Therefore it is desirable to estimate certain quantiles of this failure time
distribution as a characteristic of the reliability of the product. In this
context we discussed the existence of non-degenerate solutions for the
quantiles. The purpose of optimal experimental design is then to find the best
settings for the stress variable and/or the time variable to obtain most
accurate estimates for these quantities.

In the present model for accelerated degradation testing, it is further
assumed that stress remains constant within each testing unit during the whole
period of experimental measurements but may vary between units. Hence, in the
corresponding 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.

In the present paper 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. Moreover, the marginal optimal design for the stress variables can
be chosen independently of any model parameters. Optimal time plans may depend
on the aggregate location parameters via the median failure time, but do not
depend on which quantile of the failure distribution is to be estimated. These
results were extended to a model of destructive testing in which 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.

Finding optimal designs may become more complicated when the above assumptions
are not met. In particular, the designs for stress and time variables may no
longer be optimized independently if there are only additive effects in the
model (lacking interaction terms $xt$, cf. Example 3 for a similar situation
in the marginal stress model) or when also the stress variables are
accompanied by random effects. The impact of these deviations from the model
assumptions on optimal designs are object of further research as well as the
construction of designs which are robust against misspecification of the
nominal parameters, such as maximin efficient or weighted (“Bayesian”) optimal
designs. Of further interest would be to consider optimality criteria
accounting for simultaneous estimation of various characteristics of the
failure time distribution.

## Acknowledgement

The work of the first author has been supported by the German Academic
Exchange Service (DAAD) under grant no. 2017-18/ID-57299294.

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##  A Appendix

######  Lemma A.1.

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

$\mathbf{M}^{-1}=(\mathbf{F}^{T}\boldsymbol{\Sigma}_{\varepsilon}^{-1}\mathbf{F})^{-1}+\boldsymbol{\Sigma}_{\gamma}.$
(A.1)

###### Proof.

We prove the statement of the Lemma 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. ∎