STATISTICAL INFERENCE IN ACCELERATED LIFE TESTING WITH GEOMETRIC PROCESS MODEL. A Thesis. Presented to the. Faculty of. San Diego State University

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1 STATISTICAL INFERENCE IN ACCELERATED LIFE TESTING WITH GEOMETRIC PROCESS MODEL A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Statistics by Shan Huang Spring 2011

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3 iii Copyright c 2011 by Shan Huang

4 iv ABSTRACT OF THE THESIS STATISTICAL INFERENCE IN ACCELERATED LIFE TESTING WITH GEOMETRIC PROCESS MODEL by Shan Huang Master of Science in Statistics San Diego State University, 2011 In reliability analysis, accelerated life testing is the most common way to assess a product s life. Under such test settings, products are tested at higher-than-usual levels of stress to induce early failure. The goal of accelerated life analysis is to utilize the test data to extrapolate a product s life distribution and its associated parameters at a normal stress level. The current study introduces the geometric process model for the analysis of accelerated life testing with an exponential life distribution under constant stress. The geometric process describes a simple monotone process and has been applied to a variety of situations such as the maintenance problems in engineering. By assuming that the lifetime under increasing stress levels forms a geometric process, we derive the maximum likelihood estimators of the exponential parameter under three different censoring schemes: no censoring, Type I and Type II censoring. For each censoring type, we also derive confidence intervals for the parameters using both asymptotic distribution and the parametric bootstrap method. The performance of the estimators is evaluated by a simulation study with different pre-fixed parameters. The study also considers whether the assumption of geometric process is satisfied under a wide-acknowledged log-linear relationship between life and stress. Some simulated numerical examples are presented to illustrate how to pre-design the stress levels so that the geometric process model could be applied. We also use these numerical examples to compare the performance of the proposed geometric model and the traditional failure time regression model. The results of simulation studies, as well as the advantage of the proposed model are summarized in the conclusion part, and the prospect of future works is briefly discussed.

5 v TABLE OF CONTENTS PAGE ABSTRACT... iv LIST OF TABLES... vii ACKNOWLEDGEMENTS... viii CHAPTER 1 INTRODUCTION Introduction Purpose of the Study Format of the Thesis LITERATURE REVIEW Accelerated Life Testing Accelerated Life Model Parameter Estimation The Asymptotic Properties of MLE MLE of Failure-Time Regression Model Geometric Process GP MODEL IN ALT WITH COMPLETE DATA Model Description Estimation Asymptotic Confidence Interval Bootstrap Confidence Interval Simulation Study GP MODEL IN ALT WITH TYPE I CENSORED DATA Model Description Estimation Asymptotic Confidence Interval Bootstrap Confidence Interval Simulation Study GP MODEL IN ALT WITH TYPE II CENSORED DATA... 23

6 vi 5.1 Model Description Estimation Asymptotic Confidence Interval Bootstrap Confidence Interval Simulation Study JUSTIFICATION OF THE GP ASSUMPTION IN ALT Relationship Between Stress and Life Comparison of the Proposed Model and Failure Time Regression Model CONCLUSION BIBLIOGRAPHY APPENDIX R CODE FOR SIMULATION STUDIES... 41

7 vii LIST OF TABLES PAGE Table 1.1. Life Characteristic for Common Life Distributions... 2 Table 3.1. Parameter Estimation for the Complete Simulated Sample with λ = Table 3.2. Parameter Estimation for the Complete Simulated Sample with λ = Table 4.1. Parameter Estimation for Type I Censored Simulated Sample with λ = Table 4.2. Parameter Estimation for Type I Censored Simulated Sample with λ = Table 5.1. Parameter Estimation for Type II Censored Simulated Sample with λ = Table 5.2. Parameter Estimation for Type II Censored Simulated Sample with λ = Table 6.1. Comparison Between the GP and the FTR Model with Complete Data Table 6.2. Comparison Between the GP and the FTR Model with Type I Censored Data Table 6.3. Comparison Between the GP and the FTR Model with Type II Censored Data... 35

8 viii ACKNOWLEDGEMENTS I would like to express my gratitude to my thesis advisor, Professor Jianwei Chen, for his guidance throughout the development of this thesis. His patience, insights and expertise enabled me to develop an understanding of the subject. I offer my sincere thanks to Prof. Juanjuan Fan and Prof. Tao Xie for serving on my advisory committee and for their guidance and cooperation as thesis committee member. Their advice and suggestions were invaluable. I am also grateful for the help and encouragement of my parents and my boyfriend, and I thank all of those who supported me in any respect during the completion of the thesis.

9 1 CHAPTER 1 INTRODUCTION 1.1 INTRODUCTION Life testing is the most direct way to assess the life characteristics of a product, system or component. In traditional life testing and reliability experiments, researchers used to analyze time-to-failure data obtained under normal operating conditions in order to quantify the product s failure-time distribution and its associated parameters. However, such life data is more and more difficult to obtain as a result of the great reliability of today s products, and the small time period between design and release. This problem has motivated researchers to develop new life testing method and obtain timely information on the reliability of product components and materials. Accelerated life testing (ALT) is then adopted and widely used in manufacturing industries. In such testing situations, products are tested at higher-than-usual levels of stress (e.g., temperature, voltage, humidity, vibration or pressure) to induce early failure. The life data collected from such accelerated tests is then analyzed and extrapolated to estimate the life characteristics under normal operating conditions. Three types of stress loadings are usually applied in accelerated life tests: constant stress, step stress and linearly increasing stress. The constant stress loading, which is a time-independent test setting, has several advantages over the time-dependent stress loadings. For example, most products are assumed to operate at a constant stress under normal use. Therefore, a constant stress test mimics actual use. Besides, it is easier to run and to quantify a constant stress test. In the current study, we only discuss the application of constant stress in accelerated life testings. Censoring always occur in accelerated testings. Tests are stopped before all items fail in order to reduce test time and expense. In Type I censoring, items at each stress level start to operate at the same time, and the testing is terminated at a prescribed time t with a random number of observed failures. In Type II censoring, items at each stress level also start to operate at the same time, but the testing is terminated upon the r th failure with a random time t (r). Both of the two censoring schemes are widely applied and discussed, with Type I common in practice and Type II common in theoretical literature. In this study, we discuss the accelerated life testings under three different censoring schemes: no censoring, Type I and Type II censoring.

10 2 Most accelerated test models have two components: a life distribution for a unit s behavior at a constant stress level, and an assumed relationship between the life characteristics and the stress factor. The exponential distribution is one of the most extensively used life distributions in the areas of life testing and reliability. For one reason, it is often a reasonable approximation for the distribution of lifetime for a complex electronic system [22]. For another, it could provide insight into other distributions like the Weibull and lognormal. Other commonly used life distributions include normal, lognormal, Weibull and extreme value distributions. For the sake of simplicity, we assume the underlying life distribution is exponential in our discussion. The relationship between the life distribution (or the life distribution characteristic under consideration) and the stress levels is another essential component of the accelerated test model. It expresses the effect of changing factors like temperature, voltage, humidity on a product s failure time. Most relationships are formulated with a general log-linear expression between the life characteristic and the (possibly transformed) accelerating factors [23]. The selection of the life characteristic always depends on the assumed underlying life distribution. For example, when considering the exponential distribution, the mean life is chosen to be the life characteristic that is stress dependent. When considering the lognormal distribution, the median life is the typical life characteristic. Each specified accelerated life model relates life characteristic to the explanatory variable (or stress factor) in some particular way. A more detailed discussion of life and stress relationship will be given in Chapter 6. Table 1.1 lists some common life distributions and the corresponding life characteristics. Table 1.1. Life Characteristic for Common Life Distributions Distribution Notation Density Function Life Characteristic Meaning Exponential Exp(λ) f(x λ) = λe λx, x 0 1 λ Mean life Weibull W (α, β) f(x α, β) = β α β x β 1 e ( x α )β, x 0 α 63.2% life Lognormal LN(µ, σ 2 ) f(x µ, σ) = 1 2πσx e 1 2σ 2 (log x µ)2, x 0 e µ Median life Once a life distribution and a life-stress relationship have been selected, the distribution parameters that govern the life characteristics need to be determined. Various statistical methods have been applied to estimate the parameters and their confidence limits of an accelerated testing model; two of them are generally used by researchers: graphical estimation and maximum likelihood estimation. Graphical method is a simple and easy way to perform the estimation. It first estimates the parameters of the life distribution at each individual stress level, and then plots the life

11 3 characteristic vs. stress on a paper that linearizes the assumed life-stress relationship. The parameters of the life-stress relationship are then estimated from the plot. Clearly, the graphical method treats the life distribution and the life-stress relationship separately. One of the drawbacks of this method is that in some problems, lots of the life data are censored; there is even no failure occurs at lower stress levels. In such situations, the standard techniques of graphical estimation for the parameters cannot be used, since the values of the censored observations are not known. Maximum likelihood method is a better approach for fitting models to censored data. Using the MLE method, the life distribution and the life-stress relationship can be treated as one complete model that describes both. The life data that are observed to have an exact failure time and are censored to have a partially known value are included in the likelihood function. As we discuss in Chapter 2, a maximum likelihood estimator has some good properties including the asymptotic normality. Since the MLE method is straightforward, and applies to most theoretical models and censored data, researchers prefer to use it when fitting an accelerated testing model. 1.2 PURPOSE OF THE STUDY In this study, we introduce the geometric process model for the analysis of accelerated life testing with different types of censoring and an exponential life distribution under constant stress. The geometric process is a simple monotone process introduced by Lam [11, 12] and has been applied to a wide field of maintenance problems in engineering. In brief, a geometric process describes a stochastic process {X n, n = 1, 2,... where there exists a real-valued a > 0 such that{a n 1 X n, n = 1, 2,... forms a renewal process. We have reason to believe that in an accelerated testing, lifetime will be stochastically decreasing with respect to increasing stress levels. Therefore, the geometric process model is a natural approach to study such a problem. Suppose {X k, k = 1, 2,..., s are lifetimes under each stress level and X 1 follows an underlying life distribution Exp(λ). We assume that {X k, k = 1, 2,..., s is a geometric process with ratio a. It can be shown that the life distribution at k th stress level is Exp(a k 1 λ). In that case, the likelihood function of the exponential geometric process contains only two parameters: a and λ. The objective of this study is to develop inference for those two parameters of the corresponding geometric process model by the maximum likelihood estimation. The current study also concerns with the statistical properties of maximum likelihood estimators of the model parameters. The asymptotic equivalence of ML estimators based on censored data have been discussed by Escobar and Meeker [8]and Bhattachary [4] (please refer to Chapter 2 for a detailed review). We derive the asymptotic confidence intervals for

12 4 estimators of the geometric process model under the accelerated testing framework, and compare them with the CIs that are obtained by the parametric bootstrap method. Simulation experiments are also conducted based on three types of censored data to assess how the asymptotic theory applies with small to moderate sample sizes as in practice. 1.3 FORMAT OF THE THESIS The rest of the thesis is organized as follows. Chapter 2 reviews the literature related to models and methodologies used in accelerated life tests, as well as theories and applications of geometric process developed since Chapter 3 to Chapter 5 discuss the analyses of accelerated life testing with geometric process model based on complete, Type I censored and Type II censored data, respectively. In each of the three chapters, we describe the proposed model, present the maximum likelihood estimates of the parameters and then derive their asymptotic and bootstrap confidence intervals. We also evaluate the performance of confidence intervals in terms of probability coverages through a simulation study. Chapter 6 briefly describes the log-linear relationship between a life characteristic and the stress level. It also discusses how the assumption of geometric process is satisfied under this relationship. Two common life-stress relationship models: Arrhenius and Inverse power models are presented to illustrate how to pre-fix stress levels to fulfill the geometric process assumption. A numerical example compares the geometric process model and the traditional failure time regression model under certain stress levels. Chapter 7 contains our concluding remarks of the current study and a prospect of further researches.

13 5 CHAPTER 2 LITERATURE REVIEW 2.1 ACCELERATED LIFE TESTING Accelerated test experiments have been used in manufacturing industries for many decades. A vast and growing literature on statistical methods for accelerated life testing could be found since the late 1960 s. The main purpose of the statistical researches is to utilize test data that obtained under high stress levels to estimate the life of the product at lower stress levels encountered in normal use. In this section, we introduce the most commonly used accelerated life model and the maximum likelihood method for parametric estimation. The asymptotic properties of the ML estimators are discussed, and some research trends in the statistical inference of ALT are summarized Accelerated Life Model A simple, commonly used model that characterizes the effect of explanatory variables on lifetime is failure-time regression model. With this model, the life time is assumed to follow some distribution such as the exponential, the Weibull or the lognormal distribution. Given the assumed distribution, the accelerated life model is described by the dependence of distribution parameter on the applied stresses. For example, if the life time is described by a Weibull distribution with scale parameter α and shape parameter β, typically for parametric regression model, the parameter α is then modeled as a function of covariates and the parameter β is held fixed (see [10]). As with the model of exponential distribution f(x λ) = λe λx, which is a special case of Weibull distribution, 1/λ will be reparameterized with stress variables and regression coefficients. In general, if the life time is described by a stochastic model f(x θ) with parameter θ, then the life characteristic, denoted by g(θ), can be expressed as a function of the independent variables and the corresponding coefficients: g(θ) = φ(z; γ 0, γ 1,...), (2.1) where z is derived covariates; φ(.) is a specified functional form. Most published case studies of accelerated life testing choose the form φ(.) = exp(.), since it takes only positive values and thus guarantees θ > 0 for all possible z (see [10]). In that case, equation (2.1) could be

14 6 written as log [g(θ)] = γ 0 + γ 1 z, (2.2) if a single stress factor is applied in the life testing. This relationship indicates that the logarithm of a life characteristic is a linear function of the stress-related covariates. In particular, as Nelson [22] stated in his book, for the exponential life distribution with single stress, the mean life 1/λ is log 1 λ = γ 0 + γ 1 z. (2.3) Parameter Estimation Statisticians generally prefer the method of maximum likelihood to obtain parameter estimates when the life data are censored. In the current section, we first review the likelihood construction and the asymptotic theories for ML estimates applied to censored data, and then extend the discussion to failure-time regression models where explanatory variable (in our study, the stress factor) is incorporated. The construction of likelihood function of censored data has been discussed in numerous textbooks and journal articles (see [1, 7, 10, 22] for example). In a typical life test, n items are placed under observation and each failure time is recorded. The test terminates either at a pre-determined fixed time t (Type I censor) or after a fixed number of sample items fail (Type II censor). Let f(x) be the PDF and F (x) the CDF for chosen life distribution with parameter θ, and (n r) items survived beyond the time of termination. In both type I and type II censoring, the likelihood function may be written as [ r ] n! L(θ) = f(x (i) ) [1 F (x T )] n r, (2.4) (n r)! i=1 where in type I censoring, the time of termination x T = t, and in type II censoring x T = x (r). Associated with the likelihood function is the efficient score vector U(θ) = ln L(θ), θ and the maximum likelihood estimator ˆθ of the parameter θ verifies the equation: to maximize the likelihood function (2.4). U(ˆθ) = 0

15 2.1.3 The Asymptotic Properties of MLE 7 An important advantage of the maximum likelihood method is that under certain regularity conditions (usually met in practice), ML estimators are asymptotically normal [22]. The asymptotic variance-covariance matrix is obtained by inverting the Fisher information matrix, which is defined by [ ] 2 I(θ) = E θ ln L(θ). θ2 The asymptotic theory of ML can not be directly applied in the Type II censoring scheme, since lifetime are not independent due to the pre-fixed number of failures. Bhattachary [4] provided some procedures to establish the limiting normality and uniform strong convergence of ML estimators, and showed that using certain sets of regularity conditions, maximum likelihood estimation of parameters from Type II censored samples are consistent, asymptotically normally distributed and efficient. Escobar and Meeker [8] studied the case of location-scale distributions, and further showed that although the Fisher information matrices for Type I and Type II censoring proceed differently, under standard regularity conditions, the information matrices obtained are asymptotically equivalent MLE of Failure-Time Regression Model When stress factors are incorporated, as we discussed earlier in this Chapter, θ can be expressed as an assumed function of the stress factors z and their coefficients γ 0, γ 1,.... Under a certain stress level z k, the distribution parameter θ k is θ k = θ k (z k ; γ 0, γ 1,...). Consequently, the likelihood function in (2.4) can be written as [ rk ] n k! L(θ k ) = L(γ 0, γ 1,... x k ) = f(x k(i) ) [1 F (x kt )] n k r k. (2.5) (n k r k )! The analysis reduces to the inferences of γ 0, γ 1,... from life tests and the estimation of the parameter θ at normal stress level. The maximum likelihood estimates ˆγ 0, ˆγ 1,... are the coefficient values that maximize the sample log likelihood lnl(γ 0, γ 1,...), and are usually obtained by iterative numerical optimization methods (see Nelson [22]). According to the asymptotic theory of MLE, the joint sampling distribution of ˆγ 0, ˆγ 1,... converges to a joint normal distribution. Nelson [22] presented in detail the calculation of sample likelihood of a i=1 set of data and the ML estimates of model coefficients. In accelerated life studies, researchers are more concerned with the life distribution under the normal operating condition. The life characteristics g(θ) are functions of the

16 8 regression parameters γ 0, γ 1,.... Therefore, the ML estimates ˆθ of θ are ˆθ = g 1 φ( ˆγ 0, ˆγ 1,...). The approximate variance and confidence intervals for θ are obtained by delta method based on the asymptotic property of functions of ML estimators. A detailed illustration could be found in [10, 20, 22]. The failure-time regression model with maximum likelihood method has been extensively studied in the literature. Evans [9] considered the exponential distribution with λ related to stress by the Arrhenius equation, which is a special case of the log-linear relationship. He derived the ML estimates of λ at specified stress and used asymptotic theory to obtain the confidence limits. McCool [17] treated a 2-parameter Weibull distribution with the scale parameter a power function of stress. A numerical scheme was applied to solve the nonlinear simultaneous equations for the ML estimates of the shape parameter and the stress-life exponent. Watkins [25] discussed how to obtain MLE of parameters in a Weibull log-linear model and illustrated the numerical approach by two examples. More recently, Watkins [24] studied a variant of the basic Type II censoring scheme, and considered the statistical properties of maximum likelihood estimators of Weibull log-linear model under the certain design. Despite the popularity and wide application of the failure-time regression model, researchers are still seeking development spaces for the statistical methods of accelerated life test. Watkins [26] argued that even though the log-linear function is just a simple reparameterization of the original parameter θ of the life distribution, from a statistical perspective, it is preferable to work with the original parameters. Inspired by Watkins, Balakrishnan et al [2], and Balakrishnan and Xie [3] considered the exponential lifetimes under the simple step-stress experiment terminated by Type II censoring and Type I hybrid censoring schemes, respectively. Both articles implemented the maximum likelihood method to directly estimate the scale parameter θ 1, θ 2 at each of the two stress levels. Different types of confidence intervals are also derived and compared by simulation studies. Their studies are limited to two stress levels since a model with more stress levels demands an intensive computation. Although the accelerated life test with step-stress is beyond the scope of the present discussion, the above studies provide a new perspective of modeling the accelerated life data with an underlying exponential distribution. Instead of developing inferences for the parameters γ 0 and γ 1 of the log-linear link function, it is recommended to derive the MLEs of the unknown parameters of life distribution directly, and construct their confidence intervals. In the present study, we introduce the geometric process model to study the lifetime data

17 under s constant stress levels with different censoring schemes, and develop inferences of the original life parameter in a simple yet precise way GEOMETRIC PROCESS The concept of geometric process is first introduced by Lam in 1988 [11, 12] when he studied the problem of repair replacement. The geometric process is defined as follows (see Lam [14]): A stochastic process {X n, n = 1, 2,... is said to be a Geometric Process (GP), if there exists a real-valued a > 0 such that {a n 1 X n, n = 1, 2,... forms a renewal process. The positive number a is called the ratio of the GP. It is clear to see that a GP is stochastically increasing if 0 < a < 1; it is stochastically decreasing if a > 1. Therefore, the GP is a natural approach to analyze data from a series of events with trend. It can be shown that if {X n, n = 1, 2,... is a GP and the p.d.f. of X 1 is f with mean λ and variance σ 2, then the p.d.f. of X n will be given by a n 1 f(a n 1 x), with E(X n ) = λ/a n 1 and V ar(x n ) = σ 2 /a 2(n 1). Thus, a, λ and σ 2 are three important parameters of a GP. In our case of the exponential distribution, two parameters will be estimated, a and λ. The statistical inference for geometric process is studied in both nonparametric and parametric ways. Lam introduced least square and modified moment estimation of parameters for GP, and studied the asymptotic normal properties of these estimators (see [14]). Lam and Chan [15] derived the maximum likelihood estimators of a, λ and σ 2 of the GP with lognormal distribution. They showed that the maximum likelihood estimators are more efficient than the nonparametric estimators. Chan, Lam, and Leung [5] estimated the parameters of the GP with Gamma distribution by parametric methods including maximum likelihood method along with some nonparametric methods previously proposed by Lam. They also illustrated how to derive the limiting distribution of the ML estimators and construct the confidence intervals. Chen [6] provided the Bayesian approach to the estimation of parameters in a GP with several popular life distributions including the exponential and lognormal distributions. The comparison between the Bayesian estimators and ML estimators is also examined under simulation studies. The geometric process model is first applied to investigate a repair replacement model for a one-unit deteriorating system, in which the successive survival times of the system are stochastically non-increasing, while the consecutive repair times of the system after failure are stochastically non-decreasing (see [11, 12]). Since then, a lot of research work on different GP maintenance models has been done. For example, Lam and Zhang [16] introduced the geometric process model in the analysis of a two-component series system with one

18 10 repairman. Lam [13] studied the geometric process model for a multistate system and determined an optimal replacement policy to minimize the long-run average cost per unit time. Zhang [28] used the geometrical process to model a simple repairable system with delayed repair. Large amount of studies in maintenance problems and system reliability have shown that a geometric process model is a good and simple model for analysis of data with a single trend or multiple trends. So far, there is no study that utilizes the geometric process in the analysis of accelerated life test with censored data. The current study aims to introduce the geometric process model to analyze a series of life data that obtained from several increasing stress levels. Statistical inference of the parameters in the GP will be made and examined through a simulation study.

19 11 CHAPTER 3 GP MODEL IN ALT WITH COMPLETE DATA 3.1 MODEL DESCRIPTION Suppose that we are given an accelerated life test with s increasing stress levels. A random sample of n identical items are placed under each stress level and start to operate at the same time. Whenever an item fails, it will be removed from the test. We denote by k the index for stress level; k = 1, 2,..., s, and denote by i the index for test item under each stress; i = 1, 2,..., n. Each item s observed failure time can be written as x ki. The geometric model for accelerated life test is based on two assumptions: Assumption 1. Lifetime at the design stress level follows a certain distribution. In the current study, it is assumed to be exponential with failure rate λ. The probability density function (PDF) is given by { λe λx if x > 0, f X (x λ) = 0 if x 0. Assumption 2. Let the sequence of random variables X 0, X 1,..., X s denote the lifetimes under each stress level, where X 0 denotes item s lifetime under the design stress at which items will operate ordinarily. We assume {X k, k = 0, 1, 2,... s is a geometric process with ratio a > 0. Based on the definition of geometric process [14], if the density function of X 0 is f, then the probability density function of X k will be given by: a k f(a k x), k = 0, 1, 2,..., s. (3.1) In our case of exponential distribution, the PDF of a test item at the k th stress level is: { a k λe ak λx if x > 0, f Xk (x λ) = 0 if x 0. (3.2) The model in (3.2) indicates that if lifetime under a sequence of increasing stress levels form a geometric process with ratio a, and if the life distribution at the design stress level is exponential with failure rate λ, then the life distribution at the k th stress level is also exponential with failure rate a k λ. This conclusion and the above mentioned assumptions will also be utilized in other censored schemes described in Chapter 4 and Chapter 5.

20 ESTIMATION In accelerated life test, lifetime under the design stress level will not be observed. Therefore, based on the observed data in a total s stress levels, the likelihood function for the exponential geometric process is given by s n L(a, λ x 1,..., x s ) = a k λ exp{ λa k x ki = = k=1 i=1 s n {a kn λ exp{ λa k x ki (3.3) k=1 i=1 s n {a kn λ n exp{ λa k x ki. k=1 i=1 It follows that l(a, λ) = = { s n log a kn λ n exp{ λa k x ki k=1 i=1 { s kn log a + n log λ λa k k=1 i=1 n x ki. (3.4) The first and second derivatives of l(a, λ) are: s l(a, λ) = a s l(a, λ) = λ 2 s l(a, λ) = a2 2 l(a, λ) = 2 a λ k=1 k=1 k=1 { kn n a λ( x ki )ka k 1, { n λ ak ( { i=1 n x ki ), i=1 kn λk(k 1)ak 2 a2 n s λ a l(a, λ) = k=1 i=1 n x ki, i=1 ka k 1 x ki, 2 l(a, λ) = sn λ2 λ. (3.5) 2 The MLEs of a and λ exist but do not have closed forms. The Newton Iterative Method is applied to obtain â and ˆλ. Below is the algorithm of Newton Iterative Method: ( ) (0) a 1. Start with an initial value of. λ

21 13 2. For the (t + 1) th iteration, t 0, ( a λ ) (t+1) = ( a λ 3. Repeat (2) until converge. ) (t) 2 l(a, λ) ( 2 l(a, λ) a 2 a λ 2 l(a, λ) 2 l(a, λ) λ a λ 2 ) 1 ( a (a,λ) (t) λ l(a, λ) l(a, λ) ) (a,λ) (t). 3.3 ASYMPTOTIC CONFIDENCE INTERVAL Let I(a, λ) denote the Fisher information matrix ] of a and λ. The observed Fisher information matrix of a and λ is Î(a, λ) = [ Î11 Î 12 Î 21 Î 22 [ ] 2 Î 11 = l(a, λ) = a2 = ns(1 + s) 2a 2 + λ [ 2 Î 12 = Î21 = [ 2 Î 22 = l(a, λ) λ2 s k=1 l(a, λ) a λ ] a=â,λ=ˆλ, where { s kn + λk(k 1)ak 2 a2 k=1 { n k(k 1)a k 2 x ki = sn λ 2. ] = s k=1 i=1 n ka k 1 x ki, i=1 n x ki, i=1, (3.6) The variances of â and ˆλ can be obtained through the observed information matrix as [ ] [ ] 1 V ar(â) I11 ˆ =. V ar(ˆλ) Iˆ 22 The 100(1 α)% asymptotic confidence intervals for a and λ are then given by [ ] â ± Z 1 α 2 V ar(â), and [ ˆλ ± Z 1 α 2 V ar(ˆλ) ], respectively. 3.4 BOOTSTRAP CONFIDENCE INTERVAL We construct the confidence intervals for a and λ based on the parametric bootstrap, using the percentile bootstrap interval method. We describe the algorithm to obtain the coverage for bootstrap CI as below. 1. Based on the original sample X ki, obtain â and ˆλ, the MLEs of a and λ, by Equation (3.5) and the Newton iterative method.

22 14 2. For b = 1, 2,..., B, based on â and ˆλ, simulate a bootstrap sample Xki, k = 1, 2,..., s; i = 1, 2,..., n, where X ki Exp(x; âkˆλ). 3. Compute â (b) and ˆλ (b), the MLEs of a and λ, by Equation (3.5) and the Newton iterative method. 4. Repeat steps 2-3 for B times and obtain â (1), ˆλ (1), â (2), ˆλ (2),..., â (B), ˆλ (B). 5. The bootstrap percentile confidence interval endpoints for a and λ are the α/2 and 1 α/2 quantiles of â (1),..., â (B) and ˆλ (1),..., ˆλ (B), respectively. 3.5 SIMULATION STUDY To evaluate and compare the performance of the methods of inference described in the preceding section, we conduct a simulation study. The R code is listed in the Appendix. We first generate a sample X ki, k = 1, 2,..., s; i = 1, 2,..., n, where X ki Exp(x; a k λ). We choose the values of the parameters a = 1.03, 1.05, 1.07 and λ = 0.5, The ratios a are chosen to be close to 1 since the decreasing trend of lifetime in practice is usually not pronounced. The number of stress levels is chosen to be s = 4 or 6; the number of test products at each stress level is n 1 = = n s = 10 or 20. The estimators and the corresponding summary statistics are obtained by our proposed model and the Newton iteration method. For a given sample with different choices of a and λ, we list the average of the ML estimations (mean), the sample standard deviation of the estimates (SE), the average of asymptotic standard error ( ˆ SE), the square root of mean squared error ( MSE) and the coverage rate of the 95% confidence interval for a and λ. The CIs are obtained by two methods: the asymptotic distribution and the parametric bootstrap. The numerical results presented in Table 3.1 and 3.2 are based on 500 simulations and 500 bootstrap replications. Table 3.1 and 3.2 summarizes the results of the estimates for a and λ. We observe that â and ˆλ estimates the true parameters a and λ quite well with relatively small mean squared errors. The estimated standard error also approximates well the sample standard deviation of the 500 estimates. For a fixed a and λ, we compare their estimates across the four different cases and find that as n and s decreases, the mean squared errors of â and ˆλ get larger. This may be because that a larger sample size results in a better large-sample approximation for the distribution of â and ˆλ, so the inference for the parameters is more precise. We also notice that the coverage probabilities of the asymptotic confidence interval are close to the nominal level and do not change much across the four cases. The parametric bootstrap confidence intervals have similar coverage rates as the approximate CIs. This result indicates that the proposed model and the asymptotic approximation work well under the situation where no censoring occurs.

23 15 Table 3.1. Parameter Estimation for the Complete Simulated Sample with λ = 0.5 a Estimator M ean SE MSE case 1: n = 20, s = 6 ˆ SE 95% CI Coverage Approx Bootstrap 1.03 â ˆλ â ˆλ â ˆλ case 2: n = 10, s = â ˆλ â ˆλ â ˆλ case 3: n = 20, s = â ˆλ â ˆλ â ˆλ case 4: n = 10, s = â ˆλ â ˆλ â ˆλ

24 16 Table 3.2. Parameter Estimation for the Complete Simulated Sample with λ = 0.25 a Estimator M ean SE MSE case 1: n = 20, s = 6 ˆ SE 95% CI Coverage Approx Bootstrap 1.03 â ˆλ â ˆλ â ˆλ case 2: n = 10, s = â ˆλ â ˆλ â ˆλ case 3: n = 20, s = â ˆλ â ˆλ â ˆλ case 4: n = 10, s = â ˆλ â ˆλ â ˆλ

25 17 CHAPTER 4 GP MODEL IN ALT WITH TYPE I CENSORED DATA 4.1 MODEL DESCRIPTION Suppose that there are s increasing stress levels and under each level n items are inspected. We denote by k the index for stress level; k = 1, 2,..., s, and denote by i the index for test item under each stress; i = 1, 2,..., n. In the Type I censored scheme, the test at each stress level terminates at time t. An item s exact failure time is observed only if its lifetime x ki t. Assume at the k th stress level we observe r k failures before the test is suspended, 0 r k n. Correspondingly, (n r k ) units survive the entire test without failing. The observed ordered failure times under the k th stress level can be written as x k(1) x k(2)... x k(rk ). Note that t is fixed in advance and r k is random. The two assumptions of the geometric process model in accelerated life testing still need to be held. As is described in Chapter 3, the PDF of a test item at the k th stress level is: { a k λe ak λx if x > 0, f Xk (x) = (4.1) 0 if x 0. Consequently, the CDF of a test item at the k th stress level is: F Xk (x) = x 0 a k λe ak λx dx (4.2) = 1 e ak λx. The probability that an item censored at time t is: S Xk (t) = 1 F Xk (t) = e ak λt. (4.3) For those (r k ) items that fail at time t, their order statistics can be denoted by X k(i) with PDF: f Xk(i) (x) = n! (i 1)!(n i)! f X k (x)[f Xk (x)] i 1 [1 F Xk (x)] n i, (4.4) i = 1, 2,..., r k. Given (4.3) and (4.4), the likelihood function of observed data at the k th stress level can be expressed as L k = n! (n r k )! f X(x k(1) )... f Xk (x k(rk ))[S Xk (t)] n r k. (4.5)

26 ESTIMATION In this section, we derive the ML estimates of a and λ from the likelihood function presented in (4.5). The substitution of (4.1) and (4.3) into (4.5) yields { ( rk ) n! L k (a, λ) = (n r k )! (λak ) r k exp λa k x k(i) + (n r k )t, i=1 0 x k(1)... x k(rk ) t. (4.6) It follows that the likelihood function of observed data in a total s stress levels is: L(a, λ) = L 1 L 2 L s { { ( s rk ) n! = (n r k )! (λak ) r k exp λ x k(i) + (n r k )t, k=1 Then it can be shown that l(a, λ) = 0 x k(1)... x k(rk ) t; 1 k s. (4.7) { s ( ) ( rk ) n! log + r k log(λa k ) λa k x k(i) + (n r k )t. (4.8) (n r k )! k=1 The first and second derivatives of l(a, λ) are: s l(a, λ) = a s l(a, λ) = λ 2 s l(a, λ) = a2 2 l(a, λ) = 2 a λ 2 s l(a, λ) = λ2 k=1 k=1 k=1 { kr k a { r k λ ak { kr k a 2 λkak 1 s l(a, λ) = λ a k=1 i=1 i=1 ( rk ) x k(i) + (n r k )t, i=1 ( rk ) x k(i) + (n r k )t, i=1 λk(k 1)ak 2 k=1 ( rk ) x k(i) + (n r k )t, i=1 { ( rk ) ka k 1 x k(i) + (n r k )t, i=1 { r k λ 2. (4.9) The MLEs of a and λ exist but do not have closed forms. The Newton Iterative Method is applied to obtain â and ˆλ. Below is the algorithm of Newton Iterative Method:

27 19 1. Start with an initial value of ( a λ 2. For the (t + 1) th iteration, t 0, ( ) (t+1) ( a a = λ λ 3. Repeat (2) until converge. ) (0). ) (t) ( 2 a 2 l(a, λ) 2 l(a, λ) l(a, λ) λ 2 a λ 2 l(a, λ) 2 λ a ) 1 ( a (a,λ) (t) λ l(a, λ) l(a, λ) ) (a,λ) (t). 4.3 ASYMPTOTIC CONFIDENCE INTERVAL Let I(a, λ) denote the Fisher information matrix] of a and λ. The observed Fisher information matrix of a and λ is Î(a, λ) = [ Î11 Î 12 Î 21 Î 22 [ ] 2 Î 11 = l(a, λ) = a2 [ 2 Î 12 = Î21 = [ 2 Î 22 = l(a, λ) λ2 s k=1 l(a, λ) a λ ] s = k=1 { kr k ] = a 2 a=â,λ=ˆλ + λk(k 1)ak 2, where ( rk ) x k(i) + (n r k )t, i=1 { ( s rk ) ka k 1 x k(i) + (n r k )t, k=1 i=1 { rk λ 2. (4.10) The variances of â and ˆλ can be obtained through the observed information matrix as [ ] [ ] 1 V ar(â) I11 ˆ =. V ar(ˆλ) Iˆ 22 The 100(1 α)% asymptotic confidence intervals for a and λ are then given by [ ] â ± Z 1 α 2 V ar(â), and [ ˆλ ± Z 1 α 2 V ar(ˆλ) ], respectively. 4.4 BOOTSTRAP CONFIDENCE INTERVAL We construct the confidence intervals for a and λ based on the parametric bootstrap, using the percentile bootstrap interval method. We describe the algorithm to obtain the coverage for bootstrap CI as below: 1. Arrange the original sample X ki in ascending order and find r k such that x k(rk ) t and x k(rk +1) > t. Obtain â and ˆλ, the MLEs of a and λ, by Equation(4.9) and the Newton iterative method.

28 20 2. For b = 1,..., B, based on â and ˆλ, simulate a bootstrap sample Xki, k = 1, 2,..., s; i = 1, 2,..., n, where X ki Exp(x; âkˆλ). 3. Find r k such that x k(r k ) t and x k(r k +1) > t. 4. Compute â (b) and ˆλ (b), the MLEs of a and λ, by Equation (4.9) and the Newton iterative method. 5. Repeat steps 2-4 for B times and obtain â (1), ˆλ (1), â (2), ˆλ (2),..., â (B), ˆλ (B). 6. The bootstrap percentile confidence interval endpoints for a and λ are the α/2 and 1 α/2 quantiles of â (1),..., â (B) and ˆλ (1),..., ˆλ (B), respectively. 4.5 SIMULATION STUDY To evaluate and compare the performance of the methods of inference described in the preceding section, we conduct a simulation study. We first generate a sample X ki, k = 1, 2,..., s; i = 1, 2,..., n, where X k Exp(x; a k λ). Similar to the simulation study in Chapter 3, we choose the values of the parameters a = 1.03, 1.05, 1.07 and λ = 0.5, The number of stress levels is chosen to be s = 6; the number of test products at each stress level is n = 10 or 20. The termination time t is set to be half or three quarters of the expected lifetime. Therefore, for λ = 0.5, we choose t = 1 or 1.5; for λ = 0.25, we choose t = 2 or 3. In each case, we compare every x ki in the generated sample with termination time t, record the minimum one as the actual observed failure time, and sort them in ascending order as x k(1)... x k(rk ). The estimations and the corresponding statistics are obtained by our proposed model and the Newton iteration method. For different choices of a and λ in the four cases, we list the average of the ML estimations (mean), the sample standard deviation of the estimates (SE), the asymptotic standard error ( ˆ SE), the square root of mean squared error ( MSE) and the coverage rate of the 95% confidence interval for a and λ. The CIs are obtained by two methods: the asymptotic distribution and the parametric bootstrap. The numerical results presented in Table 4.1 and 4.2 are based on 500 simulations and 500 bootstrap replications. From the tables, we observe that the proposed estimators â and ˆλ perform well in all the cases studied. The estimation of a is more accurate than that of λ, as seen by its negligible bias. The proposed estimator ˆλ is consistently larger than the true value of λ, which means we tend to underestimate the mean lifetime of the product at the normal stress level. The estimated standard error approximates well the sample standard deviation of the 500 estimates, with the 95% confidence interval coverage rate close to the nominal level. When the sample size is fixed, the mean squared errors of the estimators decrease as the termination time t gets larger. This is quite expected because large values of t would lead to more failures before the test ends, and thus increase the efficiency of the estimators.

29 21 Table 4.1. Parameter Estimation for Type I Censored Simulated Sample with λ = 0.5 a Estimator M ean SE MSE case 1: n = 20, t = 1, s = 6 ˆ SE 95% CI Coverage Approx Bootstrap 1.03 â ˆλ â ˆλ â ˆλ case 2: n = 20, t = 1.5, s = â ˆλ â ˆλ â ˆλ case 3: n = 10, t = 1, s = â ˆλ â ˆλ â ˆλ case 4: n = 10, t = 1.5, s = â ˆλ â ˆλ â ˆλ

30 22 Table 4.2. Parameter Estimation for Type I Censored Simulated Sample with λ = 0.25 a Estimator M ean SE MSE case 1: n = 20, t = 2, s = 6 ˆ SE 95% CI Coverage Approx Bootstrap 1.03 â ˆλ â ˆλ â ˆλ case 2: n = 20, t = 3, s = â ˆλ â ˆλ â ˆλ case 3: n = 10, t = 2, s = â ˆλ â ˆλ â ˆλ case 4: n = 10, t = 3, s = â ˆλ â ˆλ â ˆλ

31 23 CHAPTER 5 GP MODEL IN ALT WITH TYPE II CENSORED DATA 5.1 MODEL DESCRIPTION Suppose that there are s stress levels and under each level n items are inspected. Let the random variable X ki denote the lifetime of the tested product, where k is the index for stress level and i is the index for test item under each stress. In the Type II censoring scheme, the test at each stress level terminates after r failures are observed. Under the k th stress level, only the r smallest observations of the sample will be collected, whose failure time can be written as x k(1) x k(2)... x k(r). Note that r is specified in advance; the test ends at the r th failure time x k(r), and (n r) units have survived at each stress level. The assumptions of the geometric process model described in Chapter 3 imply that the PDF of a test item at the k th stress level is: { a k λe ak λx if x > 0, f X (x) = 0 if x 0. Consequently, the CDF of a test item at the k th stress level is: F X (x) = x 0 (5.1) a k λe ak λx dx (5.2) = 1 e ak λx. In Type II censoring, the information available consists of the first r order statistics at each stress level. The PDF of the order statistic X k(i) is f X(i) (x) = n! (i 1)!(n i)! f X(x)[F X (x)] i 1 [1 F X (x)] n i, (5.3) i = 1, 2,..., r. For those (n r) units that still alive when the test terminates at each stress level, all we know under Type II censoring is that each one s lifetime exceeds x k(r). The probability that an item censored at time x k(r) is 1 p(x x k(r) ) = e ak λx k(r). (5.4) Both (5.3) and (5.4) would be used in the construction of likelihood function.

32 ESTIMATION stress level is: According to equation (2.4), the likelihood function of the observed test data at the k th L k (a, λ) = { ( r ) n! (n r)! (λak ) r exp λa k x k(i) + (n r)x k(r), i=1 0 x k(1)... x k(r). (5.5) It follows that the likelihood function of observed data in a total s stress levels is { { ( s r ) n! L(a, λ) = (n r)! (λak ) r exp λa k x k(i) + (n r)x k(r), i=1 k=1 0 x k(1)... x k(r) ; 1 k s. (5.6) Then it can be shown that { s ( ) ( r ) n! l(a, λ) = log + r log(λa k ) λa k x k(i) + (n r)x k(r). (5.7) (n r)! i=1 k=1 The first and second derivatives of l(a, λ) are: { ( s r ) kr l(a, λ) = a a λkak 1 x k(i) + (n r)x k(r), k=1 i=1 { ( s r ) r l(a, λ) = λ λ ak x k(i) + (n r)x k(r), k=1 i=1 { ( 2 s r ) l(a, λ) = kr λk(k 1)ak 2 x a2 a2 k(i) + (n r)x k(r), 2 s l(a, λ) = λ2 2 s l(a, λ) = a λ k=1 k=1 k=1 { r λ 2, { ka k 1 ( r i=1 i=1 x k(i) + (n r)x k(r) ) Again, the Newton Iterative Method is applied to obtain â and ˆλ.. (5.8)