with two stress factors
Von der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Simon Maria Pitzen, M.Sc.
aus Viersen
Berichter: Universit¨atsprofessorin Dr. Maria Kateri Universit¨atsprofessor Dr. Udo Kamps
Tag der m¨undlichen Pr¨ufung: 06. Juli 2021
Diese Dissertation ist auf den Internetseiten der Universit¨atsbibliothek verf¨ugbar.
First and foremost, I want to express my gratitude to my supervisor Professor Maria Kateri for her constant support and for giving me the chance to research an interesting topic.
Thank you for your trust, the opportunity to work at the Institute of Statistics in teaching and in research projects, and for your guidance and scientific advice that substantially shaped my thesis.
I want to thank Professor Udo Kamps for agreeing to be my second supervisor. The fruitful discussions and his conducive comments helped to improve this thesis significantly.
Furthermore, I would like to express my gratefulness to Professor Erhard Cramer and Professor Marco Burkschat for sparking my interest in the field of stochastics and statistics and for their encouragement to pursue a Ph.D. in the first place. I am especially thankful to Professor Burkschat as my initial supervisor for helping me to gain my first experiences in research and for the respectful and appreciative working atmosphere.
Sincere thanks go to my colleagues at the Institute of Statistics who accompanied me during the last years for the rewarding exchange of ideas and the enjoyable atmosphere at work, attending conferences, or in our leisure time. In particular, I want to thank Dr. Wolfgang Herff, Dr. Benjamin Laumen, Mia Kornely, Anastasia Gaponik, and my former office mate Christian Kohl for their helpful advice and for their moral support.
I would also like to thank Bernd Lenz and Dr. Bernd Ohligs for their motivated mathematics classes during my school time, which awakened my enthusiasm for this wonderful subject and equipped me with the basic knowledge as well as the confidence for my studies and Ph.D.
Without the scholarships provided by the Cusanuswerk and the Studienstiftung des deutsch- en Volkes during the course of my studies, I would have missed many valuable experiences and inspiring encounters. Thank you for broadening my mind.
I can always rely on my family and my friends at home, in Aachen, or anywhere else, who have been by my side in good and hard times. I owe them my deepest gratitude. Especially, I want to thank my parents Jutta and Helmut Pitzen for their overwhelming unconditional support and for encouraging me to follow my interests in any way. I can not thank them enough for their backing and for all the small and big signs of their affection. Special thanks are due to my mother Jutta for her meticulous proof-reading and her tireless efforts to give detailed comments on my thesis.
I am eternally grateful to my love Alex for being my companion throughout the last years of my Ph.D. She gave me fresh courage to continue, took the time to structure my thoughts and plans, and reminded me to take care of myself. I want to thank her for her patience and understanding, her compassion, her interest in my work and my worries, and for believing in me when I did not. Her support and the support of all my family have provided me with the perseverance to complete this thesis. It would not have been possible without you.
Contents iv
Introduction 1
1 Step-stress testing with multivariate stress factors 5
1.1 Step-stress testing with two stress factors and stress levels . . . 5
1.1.1 The model . . . 5
1.1.2 Cumulative exposure assumption and exponential distribution . . . 6
1.1.3 Inference for exponential simple step-stress under type-II censoring . . . 9
1.2 Step-stress testing with an arbitrary number of stress factors and stress levels . 13 1.2.1 The model . . . 13
1.2.2 Distribution, life-stress model, and inference . . . 14
1.3 Optimality criteria for test plans . . . 16
1.3.1 Fisher information and asymptotic variance . . . 17
1.3.2 The delta method . . . 20
1.3.3 Optimality criteria . . . 24
2 Statistical equivalence of bivariate simple SSALT plans 25 2.1 Statistical equivalence . . . 27
2.2 Model and assumptions . . . 28
2.3 Equivalent bivariate simple SSALT plans . . . 36
2.4 Extension to higher dimensions and censoring situations. . . 77
3 Optimal change points for bivariate simple SSALT plans and their robustness 81 3.1 Model and assumptions . . . 86
3.2 Optimal change points in closed form . . . 87
3.3 Robustness under parameter misspecification. . . 111
4 Comparison of different estimation methods in a bivariate step-stress setting 123 4.1 Different methods of estimation under NOC . . . 125
4.2 MLEs of the link function parameters in bivariate SSALT under type-II censoring 126 4.3 Alternative closed form estimators . . . 136
4.3.1 Introduction and motivation . . . 136
4.3.2 Theoretical properties . . . 139
4.3.3 Examples and simulation study . . . 154
4.4 Extension to higher dimensions and advanced application . . . 158
Conclusion and outlook 161
iv
A Distributions 163
B Further calculations 165
B.1 Fisher information matrix for a two-dimensional stress function . . . 165 B.2 Fisher information matrix for a two-dimensional step-stress stress function . . . 168 B.3 Unique solution for an equivalent bivariate simple SSALT . . . 170 B.4 Minors of the Fisher information matrix for a bivariate step-stress function . . . 177 B.5 MLEs of the log-link parameters in a bivariate step-stress setting . . . 181 B.6 Alternative likelihood motivated estimators for the intercept . . . 187 B.7 Log-link parameter estimation in an equidistant bivariate step-stress setting . . 189
Notation 193
List of Figures 197
List of Tables 199
References 201
In modern manufacturing industry, the development of highly reliable products is common.
To ensure this quality criterion, the need for upfront testing of materials, components, and entire devices has increased, e.g., for warranty prediction or quality control. High reliability implies large mean times to failure of the tested objects under normal operating conditions (NOC), so conventional methods of life testing fail to gather information about the lifetime distribution and the related parameters: In a test of practical length, only a few or even no failures at all occur.
Since conducting tests of very long duration is often not possible and economically unrea- sonable, accelerated life testing (ALT) is used to obtain the required information. In this methodology, the test objects are exposed to a specific stress which is related to failure and severer than under NOC, e.g., temperature, voltage, use rate, load, or humidity are commonly used stress factors. The increased stress reduces the time to failure which results in more observations during the conducted life test. Data collected during an accelerated life test and assembled inferential results need to be extrapolated to NOC to estimate the lifetime distribution of interest. General introductions to the topic of ALT were given by Nelson (1990), Meeker and Escobar (1998), and Bagdonaviˇcius and Nikulin (2002). Es- cobar and Meeker (2006) reviewed prevailing accelerated test models for the relationships between lifetime and stress factors.
The applied stress can either be constant, which represents the most widely used stress loading in ALT (see, e.g., Nelson (2005) and Yang (1994)), or vary during the test proce- dure. A common approach is to keep the stress on one level and change it once or several times after a prespecified time or upon the occurrence of a fixed number of failures. This method, which is intended to yield quicker results than a constant stress test (cf. Miller and Nelson (1983)) and allows for a more efficient usage of test equipment and test objects, is called step-stress accelerated life testing (SSALT). Step-stress tests that only involve a single stress change point are called simple step-stress tests. Figure 0.1 gives a schematic impression of the different stress loadings for a constant stress test with two different test conditions and a simple step-stress test. Contrary to this, step-stress tests with more than two stress steps are called multiple (step) step-stress tests, also referred to as k-step or k-level step-stress tests (cf., e.g., Khamis (1997a), Han (2015)). If the test units are ex- posed to different stress levels consecutively, a connection of the specific underlying lifetime distributions on the respective levels is necessary. Known approaches to this issue are the cumulative exposure (CE) model proposed by Sedyakin (1966) and popularized by Nelson (1980), the proportional odds (PO) approach by Brass (1974), the tampered failure rate (TFR) model by DeGroot and Goel (1979), which is a generalization of the Cox or propor- tional hazards (PH) model by Cox (1972), the tampered random variable (TRV) approach by Bhattacharyya and Soejoeti (1989), and most recent the hazard rate (HR) approach by Kateri and Kamps (2017), which is more flexible than the TFR model.
In order to enable the desired extrapolation of information gathered under intensified test conditions to NOC, assumptions on the relationship between the parameters of the lifetime
1
time stress
x2
x1
x0
0
Constant stress
time stress
x2
x1
x0
0 τ1
Step-stress
Fig. 0.1:Schematic representation of the conduction of a constant stress experiment (left) and a step-stress experiment (right) with two increased stress levelsx1,x2in comparison to the NOCx0. The crosses indicate observed failure times.
model and the stress factors are necessary in ALT experiments in general. These life-stress assumptions, which contribute to the ALT model as parametric link function represen- tations of the model parameters or transformations thereof, are typically motivated by physical principles like the Arrhenius equation or the inverse power law. Common choices of link functions are linear, log-linear, quadratic, or log-quadratic. Meeker and Escobar (1998) gave a detailed overview on different link functions and their physical motivation including several examples from practical application.
In current practice, the described methodology of SSALT and ALT in general meets two major difficulties:
1. Due to further advanced technology, the increased stress levels necessary to signifi- cantly decrease the time on test often need to be extremely high (cf. Park and Yum (1996), Zhu and Elsayed (2013), Li and Fard (2007)). On the one hand this requires a powerful and therefore cost-intensive test equipment, on the other hand extreme lev- els of stress can evoke failure modes that differ considerably from the failure behavior of the test object under NOC.
2. If a test object is exposed to several different kinds of stress in the field situation, a test applying only one type of stress may not be able to realistically reproduce the failure behavior and lead to severe fallacies in the estimation process (cf. Zhu and Elsayed (2013)).
These issues can be solved by running step-stress tests under the application of multiple experimental factors. The combination of several stresses results in a higher strain on the test object which could even be further enhanced by a possible interaction between the stresses, while the respective levels can be held low. Furthermore, a multi-stress test usually offers a better simulation of the operating conditions met in the field and consequently provides a better reproduction of the aging process. Artificial failure modes are avoided and the investigation is not focused on a single aspect of product failure.
Even though the conduction of accelerated tests involving two or more stress types si- multaneously is common in practice (cf. Escobar and Meeker (1995)), there is only unsat- isfactory research on the theory of planning these tests in order to optimize the reliability
estimation (cf. Zhu and Elsayed (2013)). Only a few authors have studied different sit- uations with various methods, optimality criteria, distributional assumptions, censoring schemes, link functions, and objects of optimization. Here, we focus on SSALT with two stress factors for exponentially distributed lifetimes of the test objects. The following aspects are considered in detail: the comparison of simple step-stress tests to other stress loadings with two stress factors, the derivation of optimal change points for bivariate simple step-stress designs for given stress levels, and the impact of the chosen estimation method on the performance of the estimation under NOC.
Chapter 1: We introduce the step-stress model with multivariate stress factors and estab- lish the notation for the following chapters. The test procedure and the necessary assump- tions for a stochastic model are explained in detail for the case of two stress variables. After elaborating on the general failure time distribution in a bivariate step-stress experiment, for exponentially distributed lifetimes of the test objects, inferential results for the parameters of the underlying distributions are derived and conditions necessary for the existence of the maximum likelihood estimators (MLEs) are discussed assiduously. The model and the corresponding observations are extended to arbitrary numbers of stress factors and stress levels with necessary adjustments. Finally, a brief discussion of the asymptotic variance and the delta method is used to motivate common optimality criteria for ALT plans.
Chapter 2: Step-stress designs are often conducted to simplify the test procedure, shorten the test duration, and ensure efficient usage of the test equipment. In order to justify choosing simple step-stress tests over any other stress pattern in ALT with two stress factors from a statistical point of view, different stress functions are investigated with regard to statistical equivalence of the resulting experiments. Under the assumption of exponentially distributed lifetimes, we prove that for a major class of arbitrary stress functions there is always at least one bivariate simple step-stress plan such that the Fisher information matrices of the respective tests coincide and the tests are equivalent with respect to all criteria depending on the asymptotic variance matrix. This implies that bivariate step- stress designs are the optimal design choice for ALT experiments including two stress factors if the performance measure is defined in terms of the asymptotic variance matrix only.
The behavior of the constructed equivalent step-stress designs for various sample sizes is investigated in an extensive simulation study and compared to other common designs likek- level parallel constant stress or multiple step-stress experiments but also unusual continuous stress loadings.
Chapter 3: Since the reliability data collected in SSALT experiments are used to gather information under NOC via extrapolation, even small changes of the tests set-up can have strong effects on the behavior of estimators. Consequently, optimal test design is a topic of great interest to improve the efficiency of experiments. In a situation where the stress levels of an exponential bivariate simple step-stress test are given, we are able to determine the optimal change points minimizing the asymptotic variance of the logarithmic MLE of the mean expected lifetime under NOC for step-up and step-down tests. The change points are derived as closed form solutions of an optimization problem which allows to determine an overall lower bound for the asymptotic variance of bivariate simple step-stress experiments that only depends on the choice of stress levels for one of the two stress factors. This also enables a different approach to design bivariate simple SSALT experiments: The stress levels for both stress factors can be chosen such that a target asymptotic variance is met and the probability of non-existence of the estimator is minimized. The optimal test designs are
compared to test designs from Chapter2in simulation studies, and it is observed that the optimal designs provide better estimations even for small sample sizes despite optimizing an asymptotic criterion.
The provided optimal change points are dependent on pre-estimates of the model param- eters. The asymptotic variance of the estimator under a non-optimal test design is deter- mined as a function of the relative misspecification of these parameters. The respective influences of errors in the two pre-estimations on the asymptotic variance of the resulting test design are compared. Previous simulation studies are repeated under various misspec- ification scenarios.
Chapter 4: The aim of ALT experiments in general is to gather information on a failure time distribution under NOC although the test is conducted under different, more strain- ing conditions. For this transfer, an assumption on the link between the current lifetime distribution and the stress variables is inevitable. The parameter estimate of interest is derived by estimating the parameters of the link function based on the data generated in the ALT experiment. A fundamental consideration is the decision upon the method of estimation, and different approaches are conceivable. In an exponential step-stress set-up with multiple stress factors, we consider a least squares estimation procedure starting from the MLEs of the scale parameters on the respective increased stress levels. We show that this approach is equivalent to the standard approach based on the direct MLE of the link function parameters only under certain prerequisites and draw a comparison between both methods in a specific situation where named prerequisites are not fulfilled. In this case, we prove the general existence of the MLEs of the link function parameters and derive a condition to receive the estimators in closed form. Both estimation methods are analyzed in a simulation study also considering type-II censoring. Alternative consistent estimators of the log-link parameters are motivated by the MLEs for the scale parameters. Due to their closed form and their relation to the MLEs it is possible to derive exact distributional properties of the corresponding estimator of the scale parameter under NOC. Again, all estimators are compared in an extensive simulation study.
Step-stress testing with multivariate stress factors
Step-stress tests with multiple stress factors are similar to step-stress tests with a single stress and multiple stress levels, but show structural differences. Before we move on to the general case, we consider SSALT with two stress variables each having two stress levels as described by Li and Fard (2007). Step-stress tests with two stress factors are also known as bivariate step-stress plans in the literature.
1.1 Step-stress testing with two stress factors and stress levels
After an introduction of the notation and the test procedure, we derive the lifetime dis- tribution of test objects in SSALT and generalize the common concept of a log-linear link function. The derivation of the MLEs of the mean lifetimes under increased stress is dis- cussed in detail.
1.1.1 The model
We denote the stress levels of a step-stress test with two stress factors by Sk,l,k ∈ {1,2}, l ∈ {0,1,2}, where Sk,l is the l-th stress level of stress factor k. The stress levels S1,0, S2,0 represent the NOC, also referred to as “use conditions” or “operating stress” by other authors (cf., e.g., Escobar and Meeker (2006)). In the following, we will also use the notation Sklfor brevity if no confusion can arise. The vector of the current stress levels of all factors (S1,i1, S2,i2),i1,i2∈ {1,2},i1 ≥i2, is referred to as stress step S(i1,i2).
The procedure of the concerned SSALT with two stress variables can be described as follows:
At the beginning of the test, n ∈ N test units are placed at the initial first stress step S(1,1) = (S11, S21), i.e., under the stress levelsS11 and S21, until timeτ1 at which the first stress factor is changed from level S11 toS12. Afterwards, the test is carried on until time τ2, when the second stress factor is changed from level S21 to S22. If we assume type-II censoring, the test is continued until a predetermined number r ∈N, r ≤n, of units fail.
That means, the termination point of the test is Tr:n, the r-th ordered failure time of the sample ofn test objects, which is random. If r is chosen as n, the complete sample would be observed under the step-stress test. The timesτi,i∈ {1,2}, upon which the stress levels are changed, are called change points. In the literature, the alternative term “hold times”
is also common (cf., e.g., Miller and Nelson (1983)). The random number of failures that occurred on stepi,i∈ {1,2,3}, is defined asNi. Hence, Ni denotes the observations in the interval (τi−1, τi] fori∈ {1,2} withτ0 := 0 and N3 :=r−N1−N2. Such step-stress tests with predetermined change points are referred to as time-step (step-)stress tests in contrast to so-called failure-step (step-)stress tests where the stress is changed after certain numbers
5
of failures are observed (cf., e.g., Bai, Kim, and Lee (1989)). Failure-step step-stress tests are not considered in this work. Test procedures with two stress factors and a single stress change for each factor as described above are also considered as simple step-stress tests.
An exemplary test procedure with increasing stress levels/steps is shown in Figure 1.1.
In general, the structure of a step-stress experiment with multiple stress factors is similar to a multiple step step-stress experiment with a single stress factor. An ordering of the stress levels of a stress factor is not necessary for the model but common in practice to avoid disturbances of the aging process. So called step-up models that plan an increase of severity of the stress for every step are most popular (cf. Miller and Nelson (1983)). Note that an order relation between two stress levels like S11 < S12 does not necessarily imply the strain on the test object to be increased by a stress change, e.g., for thermal stress using temperatures below 0.
time stress
step 1 S(1,1)= (S11, S21)
step 2 S(2,1)= (S12, S21)
step 3 S(2,2)= (S12, S22)
τ1 τ2 tr:n
N1 N2 r−(N1+N2)
Fig. 1.1:Test procedure for step-stress testing with two stress factors.
1.1.2 Cumulative exposure assumption and exponential distribution
As described for example by Balakrishnan (2009) for the common step-stress model with one stress variable, we denote the cumulative distribution functions of the lifetimes of the tested units under combinationsS(1,1),S(2,1), andS(2,2) of constant stress levels byF1,F2, andF3 with support (0,∞), respectively. In the following, we assume that the distribution functions Fi, i ∈ {1,2,3}, are all member of the same scale family of distributions, i.e., there is a baseline distribution functionF, such that
Fi(t) =F t
ϑi
, ∀t∈(0,∞), i∈ {1,2,3}, (1.1) where ϑi ∈ (0,∞) are the scale parameters associated with Fi for i ∈ {1,2,3}. Under the CE-assumption as mentioned above, the change of stress from one stress level to an-
other results in a change of the lifetime distribution Fi to a shifted version of the lifetime distributionFi+1 such that
Fi+1(τi+hi+1) =Fi(τi+hi), i∈ {1,2}, (1.2) whereh1 := 0. If we assume absolute continuity and a strictly increasing baseline distribu- tion function F, we receive
Fi+1(τi+hi+1)(1.1)= Fτi+hi+1 ϑi+1
!
= Fτi+hi ϑi
(1.1)
= Fi(τi+hi)
⇐⇒ τi+hi+1
ϑi+1
= τi+hi
ϑi
⇐⇒ hi+1 =
1 ϑiτi+ϑ1
ihi−ϑi+11 τi 1
ϑi+1
= ϑi+1
ϑi −1
τi+ϑi+1
ϑi hi, i∈ {1,2}. This recursive relation implies the following by induction:
hi+1=ϑi+1
Xi j=1
τj−τj−1
ϑj −τi, i∈ {1,2},
whereτ0 = 0. Hence, we receive the cumulative distribution function under the cumulative exposure model as
G(t) =
F1(t), 0< t≤τ1,
F2
t+ ϑϑ2
1τ1−τ1
, τ1< t≤τ2, F3
t+ ϑϑ3
1τ1+ϑϑ3
2(τ2−τ1)−τ2
, τ2< t <∞,
(1.3)
and the corresponding probability density function as
g(t) =
f1(t), 0< t≤τ1,
f2
t+ϑϑ2
1τ1−τ1
, τ1 < t≤τ2, f3
t+ϑϑ3
1τ1+ϑϑ3
2(τ2−τ1)−τ2
, τ2 < t <∞,
(1.4)
where fi denotes the probability density function of Fi, i ∈ {1,2,3}. We can see that the cumulative exposure assumption finds expression in location shifts of the considered distributions. The CE-model is based on the assumption that the remaining lifetime of a test object is only dependent on the so far accumulated amount of damage, i.e. the cumulative exposure to stress, without regard of how this exposure has been accumulated (cf. Miller and Nelson (1983)). Figure 1.2 illustrates the CE-assumption in a simple step- stress model. The transparent curves correspond to the distribution functions F1,F2 and the opaque curve gives the resulting compound distribution function for the step-stress model.
If we consider the distribution functionsFias exponential distributions with scale parameter
t F(t)
1
h
0 τ1
Fig. 1.2:Illustration of the CE-assumption for a simple step-stress model. The blue curve corre- sponds to the distributionF1on the first step and the red curve to the distributionF2on the second step. The dashed curve gives the distribution function under NOC.
ϑi,i∈ {1,2,3}, i.e., ϑi is the expected value (cf. AppendixA), the cumulative distribution function and the probability density function in equations (1.3) and (1.4) come to be
G(t) =
1−e−ϑt1, 0< t≤τ1, 1−e−ϑ12 t+
ϑ2 ϑ1τ1−τ1
, τ1 < t≤τ2, 1−e−ϑ13 t+
ϑ3 ϑ1τ1+ϑϑ3
2(τ2−τ1)−τ2
, τ2 < t <∞,
(1.5)
and
g(t) =
1
ϑ1e−ϑt1, 0< t≤τ1,
1
ϑ2e−ϑ12 t+
ϑ2 ϑ1τ1−τ1
, τ1< t≤τ2,
1
ϑ3e−ϑ13 t+
ϑ3 ϑ1τ1+ϑϑ3
2(τ2−τ1)−τ2
, τ2< t <∞.
(1.6)
It is usually assumed that the change pointsτi,i∈ {1,2}, between the stress levels differ, i.e., τ1̸=τ2, because the model could be reduced otherwise.
Since the idea of ALT consists in testing under intensified conditions to extrapolate the generated information on the failure behavior of the test object to NOC, a connection between the lifetime distributions in different stress situations is necessary. A common approach for exponential ALT to model the relation between lifetime and stress is to assume that the mean lifetime, i.e., the scale parameterϑ, is a log-linear function of the prevailing stress or a transformation thereof (cf., e.g., Miller and Nelson (1983)): On stress step Sx, x = (i1, i2) ∈ {(1,1),(2,1),(2,2)}, it holds for the mean lifetime ϑ of the current distribution that
log ϑ Sx
=β0+β1S1,i1+β2S2,i2, (1.7)
where β = (β0, β1, β2)⊤ ∈ R3 is a constant parameter vector dependent on the specimen and the test situation. Therefore, we receive the outline of the test as
Step 1 S(1,1)
: log(ϑ1) =β0+β1S1,1+β2S2,1, Step 2
S(2,1)
: log(ϑ2) =β0+β1S1,2+β2S2,1, Step 3
S(2,2)
: log(ϑ3) =β0+β1S1,2+β2S2,2.
(1.8)
The life-stress relationship in (1.7) and (1.8) is a well-studied model motivated by several physical principles like the Arrhenius equation, the inverse power law, or the Eyring equa- tion (see, e.g., Nelson (1990, pp. 71–105) or Meeker and Escobar (1998, pp. 471–488) for details). The link function can be used to reparametrize the cumulative distribution func- tion and the corresponding density in (1.3)–(1.6), e.g., to reduce the number of parameters in some cases or to perform maximum likelihood estimation on the parameters of the link function directly. The reparametrization is used in Chapter 2 and Chapter3. A detailed discussion and a comparison of the approaches with and without reparametrization can be found in Chapter 4.
1.1.3 Inference for exponential simple step-stress under type-II censoring Inference on the scale parameters of the exponential lifetime distributions on the increased stress levels in a step-stress set-up was considered first by Balakrishnan, Kundu, Ng, and Kannan (2007) for simple step-stress tests involving a single stress factor. Here, we want to derive the MLEs ˆϑi,i∈ {1,2,3} in a simple step-stress test for two stress factors in order to illustrate the structure of the estimators and to emphasize the necessity of existence conditions for all estimators.
If we start the experiment with a sample ofn∈Ntest objects and follow the test procedure described in Section 1.1.1, i.e., for failure times Tiiid∼G, i ∈ {1, . . . , n}, we receive the observed data of the form
t1:n<· · ·< tN1:n≤τ1 < t(N1+1):n<· · ·< t(N1+N2):n≤τ2 < t(N1+N2+1):n<· · ·< tr:n, where tj:n, j ∈ {1, . . . , n}, denotes the j-th ordered observed failure time in a realization t:= (t1, . . . , tn) of the random vector T := (T1, . . . , Tn). The general likelihood function in this setting of ordered data is given by
L(ϑ1, ϑ2, ϑ3) =g1,...,r:n(t1:n, . . . , tr:n) = n!
(n−r)!g(t1:n, . . . , tr:n) 1−G(tr:n)n−r
= n!
(n−r)!
Yr i=1
g(ti:n) 1−G(tr:n)n−r
, (1.9)
since we assume the failure times of the n objects as independent. For the exponential simple step-stress model in (1.5) and (1.6), we have to consider seven different cases:
L(ϑ1, ϑ2, ϑ3) =
c(n,r)
ϑr1 eA1(ϑ1), ifN1=r,
c(n,r)
ϑr2 eA2(ϑ1,ϑ2), ifN2=r,
c(n,r)
ϑr3 eA3(ϑ1,ϑ2,ϑ3), ifN1=N2= 0,
c(n,r)
ϑN11ϑN22eA4(ϑ1,ϑ2), ifN1+N2 =r, N1, N2̸= 0,
c(n,r)
ϑN11ϑr−N3 1eA5(ϑ1,ϑ2,ϑ3), if 1≤N1≤r−1, N2 = 0,
c(n,r)
ϑN22ϑr−N3 2eA6(ϑ1,ϑ2,ϑ3), ifN1= 0,1≤N2 ≤r−1,
c(n,r)
ϑN11ϑN22ϑr−N3 1−N2eA7(ϑ1,ϑ2,ϑ3), if 1≤N1, N2, N1+N2 ≤r−1, where
c(n, r) := n!
(n−r)!, A1(ϑ1) :=− 1
ϑ1
" r X
i=1
ti:n+ (n−r)tr:n
# ,
A2(ϑ1, ϑ2) =−nτ1
ϑ1 − 1 ϑ2
" r X
i=1
ti:n−nτ1+ (n−r)tr:n
# ,
A3(ϑ1, ϑ2, ϑ3) :=−nτ1
ϑ1 −n(τ2−τ1) ϑ2 − 1
ϑ3
" r X
i=1
ti:n−nτ2+ (n−r)tr:n
# ,
A4(ϑ1, ϑ2) =− 1 ϑ1
"N X1
i=1
ti:n+ (n−N1)τ1
#
− 1 ϑ2
Xr i=N1+1
ti:n−(n−N1)τ1+ (n−r)tr:n
,
A5(ϑ1, ϑ2, ϑ3) :=− 1 ϑ1
"N1 X
i=1
ti:n+ (n−N1)τ1
#
−(n−N1)(τ2−τ1) ϑ2
− 1 ϑ3
Xr i=N1+1
ti:n−(n−N1)τ2+ (n−r)tr:n
,
A6(ϑ1, ϑ2, ϑ3) :=−nτ1
ϑ1 − 1 ϑ2
"N X2
i=1
ti:n−nτ1+ (n−N2)τ2
#
− 1 ϑ3
Xr i=N2+1
ti:n−(n−N2)τ2+ (n−r)tr:n
,
A7(ϑ1, ϑ2, ϑ3) :=− 1 ϑ1
"N1 X
i=1
ti:n+ (n−N1)τ1
#
− 1 ϑ2
N1X+N2
i=N1+1
ti:n−(n−N1)τ1+ (n−N1−N2)τ2
− 1 ϑ3
Xr i=N1+N2+1
ti:n−(n−N1−N2)τ2+ (n−r)tr:n
.
As presented below, only in one of the seven named cases MLEs of ϑ1, ϑ2, and ϑ3 exist simultaneously, which is similar to the result presented by Balakrishnan, Kundu, Ng, and Kannan (2007) or rather Balakrishnan (2009):
Case 1: N1 =r It holds that
∂
∂ϑ2log L(ϑ1, ϑ2, ϑ3)
= ∂
∂ϑ3log L(ϑ1, ϑ2, ϑ3)
= 0, hence ˆϑ2 and ˆϑ3 do not exist.
Case 2: N2 =r It holds that
∂
∂ϑ1 log L(ϑ1, ϑ2, ϑ3)
= nτ1
ϑ21 >0 ∀ϑ1∈(0,∞),
∂
∂ϑ3
log L(ϑ1, ϑ2, ϑ3)
= 0, hence ˆϑ1 and ˆϑ3 do not exist.
Case 3: N1 =N2 = 0 It holds that
∂
∂ϑ1
log L(ϑ1, ϑ2, ϑ3)
= nτ1
ϑ21 >0 ∀ϑ1 ∈(0,∞),
∂
∂ϑ1
log L(ϑ1, ϑ2, ϑ3)
= n(τ2−τ1)
ϑ21 >0 ∀ϑ2 ∈(0,∞), hence ˆϑ1 and ˆϑ2 do not exist.
Case 4: N1+N2 =r, N1, N2 ̸= 0 It holds that
∂
∂ϑ3
log L(ϑ1, ϑ2, ϑ3)
= 0, hence ˆϑ3 does not exist.
Case 5: 1≤N1 ≤r−1, N2 = 0 It holds that
∂
∂ϑ2
log L(ϑ1, ϑ2, ϑ3)
= (n−N1)(τ2−τ1)
ϑ22 >0 ∀ϑ2∈(0,∞), hence ˆϑ2 does not exist.
Case 6: N1= 0, 1≤N2 ≤r−1 It holds that
∂
∂ϑ1
log L(ϑ1, ϑ2, ϑ3)
= nτ1
ϑ21 >0 ∀ϑ1 ∈(0,∞), hence ˆϑ1 does not exist.
Case 7: 1≤N1,N2 andN1+N2≤r−1
As mentioned above, we receive in this case withϑ:= (ϑ1, ϑ2, ϑ3)⊤
∂
∂ϑ1 log L(ϑ)
=−N1
ϑ1 + 1 ϑ21
"N1 X
i=1
ti:n+ (n−N1)τ1
#
= 0! ϑ1>0
⇐⇒ ϑ1= PN1
i=1ti:n+ (n−N1)τ1
N1 =:ϑ∗1,
∂
∂ϑ2
log L(ϑ)
=−N2 ϑ2
+ 1 ϑ22
N1X+N2
i=N1+1
ti:n−(n−N1)τ1+ (n−N1−N2)τ2
= 0!
ϑ2>0
⇐⇒ ϑ2=
PN1+N2
i=N1+1ti:n−(n−N1)τ1+ (n−N1−N2)τ2 N2
=:ϑ∗2,
∂
∂ϑ3
log L(ϑ)
=−r−N1−N2 ϑ3
+ 1 ϑ23
Xr i=N1+N2+1
ti:n−(n−N1−N2)τ2+ (n−r)tr:n
= 0!
ϑ3>0
⇐⇒ ϑ3= Pr
i=N1+N2+1ti:n−(n−N1−N2)τ2+ (n−r)tr:n
r−N1−N2
=:ϑ∗3.
If we denote the Hessian matrix of a functiong:Rr→Rina∈Rr,r ∈N,g∈C2(Rr), by (Hessg)(a)∈Rr×r, i.e.
(Hessg)(a)j,k := ∂2g(y)
∂yj∂yk
y=a forj, k∈ {1, . . . , r}, a∈Rr, we receive that the Hessian inϑ∗ := (ϑ∗1, ϑ∗2, ϑ∗3)⊤ becomes
Hess log(L) (ϑ∗) =
−(AN(1)13)2 0 0 0 −(AN(2)23)2 0 0 0 −(r−N(A1(3)−N)22)3
, (1.10)
where
A(1) :=
N1
X
i=1
ti:n+ (n−N1)τ1 >0,
A(2) :=
NX1+N2
i=N1+1
ti:n−(n−N1)τ1+ (n−N1−N2)τ2 >0, A(3) :=
Xr i=N1+N2+1
ti:n−(n−N1−N2)τ2+ (n−r)tr:n>0.
Therefore, the Hessian in (1.10) is negative definite, which ensures the existence of local maximum inϑ∗. Furthermore, it holds that
log L(ϑ)
= log
n!
(n−r)!
−N1log(ϑ1)−N2log(ϑ2)−(r−N1−N2) log(ϑ3) +A7(ϑ)
ϑ1,ϑ2,ϑ3→0
−→ −∞,
ϑ1,ϑ2,ϑ3→∞
−→ −∞,
which implies that the likelihood function attains a global maximum in ϑ∗ and we receive the MLEs as
ϑˆ1= PN1
i=1ti:n+ (n−N1)τ1
N1
, ϑˆ2=
PN1+N2
i=N1+1ti:n−(n−N1)τ1+ (n−N1−N2)τ2
N2 ,
ϑˆ3= Pr
i=N1+N2+1ti:n−(n−N1−N2)τ2+ (n−r)tr:n r−N1−N2
,
(1.11)
which coincides with the results for exponential step-stress tests with a single stress factor by Balakrishnan (2009) and especially by Kateri and Kamps (2017).
1.2 Step-stress testing with an arbitrary number of stress factors and stress levels
Now, we want to generalize the model of Section1.1 to arbitrary numbers of stress factors as well as to arbitrary numbers of respective stress levels. The occurrence of nonuniform numbers of stress levels per factor in particular makes a more complicated notation neces- sary.
1.2.1 The model
In a step-stress experiment applying nsf∈N different kinds of stress with n(k)L ∈N stress levels for every stress k ∈ {1, . . . , nsf}, we will again denote the respective stress level by Sk,l, k ∈ {1, . . . , nsf}, l ∈ {1, . . . , n(k)L }, and the stress step, i.e., all current stress levels combined, as
S(i1,...,insf):= S1,i1, . . . , Snsf,in
sf
, (i1, . . . , insf)∈∆⊆
nsf
×
k=1n1, . . . , n(k)L o ,
where ∆ denotes the index set of all stress level combinations of the experiment. The total number of stress steps is denoted by nstep. Just like in the bivariate case, we start the experiment with n ∈N test objects and successively alter the stress level of one factor at predefined change points τi, i ∈ {1, . . . , nstep−1}, with τi < τi+1, i ∈ {1, . . . , nstep−2}, starting with the first stress factor. Then, every other stress factor is changed, before the first stress factor is changed again. Stress factors for which the maximal stress level is reached remain constant. The test proceeds until the termination point Tr:n, the r-th failure time, if we consider type-II censoring after r observed failures, r ∈ N, r ≤n. The termination point is random in this case. Note that ∆ is a proper subset of the Cartesian
product since not all tuples occur as indices of stress level combinations like the tuple (1,2) for the bivariate step-stress. Since we assume that the stress levels are increased successively as it is common practice in the conduction of experiments, the total number of stress steps is given by
nstep=|∆|=
nsf
X
k=1
n(k)L −1 + 1.
Furthermore, we state a practical scenario where the stresses are ordered from high to low by their number of stress levels, i.e.,n(1)L ≥n(2)L ≥ · · · ≥n(nLsf). The stress factor for which the most stress levels are scheduled is considered as stress 1 and so on. In that case, a suitable enumeration for the stress steps is given by the function
ζ: ∆→ {1, . . . , nstep},(i1, . . . , insf)7→
nsf
X
k=1
(ik−1) + 1 =
nsf
X
k=1
ik−nsf+ 1, which is bijective with inverse
ζ−1:{1, . . . , nstep} →∆, m7→jm−1 nsf
k + 1
· 1 1 . . . 1 +
(m−1) modX nsf
k=1
ek, whereek := 0 · · · 0 1 0 · · · 0
∈R1×nsf denotes the k-th unit vector, 1≤k≤nsf. The random number of failures on step i, i ∈ {1, . . . , nstep}, is defined as Ni, and we set the cumulated number of observed failures over the steps as
ρi:=
Xi j=1
Nj, i∈ {1, . . . , nstep−1}, and ρ0:= 0, ρnstep :=r.
A schematic test procedure for the generalized case can be found in Figure1.3.
1.2.2 Distribution, life-stress model, and inference
The CE-assumption from Section 1.1can directly be transferred to the case of step-stress tests with an arbitrary number of stress factors and levels. We denote the cumulative distribution function of the lifetimes of the tested units on stress stepSx,x∈∆, byFζ(x) with support (0,∞) and assume that all distribution functions Fi, i ∈ {1, . . . , nstep}, are members of the same scale family of distributions as seen above. Then, we receive the cumulative distribution function of the lifetime of a test object in a multivariate step-stress test under the CE-model as
G(t) =
F1(t), 0< t≤τ1,
Fi
t+ϑi
i−1P
j=1
τj−τj−1
ϑj −τi−1
, τi−1 < t≤τi, 2≤i≤nstep−1, Fnstep
t+ϑnstep
nstepP−1 j=1
τj−τj−1
ϑj −τnstep−1
, τnstep−1 < t <∞.
The corresponding density function can be derived analogously. If we consider the distribu- tion functions Fi as exponential distributions with scale parametersϑi, i∈ {1, . . . , nstep},
time stress
step 1 S(1,...,1)
step 2 S(2,1,...,1)
step 3 S(2,2,...,1)
stepζ(x)
Sx=S(i1,...,insf)= (S1i1, . . . , Snsfinsf)
stepnstep
S
n(1)L ,...,n(nLsf)
τ1 τ2 τ3 τζ(x)−1 τζ(x) τnstep−1 tr:n
· · ·
· · ·
· · ·
· · ·
N1 N2 N3 Nζ(x) r−ρnstep−1
Fig. 1.3:Test procedure for step-stress testing with nsf stress factors withn(k)L stress levels for the k-th stress factor.
we receive the cumulative distribution function and the probability density function of the lifetime distribution in the arbitrary multivariate step-stress model as
G(t) =
1−e−ϑt1, 0< t≤τ1,
1−e−
1 ϑi
t+ϑiPi−1 j=1
τj−τj−1 ϑj −τi−1
, τi−1 < t≤τi, 2≤i≤nstep−1, 1−e−
1 ϑnstep
t+ϑiPnstep−1 j=1
τj−τj−1
ϑj −τnstep−1
, τnstep−1 < t <∞,
(1.12) and
g(t) =
1
ϑ1e−ϑt1, 0< t≤τ1,
1 ϑie−
1 ϑi
t+ϑiPi−1 j=1
τj−τj−1 ϑj −τi−1
, τi−1 < t≤τi, 2≤i≤nstep−1,
1 ϑnstepe−
1 ϑnstep
t+ϑiPnstep−1 j=1
τj−τj−1
ϑj −τnstep−1
, τnstep−1 < t <∞.
(1.13) The assumption of a log-linear link function from (1.7) or rather (1.8) can be generalized as follows. For the enumeration functionζ: ∆→ {1, . . . , nstep}, the steps of the conducted test in terms of the respective log-life relationship between the stress levels and the scale parameter of the exponential distribution modeling the failure time under the prevailing
stress can be written as
Step 1
S(1,...,1)
: log(ϑ1) =β0+
nsf
X
k=1
βkSk,1, ...
Stepζ(x)
Sx=(i1,...,insf)
: log(ϑζ(x)) =β0+
nsf
X
k=1
βkSk,ik, ...
Step nstep
S
n(1)L ,...,n(nLsf)
: log(ϑnstep) =β0+
nsf
X
k=1
βkSk,n(k) L
,
(1.14)
where β = (β0, β1, . . . , βnsf)⊤ ∈ Rnsf+1 denotes the constant parameter vector of the life- stress model.
Inferential results on the scale parameters of the exponential lifetime distributions on the respective stress steps follow the same structure as deduced in Section 1.1. Using the representations in (1.12) and (1.13), we are able to derive the likelihood function analogously to (1.9) for the sample of ordered failure times t1,...,r:n. The MLE ˆϑ= ( ˆϑ1, . . . ,ϑˆnstep)⊤ of ϑ= (ϑ1, . . . , ϑnstep)⊤ exists if and only if it holds that
1≤N1, . . . , Nnstep−1≤r−nstep+ 1 and ρnstep−1 ≤r−1.
With
φ1(t1,...,ρ1:n) :=
ρ1
X
j=1
tj:n+ (n−ρ1)τ1,
φi(tρi−1+1,...,ρi:n) :=
ρi
X
j=ρi−1+1
tj:n−(n−ρi−1)τi−1+ (n−ρi)τi, 2≤i≤nstep−1,
φnstep(tρnstep−1+1,...,r:n) :=
Xr j=ρnstep−1+1
tj:n−(n−ρnstep−1)τnstep−1+ (n−r)tr:n, we receive the MLE as
ϑˆi = φi(tρi−1+1,...,ρi:n) Ni
, 1≤i≤nstep−1, ϑˆnstep = φnstep(tρnstep−1+1,...,r:n) r−ρnstep−1
, (1.15) which again coincides with the results by Kateri and Kamps (2017). The proof can be adapted from the situation of two stress factors.
The criteria necessary to assess the quality of a specific test design are topic of the following section. After an introduction to the theory on the asymptotic variance and the delta method, the most important optimality criteria for ALT plans are presented.
1.3 Optimality criteria for test plans
In order to compare different test plans and to find optimal test designs in specific situa- tions, relevant criteria are necessary. Since the aim of ALT is to estimate some unknown quantity of interest in the underlying model, e.g., parameters, expected values, or quantiles,
