiency of $\tau$ in the second marginal model,

\[
	\mathrm{eff}_{\mathrm{aVar}}(\xi^* \otimes \tau; \boldsymbol\theta) = \mathrm{eff}_{c 2}(\tau; \boldsymbol\theta) = \frac{\mathbf{c}_2(\boldsymbol\theta)^T \mathbf{M}_2(\tau^*_{\boldsymbol\theta}; \boldsymbol\theta)^{-1} \mathbf{c}_2(\boldsymbol\theta)}{\mathbf{c}_2(\boldsymbol\theta)^T \mathbf{M}_2(\tau; \boldsymbol\theta)^{-1} \mathbf{c}_2(\boldsymbol\theta)} \,.
\]
The vector $\mathbf{c}_2(\boldsymbol\theta) = \mathbf{f}_2(t_{0.5})$ depends on the location parameters $\boldsymbol\beta$ through $t_{0.5}$, but not on $\boldsymbol\varsigma$,
while the information matrices $\mathbf{M}_2(\tau; \boldsymbol\theta)$ are only affected by the variance parameters $\boldsymbol\varsigma$, but not by $\boldsymbol\beta$.
For straight line regression in the time variable (cf.\ Example~\ref{ex-2}), only the ratio $\sigma(1) / \sigma(0)$ of the standard deviations for measurements at the endpoints of the experimental time interval has an effect on the information matrix $\mathbf{M}_2(\tau; \boldsymbol\theta)$, when the design $\tau$ is supported by these endpoints, as is the case for the locally $c$-optimal design $\tau^*_{\boldsymbol\theta}$.
 In the following we revisit Example~\ref{ex-2} for obtaining an optimal design in regards to the current model of cross-sectional time plans.

\begin{example*}[Example~\ref{ex-2} cont.]
 \label{ex-corss}
Under the current case of destructive testing, we provide in this example a $c$-optimal design of the time variable for the degradation model presented in Example~\ref{ex-2}.
	For the stress variable, the $c$-optimal marginal design $\xi^*$ is derived in  \citep{shat2021experimental}. In particular, the optimal marginal design $\xi^*$ is concentrated on the endpoints $0$ and $1$ of the standardized design region with weight $w^*=0.05$ at $x=1$.
	For the time variable $t$, the $c$-optimal marginal design $\tau^*$ for extrapolation at $t_{0.5} > 1$ can be similarly obtained by Elfving's theorem (\citep{elfving1952}) applied to the vector of weighted regression functions $\tilde{\mathbf{f}}_2(t)= (1,t)^T / \sigma(t)$, where the variance function $\sigma^2(t)$ is defined here by  $\sigma^2(t) = \sigma_1^2+\rho\sigma_1\sigma_2 t+\sigma_2^2 t^2+\sigma_\varepsilon^2$.
	The shape of the Elfving set is exhibited in Figure~\ref{fig-elfving-set_tau}.
	\begin{figure}
		\centering
		\includegraphics[width=0.45\textwidth]{fig-elfving-set-ex1}
		\caption{Elfving set for $t$ in Example~\ref{ex-2}: Induced design region (right solid line), negative image (left solid line), boundary (dashed lines), $\mathbf{c}_2 = (1, t_{0.5})^T $ (arrow) and corresponding ray (dotted line)}
		\label{fig-elfving-set_tau}
	\end{figure}

	The ray $\lambda \mathbf{f}_2(t_{0.5})= \lambda (1,t_{0.5})^T$ intersects the boundary of the Elfving set at the upper line segment connecting $\tilde{\mathbf{f}}_2(1) = (1, 1)^T / \sigma(1)$ and $ - \tilde{\mathbf{f}}_2(0) = (-1, 0)^T / \sigma(0)$ when $t_{0.5} > 1$.
	Hence, the $c$-optimal design $\tau^*$ is supported by the endpoints $0$ and $1$ of the standardized experimental time interval, and the optimal weights can be calculated by Elfving's theorem to $\pi^* = t_{0.5} \sigma(1) / (t_{0.5} \sigma(1) + (t_{0.5} - 1) \sigma(0))$ at $t = 1$ and $1 - \pi^* $ at $t = 0$.

	By Theorem~\ref{th-single-obs-prod-design}, the design
	\[
		\zeta^* = \xi^* \otimes \tau^* = \left(
			\begin{array}{cccc}
				(0, 0) & (0, 1) & (1, 0) & (1, 1)
				\\
				(1 - w^*) (1 - \pi^*) & (1 - w^*) \pi^* & w^* (1 - \pi^*) & w^* \pi^*
			\end{array}
		\right)
	\]
	is optimal for estimating the median failure time $t_{0.5}$.
	Under the nominal values of Table~\ref{tab-ex-intro-nominal-values} the optimal weights are $w^* = 0.05$ and $\pi^* = 0.77$.
	Then, the optimal design for estimating the median failure time is given by
	\[
		\zeta^* = \left(
		\begin{array}{cccc}
			(0, 0) & (0, 1) & (1, 0) & (1, 1)
			\\
			0.22 & 0.73 & 0.01 & 0.04
		\end{array}
		\right) \, .
	\]

In Figure~\ref{fig-weight-t-ex1}, the optimal weight $\pi^* = t_{0.5} \sigma(1) / (t_{0.5} \sigma(1) + (t_{0.5} - 1) \sigma(0))$ is plotted  as a function of $t_{0.5}$ while the ratio $\sigma(1) / \sigma(0)$ is held fixed to $1.22$ induced by the nominal values of Table~\ref{tab-ex-intro-nominal-values}.
\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.396395\textwidth}
    \includegraphics[width=\textwidth]{fig-weight_versus_tmed-ex1.pdf}
    \caption{Optimal weights $\pi^*$ in dependence on $t_{0.5}$ for Example~\ref{ex-2}}
\label{fig-weight-t-ex1}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.39639\textwidth}
    \includegraphics[width=\textwidth]{fig-weight_versus_stdratio-ex1}
   \caption{Optimal weights $\pi^*$ in dependence on $\sigma(1) / \sigma(0)$ for Example~\ref{ex-2}}
\label{fig-weight-sigma-ex1}
  \end{minipage}
\end{figure}
When $t_{0.5}$ increases, the optimal weight $\pi^*$ decreases from $1$ for $t_{0.5}$ close to the maximal experimental time $t_{\max} = 1$ to $\sigma(1) / (\sigma(1) + \sigma(0)) = 0.55$ for $t_{0.5} \to \infty$.
	In Figure~\ref{fig-weight-sigma-ex1} the optimal weight $\pi^*$ is plotted as a function of the ratio $\sigma(1) / \sigma(0)$ while $t_{0.5}$ is held fixed to $1.583$. On the other hand the optimal weight $\pi^*$ increases in the ratio $\sigma(1) / \sigma(0)$ from $0$ for the ratio close to $0$ to $1$ when the ratio tends to infinity.
	Those limiting values are indicated in Figures~\ref{fig-weight-t-ex1} and \ref{fig-weight-sigma-ex1} by horizontal dashed lines, and the nominal values for $t_{0.5}=1.583$ and $\sigma(1) / \sigma(0)=1.22$ are indicated by vertical dotted lines, respectively.

To assess the sensitivity of the optimal design with respect to parameter misspecification, we plot the efficiency of the locally optimal design $\zeta^* = \xi^* \otimes \tau^*$ in Figures~\ref{eff-t-ex1} and \ref{eff-sigma-ex1}.
For a comparison with standard marginal designs on the time scale, we also plot there the efficiency of the  designs $\xi^* \otimes \bar\tau_2$ (dashed line) and $\xi^* \otimes \bar\tau_6$ (dashed and dotted line), the marginal design $\bar\tau_k$ is uniform on $k$ equally spaced time points, i.\,e.\ $\bar\tau_2$ assigns weight $1 / 2$ to the endpoints $0$ and $1$ of the time interval, and $\tau_6$ assigns weight $1 / 6$ to the time points $t=0.0$, $0.2$, $0.4$, $0.6$, $0.8$, and $1.0$

\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.413996395\textwidth}
    \includegraphics[width=\textwidth]{fig-efficiency_versus_tmed-ex1.pdf}
    \caption{Efficiency of $\zeta^*=\xi^*\otimes\tau^*$ (solid line), $\xi^* \otimes \bar\tau_2$ (dashed line) and $\xi^* \otimes \bar\tau_6$ (dashed and dotted line)  in dependence on $t_{0.5}$ for Example~\ref{ex-2}}
\label{eff-t-ex1}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.410396395\textwidth}
    \includegraphics[width=\textwidth]{fig-efficiency_versus_stdratio-ex1}
   \caption{Efficiency of $\zeta^*=\xi^*\otimes\tau^*$ (solid line), $\xi^* \otimes \bar\tau_2$ (dashed line) and $\xi^* \otimes \bar\tau_6$ (dashed and dotted line)  in dependence on $\sigma(1) / \sigma(0)$ for Example~\ref{ex-2}}
\label{eff-sigma-ex1}
  \end{minipage}
\end{figure}

	In Figure~\ref{eff-t-ex1} the efficiency is displayed in dependence on the true value of the median failure time $t_{0.5}$ while the ratio $\sigma(1) / \sigma(0)$ of standard deviations is held fixed to the nominal value $1.22$.
	In Figure~\ref{eff-sigma-ex1} the efficiency is shown in dependence on the true value of the ratio $\sigma(1) / \sigma(0)$ of standard deviations while the median failure time $t_{0.5}$ is held fixed to $1.583$ derived from the nominal values.
	Also here the nominal values for $t_{0.5}$ and $\sigma(1) / \sigma(0)$ are indicated by vertical dotted lines in the corresponding panel, respectively.

	The efficiency seems to be more sensitive with respect to deviations in the ratio of the standard deviations than in the median failure time $t_{0.5}$.
	However, commonly neither small values of $t_{0.5}$ nor small values of $\sigma(1) / \sigma(0)$ seem to be reasonable.
	In particular, we may expect that the variance $\sigma^2(1)$ at the end of the experimental time interval is larger than the variance $\sigma^2(0)$ at the initial time, $\sigma(1) / \sigma(0) \geq 1$.
	This is satisfied in the case of positive or no correlation, and it also holds for slight to moderate negative corrleations, $\rho \geq - \sigma_1 / (2\sigma_2)$.
	In total, the locally optimal design $\zeta^*=\xi^*\otimes\tau^*$ under the nominal values performs quite well over a wide range of parameters while the design $\xi^* \otimes \bar\tau_2$ which is concentrated on the same time points is only preferable for larger values of the median failure time $t_{0.5}$.
	The design $\xi^* \otimes \bar\tau_6$ with six equally spaced time points performs substantially worse throughout.
	It has to be noted for completeness that the efficiency of $\bar\tau_6$ depends on the variance parameters $\boldsymbol\varsigma$ not only through the ratio $\sigma(1) / \sigma(0)$ because measurements are also to be taken in the interior of the interval.
	To accomplish for this, the plot in Figure~\ref{eff-sigma-ex1} has been generated by fixing $\sigma_1^2 = \sigma_2^2 + \sigma_{\varepsilon}^2$ and letting $\rho$ vary.
	However, other choices of the variance parameters do not change much in the performance of the design.
\end{example*}

\section{Discussion and conclusion}
\label{sec-discussion}

An elaborate assessment of the reliability related characteristics is of importance during the design stage of highly reliable systems.
The variability between units in accelerated degradation testing leads to a degradation function that can be described by a mixed-effects linear model.
This also results in a nondegenerate distribution of the failure time under normal use conditions such that it is desirable to estimate quantiles of this failure time distribution as a characteristic of the reliability of the product.
The purpose of optimal experimental design of the time plan is then to obtain the best settings for the time variable to obtain most accurate estimates for these quantities.

In the present model for accelerated degradation testing, it is assumed that in the experiment a cross-sectional design between units has to be chosen for the stress variable while for repeated measurements the time variable varies according to a longitudinal design within units.

Here we assumed a model with complete interactions between the time and the stress variables and random effects only associated with time but not with stress.
Then the cross-sectional design for the stress variables and the longitudinal design for the time variable can be optimized independently, and the resulting common optimal design can be generated as the cross-product of the optimal marginal designs for stress and time, respectively.
In particular, the same time plan for measurements can be used for all units in the test.

In contrast to that, in a model of destructive testing also the time variable has to be chosen cross-sectionally.
There the optimal choice of measurement times may also be affected by the variance covariance parameters of the random effects.
In both cases (longitudinal and cross-sectional time settings) the efficiency of the designs considered factorizes which facilitates to assess their performance when the nominal values for these parameters are misspecified at the design stage.

\renewcommand{\thesection}{\Alph{section}}
\setcounter{section}{0}

\section{Appendix}

\begin{proof}
	We prove the statement of the Lemma \ref{lem-repr-minv} by showing that multiplication of $\mathbf{M}$ with the right hand side $\mathbf{C} = (\mathbf{F}^T \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F})^{-1} + \boldsymbol\Sigma_\gamma$ results in the $k \times k$ identity matrix $\mathbf{I}_k$.
	For this note first that after premultiplication with $\mathbf{F}$  the right hand side can be expanded to
	\[
		\mathbf{F} \mathbf{C} = \boldsymbol\Sigma_\varepsilon \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F} (\mathbf{F}^{T} \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F})^{-1} + \mathbf{F} \boldsymbol\Sigma_{\gamma} (\mathbf{F}^{T} \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F}) (\mathbf{F}^{T} \boldsymbol\Sigma_\varepsilon^{-1} \mathbf{F})^{-1} = \mathbf{V} \boldsymbol\Sigma_{\varepsilon}^{-1} \mathbf{F} (\mathbf{F}^{T} \boldsymbol\Sigma_{\varepsilon}^{-1} \mathbf{F})^{-1}.
	\]
	Hence, by straightforward multiplication
	\[
		\mathbf{M} \mathbf{C} = \mathbf{F}^{T} \mathbf{V}^{-1} \mathbf{F} \mathbf{C} = \mathbf{F}^{T} \boldsymbol\Sigma_{\varepsilon}^{-1} \mathbf{F} (\mathbf{F}^{T} \boldsymbol\Sigma_{\varepsilon}^{-1} \mathbf{F})^{-1} = \mathbf{I}_k
	\]
	which proofs the lemma.
\end{proof}
\section*{Acknowledgement}
The work of the first author has been supported by the German Academic Exchange Service (DAAD) under grant no.~2017-18/ID-57299294.

\section*{References}
 \bibliographystyle{apa}

redpajama_set_name: RedPajamaArXiv


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text:
THE WRAP: The best of times and the worst of times
Oct 2, 2008 8:00pm
It rained hard in Brussels this week, soaking visitors to the IEC's Broadband World Forum, but curiously not dampening spirits.
Never mind that the Belgian government had just had to bail out the country's biggest bank or that service providers and vendors alike are struggling to find sustainable business models (see today's analystwire), the Forum demonstrated the industry's terrific determination to overcome all obstacles.
It was great to see such a diversity of opinion about timing as well as approaches, with the emphasis very much on Layer 1, and at the same time on the entire seven layer OSI stack. Speed and integration were the watchwords.
Sweden's privately held Transmode hasn't been touched by the economic conditions - it has been in profit for the last seven quarters. It has prospered by concentrating on providing straightforward, no frill layer one Wave Division Multiplexing (WDM) transport to Tier 2 and 3 carriers, cable operators (most notably Virgin Media in the UK) and metro-type networks - in other words, sticking to its knitting.
CEO Karl Thedéen points out that there is no need to have all singing and dancing layer 2 and 3 intelligence at every aggregation point as it slows everything down and consumes huge amounts of power - a subject close to every operator's heart as energy costs surge.
Thedéen says, "We can carry 80 channels/colours per fibre pair, meaning we can carry a total of 800Gbps per fibre pair, whereas GPON is restricted to 60 - 70Mbps downstream to the customer and even less upstream."
The beauty is that Transmode's stripped down for speed approach also means its typical power consumption is less than 10 Watts per 10 Gigabit Ethernet link. No wonder it has signed up its first, unnamed, Tier 1 customer in Europe.
Transmode's much larger competitor ADVA also caused quite a stir with its new Wave Division Multiplexing Passive Optical Network (WDM-PON) technology, Flexible Remote Node.
According to Dr Stefan Neidlinger, director, application and solution manager, ADVA, It means that the PON can cover distances of up to 100km between the central office/point of presence (POP) and the end point, again with minimal power consumption and enabling operators to get rid of some exchanges/central offices.
Big potatoes given that at last week's Carrier Ethernet World Congress in Berlin, Verizon commented that eliminating active equipment from transport wherever possible and getting rid of central offices would save it US$100 (€72.22) per year per customer.
AVDA argues that the time to deploy WDM PON is now, although one of its rivals, ECI, insisted that widespread commercial developments wouldn't happen until 2015 and in the meantime, Gigabit PON (GPON) would be the dominant technology to bring fibre closer to the user.
ECI's associate vice president, product marketing, network solutions division, Ron Levin, did acknowledge that in Europe operators had, for the time being, moved away from any plans to bring fibre to the home (FTTH) in favour of fibre to the [basements of multi-occupancy buildings] (FTTB) and fibre to the curb [kerb or street cabinet] (FTTC).
Nokia Siemens Networks pulled out of GPON earlier this year because it wasn't making much headway (Ericsson, Alcatel-Lucent and Huawei are all streets ahead of it).
This also cast doubt on the future of Provider Backbone Transport (referred to as PBT or PBB-TE) as Ethernet's not yet standardised alternative to Multiprotocol Label Switching (MPLS).
However, Fujitsu evidently has other ideas, championing its 9500 multiservce optical solution in Europe for the first time. Flashwave technology has been deployed by Verizon to migrate traffic synchronous digital hierarchy (SDH) traffic to Ethernet. According to Fujitsu, the only way to scale Ethernet over SDH is using PBT.
Sanjay Mewada argued that the only way for carriers to make money is from new charged-for services that are possible because of the increased bandwidth they supply. He is vice president of strategy, of software as services company NetCracker and pointed to the on-demand Microsoft Net Meeting service it provides for Telefonica.
As befits the industry at the moment, this involves two apparently contradictory elements: the separation of services from infrastructure and the simultaneous integration of all the relevant aspects of layers one to seven, from provisioning of a service to the allocation of sufficient bandwidth.
The trick is to avoid all the Frankenstein OSS/BSS, using meta data to extract only what is necessary for the task in hand, and billing, of course.
Next year's Broadband World Forum is in Paris. We're looking forward to it already, even if it rains.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Why would a Conditional Compilation Symbol not be detected in Release build? I have a WPF solution containing a sub-project component with WPF and Silverlight code. If the "WPF" Conditional Compilation Symbol is defined, the WPF code is built and if it is not defined then the Silverlight code is built. This works for my Debug builds just fine. However, when I attempt to do a Release build, the build does not detect the "WPF" Symbol and because it then incorrectly compiles the silverlight code I get a build that will not work. Any ideas why the WPF symbol is not being detected by the build?
Here is the relevant section of the .csproj. Note the "WPF" symbol is defined in DefineConstants for both Debug and Release, yet its not being detected in release builds. Can anyone help?
<PropertyGroup Condition=" '$(Configuration)|$(Platform)' == 'Debug|AnyCPU' ">
  <DebugSymbols>true</DebugSymbols>
  <DebugType>full</DebugType>
  <Optimize>false</Optimize>
  <OutputPath>bin\Debug\</OutputPath>
  <DefineConstants>TRACE;DEBUG;WPF</DefineConstants>
  <ErrorReport>prompt</ErrorReport>
  <WarningLevel>4</WarningLevel>
  <NoWarn>1607</NoWarn>
</PropertyGroup>
<PropertyGroup Condition=" '$(Configuration)|$(Platform)' == 'Release|AnyCPU' ">
  <DebugType>pdbonly</DebugType>
  <Optimize>true</Optimize>
  <OutputPath>bin\Release\</OutputPath>
  <DefineConstants>TRACE;WPF</DefineConstants>
  <ErrorReport>prompt</ErrorReport>
  <WarningLevel>4</WarningLevel>
</PropertyGroup>

A: From what I've read, I think you may need to update your build definition to include something like:
/p:DefineConstants="WPF"

This is in the Build definition under Process > Advanced > MSBuild Arguments.

A: Make sure your definitions are specified for all configurations DEBUG and RELEASE and any others you may have.

redpajama_set_name: RedPajamaStackExchange


text:
overtheshouldermlb
Dollar Time
Eight Eyes Fail Site Test
The Curious Case Of A Golden Glove
Posted on October 27, 2013 by overtheshouldermlb
It's not often that a person wins the game's highest honor for defense and loses it. No. The player didn't lose his Golden Glove but he simply 'lost it' a lot. Jimmy Piersall ('Fear Strikes Out') was one of the most hilarious sidekick baseball announcers of all time. His choice of words and phrases to describe what he just saw was far funnier than anything Jack Buck could utter. To listen to him on a telecast of the Chicago White Sox from 1977 through 1981 was worth the price of admission. Teamed with legendary Harry Caray, no slouch himself when it came to 'wacky', Piersall set the standard for 'losing it' as an announcer when he was fired after excessive on-air criticism of team management. That was only one instance of Piersall's conflict with the establishment.
Here was a guy who actually came to the plate in a major league game while wearing a Beatles wig and playing an air guitar with his bat as he stepped into the batter's box of old Shea Stadium. Of course he was playing for the Mets in their early years of wackiness. Again as a Met, in another game, hit his 100th career home run and celebrated by facing backwards as he trotted around the bases. Now you might ask yourself, why is that unusual? Really?
My favorite Piersall moment however was his wacky and inventive manner of distracting Ted Williams in a game at Fenway, July 20, 1961. There was the Splendid Splinter at the plate, a formidable figure if there ever was one in baseball. Here was a man who was the last player in baseball to hit .400+ in a single season. This was his last season. But he was still a force. He stepped into the box, tapped the plate…took those two perfect practice swings, and he was ready. All of a sudden, here was Piersall racing back and forth in the outfield in an attempt to distract the future Hall of Fame hitter. Close your eyes and think about it. Your imagination runs wild. While at the plate, Williams, a notorious left-handed hitter with supreme concentration, out of the corner of his right eye was seeing this blur moving toward and then rescinding, back and forth, in his vision of sight. Today various managers, using computer information that is updated after every pitch, design defenses of over shifts to combat the strengths of various hitters. But here was a guy who was way before Univac. He was creating the distraction of all distractions. What could Williams do about it? He stepped out of the box and asked the plate umpire what he was going to do about it. What could the manager do about it? The umpire was chastising the manager for Piersall's actions. What could umpires do about it? There were no rules covering this in the rule book of baseball. He was a whirling dervish all by himself. Solution? He was banished….BANISHED, not ejected but banished, from not only the game, but from baseball. His subsequent meeting with American League president, Joe Cronin, brought him back into baseball.
Here was a guy who would make Reggie Jackson smile because he was the guy who got into a fight with Billy Martin BEFORE a game. An All-Star in 1954 & 1956 for Boston he won the Golden Glove in 1958 and again 1961 for Cleveland. He had 95 assists in his career with 15 in a single season. He was #1 in the league in assists with 12 in 1957. From 1955 through 1961, he was #1 or #2 in range factor for 9 innings as a center fielder. In 1961 he had a .322 batting average, 3rd best in the American League. He was 3rd in the American League in scoring (103 runs) in 1957. He was a player.
There are few in the history of the game who have ever accomplished so much. Yet this magnificent player, who suffered from Bipolar disorder, will never make it to Cooperstown. Baseball doesn't get guys who do strange things like wearing wigs, playing air guitar, running backwards, creating havoc in the outfield to distract and who openly criticized management for not doing all they could to improve the team. Jimmy loved baseball. Jimmy loved his teams.
Now you know the curious case of a golden glove winner. As people are considering who is going to win one for this past season's play, it enjoyable to remember one of the greats who ever played the game. In 2010, at least Boston recognized his greatness as Jimmy Piersall was inducted into the Boston Red Sox Hall of Fame. As for Cooperstown, he should be placed in it and enshrined for the sole reason that he punched out Billy Martin BEFORE the game began.
For more daily doses on baseball, if there is something that is happening, go to: http://www.facebook.com/overtheshouldermlb. Heck, go there even when something isn't happening.
This entry was posted in Uncategorized and tagged baseball, Billy Martin, Boston Red Sox, Chicago White Sox, Cleveland Indians, Cooperstown, Golden Glove, Hall of Fame, Harry Caray, Jimmy Piersall, Lance Hanish, MLB, Ted Williams, www.facebook.com/overtheshouldermlb by overtheshouldermlb. Bookmark the permalink.


Xander Time is an English educational app for young children to learn to tell the time through healthy technology. Children are introduced to the clock and each hour before they practice to tell the time of the day through an entertaining game.
Xander Prepositions is a educational app for young children to learn about prepositions through healthy technology. Children are introduced to terms and visible placement features after being presented with the meaning of prepositions. Through an interactive and highly entertaining hide-and-seek game with Xander and friends, your child with be able to use prepositions with more confidence.
Xander Time part 2 is a educational app for young children to learn to tell the time through healthy technology. Children are introduced to more advanced features after learning the basics of time telling in Part 1. Here children learn about each half hour and quarter hour before they practice their new skills through an entertaining game of stop-the-clock.
Zulu 123 is an Zulu educational app for young children that teaches numeracy using child-friendly counting beads and characters from the animal kingdom. Xander, the lovable narrator, guides your child through four learn-through-play activities involving vision, hearing and touch to teach them how to count to 10 in Zulu.
Zulu Shapes and Colours is a educational app for your children that teaches six basic shapes and the primary and secondary colours using age appropriate games. Xander, the lovable narrator, guides your child through six learn-through-play activities involving vision, hearing and touch to teach them to identify shapes and colours.


Helmerich & Payne lost a $69 million in Tulsa County District Court to royalty owners, but the company said the verdict is not final and subject to modification by the Court. The Court is expected to enter judgment in the lawsuit in the near future.
Late last month, H.B. Krug won a decision relating to the exploration and production business that separated from Helmerich & Payne following a spin-off transaction completed in 2002. As part of the spin-off, Cimarex assumed all of the assets and liabilities relating to the E&P business and insured Helmerich & Payne for E&P-related liabilities — including the Krug litigation.
Since then, Cimarex has conducted the defense of the Krug litigation.


text:
Rishi Sunak Plays The Long Game As He Prepares To Unpick The Triple Lock
You're reading The Waugh Zone, our daily politics briefing. Sign up now to get it by email in the evening.
Ever since Gordon Brown blundered into giving pensioners a measly 75p rise in their pension, politicians have been terrified of upsetting the grey vote. Back in 1999, the Treasury was so pleased with low inflation that it had failed to spot the PR disaster of the tiny increase it entailed in all index-linked benefits.
Brown never repeated the error, and indeed paid his penance by swiftly making his temporary winter fuel allowance a permanent fixture, as well as introducing free TV licences for the elderly and pension credit.
During the 2010 election, David Cameron was bounced by Brown into keeping the measures. The Tory leader went further by bringing in the now infamous 'triple lock' that guarantees to uprate the basic and state pension in line with earnings, prices or 2.5%, whichever is the higher.
In 2015, the Conservatives did dip a toe in the dangerous waters of cutting "pensioner perks", as some called them, announcing they would withdraw state funding of the free TV licence for over-75s and ask the BBC to foot the bill instead. The blame game still plays out, but a survey by AgeUK found that Tory voters will be hit hardest.
Fast-forward to today and it appears that Rishi Sunak is preparing to think the unthinkable and not honour the triple lock pledge that was reaffirmed in the last Tory manifesto. The chancellor has in some ways the perfect cover in the form of the pandemic and the huge costs it has inflicted on the whole nation.
The young in particular have been hit hard by joblessness, on top of student debt and the UK's chronic inability to offer them affordable housing. With the third wave of Covid powering a fresh tsunami of cases among the under-30s, it seems they are in for their share of ill-health too. While ministers seem bent on a form of herd immunity, long Covid is the fear that stalks many a young person's Whatsapp chat.
The demands for "intergenerational fairness" have been getting louder and Sunak seems to have listened. Of course, he was subtle about it on the breakfast sofa shows, but the message seemed pretty clear: the triple lock will be tweaked, amended, possibly suspended to save valuable cash.
In a masterclass of political hint-dropping, the chancellor said that yes, the triple lock was still government policy "but I very much recognise people's concerns". "We want to make sure the decisions we make and the systems we have are fair, both for pensioners and for taxpayers." It was straight out of the overseas aid playbook: the overall aim is to stick to the manifesto but the pandemic has forced a rethink.
That's why the chatter in Whitehall and among MPs is that the Treasury is considering a possible suspension of the triple lock for one year only, just as happened on the pledge to spend 0.7% of GDP on aid. The hint was all but official when even Boris Johnson refused to kill off the speculation on Thursday. "We've got to have fairness for pensioners and the taxpayers," he said.
Whereas Brown was trapped into a PR nightmare by linking pensions to (low) inflation, Sunak is obviously keen not to suffer from a similarly self-inflicted, locked-in syndrome. With pensions now linked earnings (which are soaring at 8%), even a temporary way of avoiding the £3bn cost would save the Treasury serious money. Pensioners could still get a rise, just not a mega rise.
Insiders stress that nothing is decided and it will all depend on how the numbers look later in the year ahead of the spending review, but no one is killing the idea of a brief suspension of the full triple lock. The fact that both the PM and chancellor sounded like they were coordinating their message (on various outlets) shows there is a softening up exercise going on at the very least.
With Keir Starmer's shift from constructive to destructive opposition, Labour will inevitably try to seize on any change as an outrageous theft from pensioners' purses. But what everyone will be trying to gauge is just how many Tory backbenchers try to prevent any unpicking of the lock. Will those who were gung ho about a temporary cut to aid now bite their tongues?
It will certainly be a test of Sunak's political skills, though he does perhaps have time on his side. Just as with the international development 'cut', opting to do this in the middle of a parliament (and the middle of a pandemic) gives him room to make up for it in the run up to the next election.
It could also be that on the triple lock the chancellor realises that over the long term he could burnish rather than tarnish his reputation, not least among younger voters who see this as a fairness issue.
The furlough scheme has already helped make him the most popular politician in the UK and he may think he can afford to burn a bit of that capital now. Ending the £20 uplift in Universal Credit and jacking up corporation tax prove he is unafraid of making unpopular decisions.
If he does go ahead and take the bold option on pensions, it would certainly signal that Sunak really does have his eyes on the main prize. Unlike his 'buy now, pay later' boss, the chancellor cannot be seen to rely on borrowing as his way of balancing the books. If he can project himself to Tory MPs as firm but fair on tax and spend, he may have a decent crack at No.10 himself.
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text:
U2 returns to new relevance with sloganeering rock
Published November 30,2017
U2 has rarely been accused of lack of ambition, from record-breaking tours to headline-grabbing human rights campaigns, yet after four decades together the band inevitably has wondered how to move forward.
If 2017 has been chaotic for the world, the year has given fresh relevance to U2. The group's first album in three years, "Songs of Experience," sees Bono and Company returning to U2's founding faith in rock as a means of inspiration, at a historical moment when many listeners may be better disposed to activism than cynicism.
President Donald Trump's shadow hovers over much of the album, which comes out Friday after a delay of a year, which U2 said it needed to fine-tune its message after the right-wing tycoon's shock election victory.
"Don't take it lying down / You got to bite back / The face of Liberty's starting to crack," Bono sings on "Get Out of Your Own Way."
On "American Soul," the Irishman longs for the United States as an ideal higher than a messy, divided nation, singing: "It's not a place / This country to me is a thought that offers grace / For every welcome that is sought."
Kendrick Lamar, one of the most venerated voices of hip-hop, brings together the two tracks as he flips around the words of the Gospel of Matthew to ironize: "Blessed are the bullies / For one day they will have to stand up to themselves."
- FLASHES OF EARLY U2 -
"Songs of Experience," U2's 14th studio album, is a companion of sorts to the group's last album, 2014's "Songs of Innocence," the juxtaposed concepts in the titles an allusion to a poetry collection of William Blake.
"Songs of Experience" maintains the confessional approach of the previous album, with Bono, when not taking up the state of the world, reminiscing about his life's 57 years from his gratitude for support early in his career to his enduring love for his wife.
But if "Songs of Innocence" saw U2 half-heartedly adopting the lighter pop-rock style of bands whom the Dubliners once inspired, "Songs of Experience" carries flashes of U2's earlier passion, even if few tracks seem destined to join the band's classics.
"The Blackout," the first song released off the album, brings the energy of 1980s U2 with Adam Clayton's bass line pounding the way for The Edge's heavily distorted guitar.
"The Little Things That Give You Away" and "13 (There is a Light)" show anew U2's knack for the understated, with The Edge comfortably in the background, while the band stopping short at a return to full-effect ballads.
- NO MORE FREE ALBUM -
U2 also steered clear of the most memorable feature of "Songs of Innocence" -- its much maligned format, as a free release on iTunes.
Apple sent out "Songs of Innocence" to iTunes' half-billion users as part of a promotion, delighting some fans but triggering a flurry of complaints that the band was presumptious in believing the whole world wanted to listen.
Bono still has no shortage of detractors. Weeks before the album's release, the longtime advocate for international development was among wealthy figures named in the leak of the so-called Paradise Papers showing how he parked money overseas to avoid tax.
There is no indication Bono did anything illegal and he welcomed the transparency. Still, he may find embarrassing, even out-of-context, his line on "Red Flag Day" off the new album, "Paradise is a place you can see when it's yours."
But "Songs of Experience" shows U2 thinking large, opening with an unusually futuristic meditation for a band so focused on problems on Earth.
Bono, at once reflective and triumphant, sings: "So many stars / So many ways of seeing / Hey, this is no time not to be alive."
Conspicuous consumption: Edible gold
2018 Grammy Awards: List of nominees in top categories


text:
św. Krzysztof (zm. ok. 250) – męczennik, jeden z Czternastu Świętych Wspomożycieli
 św. Krzysztof z Tlaxcala (ok. 1515–1527) – uczeń szkółki misyjnej w Tlaxcala, męczennik
 św. Krzysztof Magallanes Jara (1869-1927) – meksykański ksiądz, męczennik
 Święty Krzysztof (obraz Goi) – obraz Goi

Zobacz też:
 Błogosławiony Krzysztof

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It was a typical boring dorm party.
We were in the party room in the basement of the Sanford tower. A nice looking girl walked up and started talking to me. She said she decided to talk to me because I looked so bored. She didn't live in the dorm. She and her girlfriends were leaving and going back to the house they rented. Did I want to tag along? You bet.
Most of her girlfriends played on the University field hockey team. We sat around and listened to music. Suddenly, "Paradise by the Dash Board Light" by Meat Loaf played. They girls jumped up, cranked the stereo to eleven (Spinal Tap reference), and proceeded to play "air guitar" to the whole, long song. One took air drums, the others had air guitars, and they danced and jumped and sang. Man, I was kind of freaked out. But I respected the uninhibited nature of the performance. I think of it every time I hear that song on the radio.


text:
Hopes grow for Cancun summit success
By Hu Yue
0 Comments Print E-mail China.org.cn, December 1, 2010
Global climate negotiators gathered in the Mexican resort city Cancun hope their decisions can have a positive impact on the global environment.
The Cancun summit will be successful if parties compromise, if they make sure that in the process of getting what they want, they allow others to leave with what they need, said Christina Figueres, Executive Secretary of the United Nations Framework Convention on Climate Change.
Governments can reach a deal to launch action on adaptation, technology transfer, and forests, and they can create a new fund for long-term climate finance, she said.
Figueres also said the Cancun summit will not solve everything and the outcome needs to be pragmatic, but the participants also need to keep ambition alive, she added.
Su Wei, Deputy Director of the Chinese delegation to the summit, said China has a positive attitude toward the climate negotiations.
"Cancun is an important station in the international negotiating, and we continue to stick to the UN Framework Convention on Climate Change, the Kyoto Protocol and the Bali Roadmap," he said.
Su said Beijing expects the conference to "bear real fruit" on the key issues of mitigation, adaptation, financial support and technology transfer and thus to lay a solid foundation for future negotiations and eventually a legally binding treaty.
The developed world should take the lead in cutting emissions due to their past responsibility for climate change, Sergio B. Serra, Brazilian Ambassador for Climate Change, told Beijing Review.
He also said Brazil appreciates the constructive role China has played in the negotiations and voluntary efforts of the country to cut emissions.
The main problem of the current negotiating process is the inability of the United States administration to make a meaningful commitment to cut its national emissions adequately. It's now clear Congress will not adopt a comprehensive climate bill, according to a report by the non-governmental organization Third World Network.
The US reluctance makes other developed countries disinclined to firm up their commitments, or even retain the existing regulated system. Many of them are still dragging their feet in stating how much they would cut their emissions, individually and as a group, in the Kyoto Protocol's second commitment period that is to start in 2013, the report said.
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text:
Categories: All My Children, Another World, As The World Turns, The Bold and the Beautiful Days of our Lives, Daytime, General Hospital, History, Loving, Mary Hartman Mary Hartman, Soap of the Week Poll, Santa Barbara, The Young and the Restless
'Pretty Little Liars' Finally Reveals Identify of "A" and It's an 'As the World Turns' Alum
During Tuesday's season six summer finale, Pretty Little Liars finally unmasked "A", and it was a familiar face to As the World Turns fans.
Warning: Spoilers ahead if you haven't watched!
Vanessa Ray plays CeCe Drake on Pretty Little Liars.
"A" is Charles DiLaurentis, who later transitioned to become Charlotte DiLaurentis, whom we've known for years as CeCe Drake. CeCe is played by actress Vanessa Ray, who starred on CBS daytime soap opera As the World Turns as Teri Ciccone.
Many fans voiced their displeasure with the reveal on Twitter.
PLL executive producer Marlene King spoke with ET about what's ahead.
"There's a lot of great stuff," she teased. "It has been so much fun to make this five-year time jump and so much fun especially this season -- the season that will premiere in January. We're kind of finding out together who these girls are five years later. Because it's a mid-season for us in the [writers'] room, it's not something that we talked about for months and months and months. We knew where they were going to go to school, we knew where they were going to be in their personal lives but as we bring them back to Rosewood, there's an even that happens that brings them all back together. It's just kind of fun exploring who they are they and they're having fun exploring their characters too."
King also spoke with EW.com to clarify many of the show's mysteries.
Posted by Kevin Mulcahy Jr.
Categories: ABC Family, As The World Turns, Daytime, Pretty Little Liars, Primetime
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Q: Deploy to Azure with a reference to a project I'm deploying PrjA to Azure WebSites (via Git), everything works fine if PrjA references PrjB.dll but doesn't work if PrjA reference the project PrjB. Azure says that it cannot find PrjB (correct, I do not want to push source code & C for PrjB).
When PrjA reference project PrjB I can read in properties of References/PrjB that Copy Local is set to true, so I thought it was working.
In other words, I'd like to reference project PrjB from PrjA so that I can hit F12 and get to the right source but I would like to push to the repo/Azure just the PrjB.dll not the entire project.
Is this possibile?


Galloway, N.J. - Student leaders, Stockton University President Harvey Kesselman, Executive Vice President Susan Davenport and members of the Cabinet today agreed that the university is "Stockton Safe," providing a safe zone for all students, faculty, and staff regardless of immigration status.
"The university, to the fullest extent permitted by law, protects the identity and information of all members of the community," said President Harvey Kesselman.
"We had a very productive meeting and came to a consensus about what will be most beneficial for the students. I thank Dr. Kesselman for allowing us to be part of this process and I am confident for a brighter future for Stockton University," said Ike Ejikeme, student Board of Trustee alternate.
"Historically, colleges and universities have been presented with the opportunity to stand for at-risk students during times such as the Fugitive Slave Act or Japanese internment. Many have failed. Today, as a result of the students' voice, Stockton took a step in the right direction to protect our own," said Kaltoum Alibrahimi, an Anthropology major.
"This movement is a sway against strong currents in uncertain waters. This is a sway in the right direction. Let Stockton Safe set the standard to embody the people, the voice and the heart of this institution to do what is just in times of adversity," said Mahalia Bazile, president of Unified Black Students Society.
"I have always been a proud member of the Stockton community, but I am especially proud today after participating in the first steps to defending the rights of all students regardless of status," said Theresa McMackin, vice president of Stockton STAND.
The group met in response to a Student Senate resolution regarding protections for undocumented students.
The group agreed to a series of protections, many of which are already in place for those who are undocumented, along with measures to educate the community about immigration rights.
The measures outlined today will be further discussed at upcoming meetings of the Faculty Assembly on Feb. 21 and the Board of Trustees on Feb. 22. The group committed to having a final agreement on specific protections and procedures by Feb. 24.


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What's Your 'Milestone Age'?
Katie J.M. Baker
There are some unarguably monumental age milestones — 16, 21, 50, — but what about the not-so-obvious birthdays that we deem meaningful for one reason or another?
In a piece for The Awl, Adriane Quinlan writes about America's obsession with 27-year-olds:
...everything America mythicizes and celebrates and destroys is twenty-seven and has always been twenty-seven: Ingrid Bergman, in Casablanca; Heather Graham, in Boogie Nights; Marilyn Monroe in Gentleman Prefer Blondes; Jemima Kirke, in "Girls"; and every other actress expected to be a sexual prize for the first 89 minutes and believably settled down in the final frame.
The twenty-seven-year-old can accomplish anything: Yuri Gagarin orbited at age 27; Flannery O'Connor published Wise Blood and Hemingway The Sun Also Rises-their debuts. Think of Ryan Lochte v. Michael Phelps just last month when both were 27, or LeBron James, 27. This is the year at which baseball players ripen, like cantaloupes, their desirability on fantasy rosters spiking (think Matt Kemp, Prince Fielder). And it's not because they're so good (Delmon Young, 27) but because next season, they
settle in; because twenty-seven's home runs and "Play-it-Again-Sams" wax into twenty-eight's solid OBPs and loveless marriages.
At least that's what Julia Roberts' character thinks in My Best Friend's Wedding, which is predicated on two friends promising that if they are not engaged to others by twenty-eight, they'll marry. I watched it while eating a pint of Ben & Jerry's (founded by two twenty-seven-year-olds) and wept: twenty-seven is the last year of romance, ego, mania. It's the last year of Bold Moves. It's the age of the real man of the hour, Christian Grey, who at twenty-seven jumps out of the playpen and into the arms of boring Anastasia. How mature.
We asked some other women about their personal milestone ages:
"I feel like I need to be chasing my dreams by the time I'm 25," an almost-24-year-old said. "25 sounds thrilling but also scary." The same woman said that 32 sounded intimidatingly rock solid, "like the decisions I've made will all of the sudden make total sense, and life will stop being so full of possibility, but in a stable, positive way."
A 32-year-old said she liked her early 30s, but hated 27, because it was the official crossover from mid-twenties to late-twenties. "By 27, I didn't feel so young and special anymore," she said. She also said she wasn't looking forward to 36, "probably because the media pounds it into our heads that fertility drops at 35. I dunno. Just sounds not good to me: 36."
A 36-year-old woman felt differently:
My thought is that the milestones have changed pretty significantly in the last years, and what's a milestone at 27 might be the same thing at 36...it's not that there's a particular age at which you have to buy a house or get married, particularly in urban life. The "milestone" if there is one that's more shared is probably a sense of feeling like you truly know who you are and/or are moving toward who you want to be. And that can happen at any age, but I think it really gets going in your 30s, probably, for women, when you start letting go of some of the trappings of adulthood or who you THINK you're supposed to be, and instead embrace yourself for who you really are.
A 41-year-old woman we spoke with said people were weirder about 41 than 40, but she didn't know why. "Everyone makes a big deal out of turning 40 — even if that big deal is "40 is no longer old!" (a la Oprah, etc)," she wrote via IM. "But 40 felt like no big deal to me. But saying I'm 41, that feels like a big deal, I see people's faces change at that one. So while 41 is not a milestone internally, I see people react to it like it's a milestone, more significantly than 40."
It's reassuring to hear that people have different age milestones because it reinforces the concept that age is really just a number, what you make of it, etc. Unless you're Jennifer Aniston, of course. (KIDDING.)
Twenty-Seven [The Awl]
Image via Elena Schweitzer/Shutterstock.
DorothyBelle
As the official Oldest Lady Of Them All, I assure you, it's all good—or at least it is until you hit 60.
25 is important. 30 is grand. 40 is a whirlwind. 50 is exciting—kind of hard if you haven't done a lot of the things you meant to do, but definitely lively nevertheless, full of fine moments. You're pretty sure you look good, and in any case you feel as though on the whole you're the person you wanted to be. You've stopped worrying about a million little things. You can still go wild on occasion without feeling absolutely foolish. You know things you didn't know before, including what matters and what doesn't matter. It feels awfully good to know what doesn't matter.
Then comes 60. At 60, you cross a bridge. You can look around and see the other side; all the things you've left behind you are clearly visible; but you can never cross back over again.
Some hotel clerk calls you "young lady," and you realize in a rush that that's the kind of thing hotel clerks only say to females under 10 and over 60. It's kindly meant, but it doesn't fool anybody. What it really tells you is that your neck is going.
Someone smiles too broadly and tells you "you're not 60 years OLD, you're 60 years YOUNG!" and even though all you want to do is smack them, you have to smile and accept it, because you know they mean well—they don't want to insult you by acknowledging that you are O-L-D.
You are out of the running forever when it comes to sex and beauty. The best you can hope for is that someone says you look good for your age, and they always say it a little too gently, a little patronizingly, as if you were an anxious child who needs to be reassured and placated. You're all of a sudden becoming one of those people that the kids make vomit noises about when they associate you with sex in any way shape or form—they flap their hands and say, "ew, ew, ew!" just like you used to do, talking about old people and sex, the most ludicrous and grotesque of erotic possibilities.
A little more time goes by, and now you're the little old lady who's driving too slowly and blocking traffic. You can practically see your own young self in the car behind you, stamping on the brakes and snapping, "Oh, come ON, get out of the WAY, you old bat!" You can't very well lean out the car window and shout, "Now I get it! The eyes and the reflexes, they don't work quite the way they did! I'm not driving slowly to annoy you or to humiliate myself—it's just that my body is starting to betray me, and this is the absolute best I can do."
It takes a while to deal with all that. But then you deal with it, because what else are you going to do? And after a while, you realize that if you're going to have to be an old lady, you'd might as well be fully alive while you're being it. The capacity for joy and invention lingers for a good many years, even when all you look like is just a wrinkled old thing.
Meanwhile, do enjoy—madly enjoy—being 25, or 27, or 30, or 35, or 38, or for that matter 50 or 55 and beyond. It all really is exceptional. Don't waste a minute of it worrying about the passage of time. Time, I assure you, will rush past on its own even without your worry to hasten it along.