Some Inferential Results for One-Shot Device. Testing Data Analysis

Transcription

1 Some Inferential Results for One-Shot Device Testing Data Analysis

2 SOME INFERENTIAL RESULTS FOR ONE-SHOT DEVICE TESTING DATA ANALYSIS BY HON YIU SO, B.Sc., M.Phil. a thesis submitted to the Department of Mathematics & Statistics and the School of Graduate Studies of McMaster University in partial fulfilment of the requirements for the degree of Doctor of Philosophy c Copyright by HON YIU SO, March 2016 All Rights Reserved

3 Doctor of Philosophy (2016) (Mathematics) McMaster University Hamilton, Ontario, Canada TITLE: Some Inferential Results for One-Shot Device Testing Data Analysis AUTHOR: HON YIU SO B.Sc., Risk Management Science The Chinese University of Hong Kong, Hong Kong, China M.Phil., Risk Management Science The Chinese University of Hong Kong, Hong Kong, China SUPERVISOR: Prof. Narayanaswamy Balakrishnan NUMBER OF PAGES: xxv, 179 ii

4 To my wife Ms. Tian Feng To my parents Mr. Chun Pong So and Ms. Oi King Lui To the memory of my grandmother, Ms. Lin Hing Ho

5 Abstract In this thesis, we develop some inferential results for one-shot device testing data analysis. These extend and generalize existing methods in the literature. First, a competing-risk model is introduced for one-shot testing data under accelerated life-tests. One-shot devices are products which will be destroyed immediately after use. Therefore, we can observe only a binary status as data, success or failure, of such products instead of its lifetime. Many one-shot devices contain multiple components and failure of any one of them will lead to the failure of the device. Failed devices are inspected to identify the specific cause of failure. Since the exact lifetime is not observed, EM algorithm becomes a natural tool to obtain the maximum likelihood estimates of the model parameters. Here, we develop the EM algorithm for competing exponential and Weibull cases. Second, a semi-parametric approach is developed for simple one-shot device testing data. Semi-parametric estimation is a model that consists of parametric and nonparametric components. For this purpose, we only assume the hazards at different stress levels are proportional to each other, but no distributional assumption is made on the lifetimes. This provides a greater flexibility in model fitting and enables us to examine the relationship between the reliability of devices and the stress factors. Third, Bayesian inference is developed for one-shot device testing data under iv

6 exponential distribution and Weibull distribution with non-constant shape parameters for competing risks. Bayesian framework provides statistical inference from another perspective. It assumes the model parameters to be random and then improves the inference by incorporating expert s experience as prior information. This method is shown to be very useful if we have limited failure observation wherein the maximum likelihood estimator may not exist. The thesis proceeds as follows. In Chapter 2, we assume the one-shot devices to have two components with lifetimes having exponential distributions with multiple stress factors. We then develop an EM algorithm for developing likelihood inference for the model parameters as well as some useful reliability characteristics. In Chapter 3, we generalize to the situation when lifetimes follow a Weibull distribution with non-constant shape parameters. In Chapter 4, we propose a semi-parametric model for simple one-shot device test data based on proportional hazards model and develop associated inferential results. In Chapter 5, we consider the competing risk model with exponential lifetimes and develop inference by adopting the Bayesian approach. In Chapter 6, we generalize these results on Bayesian inference to the situation when the lifetimes have a Weibull distribution. Finally, we provide some concluding remarks and indicate some future research directions in Chapter 7. KEY WORDS: One-shot device testing, competing risk, multinomial data, censoring, accelerated life testing, exponential distribution, Weibull distribution, semiparametric method, proportional hazard model, EM algorithm, Bayesian approach, asymptotic method, parametric bootstrap, inequality constrained least square, point estimation, confidence interval, transformation approach. v

7 Acknowledgements I would like to express my deep appreciation to my supervisor, Prof. Balakrishnan, for accepting me as his student and suggesting many problems that I have worked on in this thesis and will work on in the future. His guidance and support during the preparation of this thesis, his encouragement, insightful discussion and suggestions have all made this work possible. I would also like to thank Dr. Childs and Dr. Viveros-Aguilera, for being in my thesis committee and giving valuable suggestions during the supervisory committee meetings. Gratitude is also expressed to the Department of Mathematics and Statistics at McMaster University for the financial support they provided throughout my study. I express my sincere thanks to my wife, Ms. Tian Feng, and my parents, Mr. Chun Pong So and Ms. Oi King Lui, for supporting me throughout my studies. Finally, I take this opportunity to express my earnest thanks to all my friends and fellow students who encouraged and inspired me. vi

8 Acronyms and abbreviations EM algorithm MLE MSE LSE ICLS c.d.f. p.d.f. r.v. MGF se MCMC expectation-maximization algorithm maximum likelihood estimate mean square error least-squares estimate inequality constrained least-squares cumulative distribution function probability density function random variable moment generating function standard error Markov Chain Monte Carlo vii

9 Notation and abbreviations I J R M IT i number of inspection times number of stress levels number of competing causes (risks) number of stress variables i-th inspection time, i = 1,, J IW i log-transformed i-th inspection time, log(it i ) IT the collection of IT i : {IT i, i = 1,, I} T rijk the Weibull lifetime of r-th cause of the k-th item under IT i and j-th stress level W rijk F Y f Y log(t rijk ) which follows an extreme value (Gumbel) distribution c.d.f. of r.v. Y which can be T rijk or W rijk p.d.f. of r.v. Y which can be T rijk or W rijk x mj stress value of m-th stress variable at j-th stress level, m = 0,, M, j = 1,, J and x 0j 1 x j the vector of stress variable at the j-th stress level, (x 0j, x 1j,, x Mj ) x 0 x j x 0 the vector of stress variable under normal operating condition j-th stress level in single stress model normal operating condition in single stress model viii

10 K ij D rij d rij D ij d ij ˆD ij S ij s ij M ij number of devices tested under IT i and x j number of devices failed due to the r-th Cause under IT i and j-th stress level realization of D rij number of simple one shot devices failed under IT i and j-th stress level realization of D ij expected number of failures at IT i in the j-th test group number of devices that survive under IT i and w j, S ij = K ij R r=1 D rij the realization of S ij number of devices failed under IT i and j-th stress level with masked causes of failure m ij αr0 αr1 α α j the realization of M ij intercept parameter of λ rj in exponential distribution for cause r coefficient of the stress of λ rj in exponential distribution for cause r the collection of αr0 and αr1 scale parameter of the Weibull distribution in simple one-shot device under the j-th test group a m a a coefficient of the m-th stress factor in semi-parametric setting the vector of parameters, (a 0,, a M ), in the link function for α j the vector of parameters, (a 0,, a M ), the coefficient of stress factors in semi-parameteric setting λ rj the failure rate in exponential distribution of cause r under x j given by λ rj = α r0 exp(α r1x j ) α rj scale parameter of Weibull distribution in the r-th cause at the j-th stress level, exp ( a r x j ) ix

11 η rj a r br shape parameter of Weibull distribution in the r-th cause at the j-th ( ) stress level, exp b r x j the vector of parameters, (a r0,, a rm ), in the link function for α rj the vector of parameters, (b r0,, b rm ), in the link function for η rj µ rj location parameter in Gumbel distribution, log(α rj ) = a r x j ν rij σ rj the standardized log-transformed inspection time, (IW i µ rj )/σ rj scale parameter in Gumbel distribution, η 1 rj = exp ( b r x ) j a the collection of a r : { a r, r = 1, 2} b the collection of b r : { b r, r = 1, 2} θ T rijk the collection of some parameters the lifetime of r-th cause of the k-th item under IT i and j-th stress level ijk δ ijk status indicator of the k-th device under IT i and j-th stress level realization of ijk p rij P r( ijk = r IT i, w j ) p 0i P.d1 T ( x; θ) T r Exp( ) Norm( ) Dir( ) ˆp p surivial probability at IT i under normal condition (reliability) probability of failure due to cause 1 given failure the mean lifetime of the device under the stress variable x and θ expected lifetime of cause r under normal condition estimation with an exponential prior estimation with a normal prior estimation with a Dirichlet prior prior belief on p rij based on past experience prior belief on p rij based on data only x

12 R(t; x; θ) the reliability of the device at time t under the stress variable x and some parameters θ R 0 (t) H(t) H 0 (t) c i baseline reliability function in semi-parametric setting cumulative hazard function baseline cumulative hazard function log-transformed baseline cumulative hazard function c i = log(h 0 (IT i ; β)) β i parameter of the function γ at IT i β the vector of parameter (β 1, β 2,, β I ) γ(β i ) (1 R 0 (IT i ))/(1 R 0 (IT i+1 )) G i I m=i γ(β m) Γ(t + 1) Γ(t + 1, exp(u)) η b F T (t) I complete (θ) I obs (θ) I missing (θ) V (ˆθ) complete gamma function upper incomplete gamma function constant shape parameter of the Weibull distribution parameter of η cdf of the Weibull distribution complete Fisher information matrix observed Fisher information matrix missing Fisher information matrix asymptotic variance-covariance matrix of the MLEs of the model parameters ˆR V ar( ) se( ) the MLE of reliability variance standard error xi

13 M Dist L( ) l( ) S( ) P H ACI P H T CI distance-based test statistic likelihood function log-likelihood function score function S(θ) = l(θ)/ θ asymptotic confidence interval under the proportional hazards model logit-transformation confidence interval under the proportional hazards model W ei T CI logit-transformation confidence interval under the Weibull model xii

14 Contents Abstract iv Acknowledgements vi Acronyms and abbreviations vii Notation and abbreviations viii 1 Introduction and Statement of Problem Motivation Types of Censoring Accelerated Life-Test One-Shot Device Testing Data Competing Risks Model EM Algorithm Semi-Parametric Model Bayesian Approach Scope of the Thesis xiii

15 2 EM Algorithm for One-Shot Device Testing with Two Competing Risks under Exponential Distribution Introduction Model Description EM Algorithm E-step M-Step Inequality Constrained Least-Square Estimate as Initial Estimate Simulation Study Comparison to Fisher-Scoring Method Convergence, Computational Time and Tolerance Robustness of Initial Values Missing Information Principle Masked Causes of Failure An Illustrative Example Concluding Remarks EM Algorithm for One-Shot Device Testing with Competing Risks under Weibull Distribution Introduction Model Description EM Algorithm M-Step E-Step Starting Point of the EM Algorithm xiv

16 3.3.4 Estimation of Reliability and Mean Lifetime Interval Estimation From Observed Fisher Information Matrix By Parametric Bootstrap Method Use of Transformation Approach Simulation Study Robustness of Initial Values Masked Causes of Failure A Goodness of Fit Test An Illustrative Example Concluding Remarks Likelihood Inference under Proportional Hazards Model for One- Shot Device Testing Introduction Model and Likelihood Function Point and Interval Estimation Testing of Proportional Hazard Rates Monte Carlo Simulation Study Illustrative Examples Concluding remarks Bayesian Approach for One-Shot Device Testing with Exponential Lifetimes under Competing Risks Introduction xv

17 5.2 Model Specification Bayesian Estimation Exponential Prior Normal Prior Dirichlet Prior Prior Belief on p rij Simulation Study Sensitivity Analysis on Prior Accuracy Masked Causes of Failure An Example from a Tumorigenicity Experiment Concluding Remarks Bayesian Approach for One-Shot Device Testing under Weibull Lifetimes with Non-Constant Shape and Scale Parameters Introduction Model Specification Bayesian Approach Laplace Prior Normal Prior Beta Prior Prior Belief on p i Simulation Study Sensitivity Analysis on Prior Accuracy Application to Modified Class H Insulation Life Data Concluding Remarks xvi

18 7 Concluding Remarks Summary Further Generalizations and Extensions A Appendix 149 A.1 Appendix for Chapter A.1.1 Derivations of Conditional Expectations A.1.2 Derivation of Updating Equations for Maximization A.1.3 Derivation of ICLS A.2 Appendix for Chapter A.2.1 Derivatives of M (δ ijk) rij (t) A.2.2 Derivations for Fisher Scoring Method A.2.3 Derivation of the Exact P -Value A.3 Appendix for Chapter xvii

19 List of Tables 1.1 An example of one-shot device testing data under stress level setting (-1, -0.1, 0.3) and inspection times 0.5, 1, 1.5 with 2 competing risks The conditional expected values of missing data, with λ rj = λ rj (α ), r = 1, 2, for different cases Parameter values used in the simulation of devices with high reliability The Bias and MSE of the parameters for devices with high reliability under EM algorithm and ICLS Quantities of interest under room temperature with ˆλ r = ˆα r0e 25ˆα r1, for r = 1, The Bias and MSE of various useful lifetime quantities of devices with high reliability under EM algorithm and ICLS Parameter values used in simulation of devices with moderate reliability Parameter values used in simulation of devices with low reliability The Bias and MSE of the parameters for devices with moderate reliability under EM algorithm and ICLS The Bias and MSE of the parameters for devices with low reliability under EM algorithm and ICLS xviii

20 2.10 The Bias and MSE of various useful lifetime quantities of devices with medium reliability under EM algorithm and ICLS The Bias and MSE of various useful lifetime quantities of devices with low reliability under EM algorithm and ICLS The comparison of the number of divergent cases, the average computational time and the tolerance of the Fisher-scoring method and the EM algorithm The number of divergent cases for the EM algorithm and the Fisherscoring method with different initial values The conditional expected values of missing data with masked causes corresponding to different cases The number of mice sacrificed (r = 0) and died (without tumour r = 1, with tumour r = 2) from the ED01 experiment data Estimates of various parameters under EM algorithm and ICLS for the mice data The parameter settings for Monte Carlo simulation Values of Bias and MSE for the parameters of interest in case of devices with high reliability Values of Bias and MSE for the parameters of interest in case of devices with moderate reliability Values of Bias and MSE for the parameters of interest in case of devices with low reliability xix

21 3.5 Values of coverage probabilities and average widths of 95% confidence intervals for some parameters of interest for various sample sizes in case of devices with high reliability Values of coverage probabilities and average widths of 95% confidence intervals for some parameters of interest for various sample sizes in case of devices with moderate reliability Values of coverage probabilities and average widths of 95% confidence intervals for some parameters of interest for various sample sizes in case of devices with low reliability Comparison between the Fisher scoring method (FS) and the EM algorithm (EM) in terms of average tolerance, computational time of converged cases and the number of divergent cases The number of motors censored (δ ijk = 0), failed due to Turn (δ ijk = 1) and failed due to Ground (δ ijk = 2) from modified Class-B insulation data set Point estimates (s.e.) and confidence intervals for different parameters and quantities of interest based on the data in Table The numbers of cases of convergence by Newton-Raphson method for the semi-parametric model under different simulation settings Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the semiparametric model with b = 0.5 under different simulation settings xx

22 4.3 Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the semiparametric model with b = 0 under different simulation settings Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the semiparametric model with b = 0.5 under different simulation settings Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the reliability with b = 0.5 under different simulation settings Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the reliability with b = 0 under different simulation settings Bias, root mean square error (RMSE), coverage probabilities (CP), and average widths (AW) of 95% confidence intervals for the reliability with b = 0.5 under different simulation settings Data on 120 one-shot devices subjected to 2 stress factors and inspected at 3 different times One-shot device testing data analysis by using the proportional hazards model, the Weibull distribution and the gamma distribution models Data on 120 one-shot devices subjected to 2 stress factors and inspected at 3 different times Parameter values used in the simulation study for devices with high reliability xxi

23 5.2 Bias and MSE of the estimates of the parameters for devices with high reliability under different estimation methods Bias and MSE of the estimates of some probabilities of interest for devices with high reliability under different estimation methods Bias and MSE of the estimates of mean lifetimes for devices with high reliability under different estimation methods Parameter values used in the simulation study for devices with moderate reliability Parameter values used in the simulation study for devices with low reliability Bias and MSE of the estimates of the parameters for devices with moderate reliability under different estimation methods Bias and MSE of the estimates of the parameters for devices with low reliability under different estimation methods Bias and MSE of the estimates of some probabilities of interest for devices with moderate reliability under different estimation methods Bias and MSE of the estimates of some probabilities of interest for devices with low reliability under different estimation methods Bias and MSE of the estimates of mean lifetimes of devices with moderate reliability under different estimation methods Bias and MSE of the estimates of mean lifetimes of devices with low reliability under different estimation methods The method of estimation with the least bias The method of estimation with the least MSE xxii

24 5.15 The method of estimation with the least bias among the methods based only on observed data The method of estimation with the least MSE among the methods based only on observed data Comparison of different prior variances c 2 in the estimate of α21 under Exp(ˆp) Modification with zero correction on p 0ij, p 1ij, p 2ij The number of mice sacrificed (r = 0) and died (without tumour r = 1, with tumour r = 2) from the ED01 experiment data The estimates of various quantities of interest under different estimation methods An example of the failure records under temperature x i, 30, 40, 50 (in C ), at inspection times IT i, 5, 10, 15 (in days), with number of units allocated, K i = 30, at every condition Parameter values used in the simulation study for devices with high reliability Bias and MSE of the estimates of parameters for devices with low reliability under different estimation methods Bias and MSE of the estimates of parameters for devices with moderate reliability under different estimation methods Bias and MSE of the estimates of parameters for devices with high reliability under different estimation methods xxiii

25 6.6 Bias and MSE of the estimates of mean lifetime under normal operating condition, T ( x 0 ), and the reliability under normal operating condition, R(, x 0 ), for devices with low reliability under different estimation methods Bias and MSE of the estimates of mean life time under normal operating condition, T ( x0 ), and the reliability under normal operating condition, R(, x 0 ), for devices with moderate reliability under different estimation methods Bias and MSE of the estimates of mean life time under normal operating condition, T ( x0 ), and the reliability under normal operating condition, R(, x 0 ), for devices with high reliability under different estimation methods The method of estimation with the least bias The method of estimation with the least MSE The method of estimation with the least bias among the methods with prior based only on observed data The method of estimation with the least MSE among the methods with prior based only on observed data Comparison of different prior variances c 2 in the estimate of α 1 under Norm(ˆp) The modified Class-H insulation life data The estimates of the model parameters and some quantities of interest for the modified Class-H insulation life data xxiv

26 List of Figures 1.1 An electro-explosive device designed by Thomas and Betts (1967) Reliability (Top) and probability density function (Bottom) of the one-shot device under normal operating conditions (x 1, x 2 ) = (25, 35) for Table 4.8 under the semi-parametric model, the Weibull distribution and the gamma distribution, along with 95% confidence intervals (dashed lines) Reliability (Top) and probability density function (Bottom) of the oneshot device under normal operating conditions (x 1, x 2 ) = (25, 35) for Table 4.10 under the semi-parametric model, the Weibull distribution and the gamma distribution, along with 95% confidence intervals (dashed lines) xxv

27 Chapter 1 Introduction and Statement of Problem 1.1 Motivation The study of one-shot device testing with competing risks under accelerated life-test (ALT) is motivated by the work of Fan et al. (2009) and Ling (2012). Fan et al. (2009) developed a Bayesian approach for highly reliable explosive devices, which is a typical one-shot device. In their simulation, they achieved highly accurate estimation results even in case of small sample sizes. Ling (2012) developed EM algorithms for estimating the model parameters from one-shot device data and also developed optimal test plans. However, in both their works, only one failure mode was considered for the one-shot devices and a competing risk model was not considered. However, many one-shot devices contain multiple components and thus could contribute to the failure of devices. For example, fire extinguishers contain valves, pressure sensors as well as foam chemicals; bullets contain gun powder, metallic cases and primers. 1

28 A failure of any one of the components may result in the failure of the device. In order to improve the quality of one-shot devices, it is therefore necessary to identify which component caused the failure in order to see whether the particular component could be improved or replaced for improving the reliability of the device. For this reason, we introduce competing risks into the one-shot device testing model. Estimation procedures by EM algorithm and Bayesian approach are then developed for this framework. Since we will also be interested in the reliability and the mean lifetime of such devices at working operating conditions, we also discuss inference for the these quantities as well. 1.2 Types of Censoring In lifetime analysis, complete data is always preferable as it gives entire information about the life characteristics of interest. However, due to constraints on time and budget of the experiment, nature of sampled objects, and so on, the data collected are inevitably censored. There are two kinds of censoring commonly encountered in practice. If the device under test fails before observable time, we define that to be left censored. Typical example includes failure of an alarm system to alert in an incident of fire. On the other hand, if the device under test survives beyond the observable time point, it is said to be right censored. For instance, computers are often replaced by a new one when they are still working, and cars are sent to junkyard when they are still drivable. Based on the nature and the form of the experiment, there are two common forms of right censoring: 1. Type-I censoring: Since it may take a long time for devices to fail in a lifetesting experiment, it is reasonable to set to a fixed termination time point 2

29 to perform the test. The number of failures observed during the experimental period in this situation is random. Even zero failure is possible within the test design; 2. Type-II censoring: To ensure enough number of failures are observed, the lifetest may be terminated when a pre-fixed number of failures have been observed. In this case, the stopping time of the experiment is random. These forms of censoring have been studied extensively in the literature. For example, Lynn (2001) developed maximum likelihood inference for left censored HIV RNA data; Mitra and Kundu (2008) analyzed left censored data from the generalized exponential distribution. For Type-I censoring, Meeker (1984) compared accelerated life-test plans for Weibull and lognormal distributions under Type-I censoring; Gouno et al. (2004) developed optimal step-stress test under progressive Type-I censoring. For the case of Type-II censoring, Balakrishnan and Chan (1992) provided different estimation methods for the scaled half logistic distribution under Type-II censoring, while Balakrishnan et al. (2007) derived point and interval estimation methods for a simple step-stress model under Type-II censoring. 1.3 Accelerated Life-Test Accelerated life-testing (ALT) has become a key tool in reliability engineering. Due to advancement in technology, industrial products have now become high quality and so their mean lifetimes are usually quite large. If such devices are placed in a lifetest under normal operating conditions, the observed failures will usually be scarce in a reasonable period of time. Consequently, the resulting statistical inference will 3

30 become inefficient in such a case. For this reason, an ALT plan, in which the devices are tested under higher stress levels, becomes essential in order to shorten the lifetimes of devices so that more failures can be observed which may result in more efficient inference. Some common stress factors that could be used for this purpose include air pressure, temperature and humidity which can be controlled easily in a laboratory setup. After estimating the model parameters from data obtained from such an accelerated life-test, the reliability engineer can then predict the life characteristics of the product, such as the mean lifetime and reliability at specific mission times, under normal operating conditions by choosing some appropriate link functions that connect the lifetimes of devices with the stress factors. ALT has received much attention in reliability engineering. Nelson (1972) proposed a graphical analysis of accelerated life-test data with the inverse power law. Nelson (1980) further proposed a step-stress model and derived the corresponding maximum likelihood estimates for the Weibull distributed lifetimes under inverse power law. Later, Meeter and Meeker (1994) considered the lifetimes to be Weibull distributed wherein the scale parameter was linked to the stress level by a log-linear function. Meeker et al. (1998) studied the accelerated degradation model for the effect of temperature on a failure-causing chemical reaction. They also considered mixed effects in the model and analyzed it by approximate maximum likelihood estimation. Recently, Ka et al. (2011) discussed the problem of the optimal allocation of test devices in the accelerate life-tests when the lifetimes are Weibull distributed and that Type-II censoring was allowed in the life-test. So (2012) further studied this problem under Type-I censoring. 4

31 1.4 One-Shot Device Testing Data One-shot devices are products that will get destroyed immediately after use. We can only observe whether the failure is either before or after the inspection time. Therefore, the data from life-tests of such devices consist of both left censored (failure) and right censored (success) observations at pre-fixed inspection time points. They are also known as current status data in survival analysis. A typical example is the air-bag in an automobile. Such a device would be activated once a collision is detected by a sensor. Once the airbag gets deployed, it cannot be used again. No matter whether the airbag successfully deploys or not, it is not possible to observe the exact failure time. In cases of a successful deployment, the lifetime is right censored at the inspection time since we know the failure time will be after that time. On the other hand, the lifetimes becomes left censored at the inspection time if the airbag fails to deploy since we know in this case the failure time was before the inspection time. Recently, Fan et al. (2009) developed a Bayesian approach to analyze the reliability of electro-explosive devices which are indeed one-shot devices. They found the normal prior to be the best one when the failure observations are rare, that is, when the devices are highly reliable. Balakrishnan and Ling (2012a) developed an EM algorithm for the determination of the MLEs of model parameters under exponential lifetime distribution for devices with a single stress factor. Balakrishnan and Ling (2012b) further extended this work to a model with multiple stress factors. Balakrishnan and Ling (2013) developed more general inferential results for devices with Weibull lifetimes under non-constant shape parameters. Further, Balakrishnan and Ling (2014) extended their work to devices with gamma lifetimes. 5

32 Figure 1.1: An electro-explosive device designed by Thomas and Betts (1967). 1.5 Competing Risks Model In a life-testing of devices, as mentioned earlier, there are different causes of failure and that can be described by a competing risks model. There are various examples of failure data under competing risks in the literature. Craiu and Duchesne (2004) discussed the competing risks model with masked causes of failure. Craiu and Lee (2005) studied the model selection method for competing risks. Balakrishnan and Han (2008) developed exact inference for a simple step-stress model with competing risks for exponential lifetimes under Type-II censoring. Along similar lines, Balakrishnan and Han (2010) developed exact inference for a simple step-stress model for the case of the exponential lifetime distribution, but under time constraint. Like many industrial products, there are multiple components in one-shot devices. 6

33 However, competing risks models have not been considered for one-shot devices until now. Let us consider, for example, an ordinary electro-explosive device displayed in Figure 1.1 in which many components are displayed. Failure of any one of these component may result in product malfunction. Therefore, we consider here a competing risk model for one-shot devices. Specifically, suppose the devices are placed in J stress levels and are tested in I different inspection times (IT s): 1. the tests are only conducted at inspection times IT i, for i = 1,..., I; 2. the devices are tested under J different stress level settings, denoted by x j = (x 0j, x 1j, x 2j,, x Mj ), and x 0j 1 for j = 1,..., J; 3. the number of devices failed due to the r-th cause at IT i and j-th stress level setting is denoted by D rij, for r = 1,, R; 4. the number of devices that survive (successfully detonated) at IT i and x j is denoted by S ij = K ij R r=1 D rij. For simplicity, we assume there are two independent components in the one-shot device, ie., R = 2. Of course, the theoretical framework for more than 2 competing risks can be very easily developed. We define ijk to be the indicator for the k-th device under stress level setting x j and inspection time IT i. When the device is successfully detonated, we will set ijk = 0. However, if the device fails to detonate, we will identify (by careful inspection) the specific cause responsible for the failure. If Risk r is the cause for the failure, we will denote this event by ijk = r, for r = 1, 2. 7

34 Table 1.1: An example of one-shot device testing data under stress level setting (-1, -0.1, 0.3) and inspection times 0.5, 1, 1.5 with 2 competing risks. IT 1 = 0.5 IT 2 = 1 IT 3 = 1.5 δ ijk = 0 δ ijk = 1 δ ijk = 2 x 11 = 1 S 11 = 18 d 111 = 30 d 211 = 52 x 12 = 0.1 S 12 = 92 d 112 = 6 d 212 = 2 x 13 = 0.3 S 13 = 100 d 113 = 0 d 213 = 0 x 11 = 1 S 21 = 8 d 121 = 24 d 221 = 68 x 12 = 0.1 S 22 = 67 d 122 = 18 d 222 = 15 x 13 = 0.3 S 23 = 98 d 123 = 2 d 223 = 0 x 11 = 1 S 31 = 4 d 131 = 34 d 231 = 62 x 12 = 0.1 S 32 = 39 d 132 = 42 d 232 = 19 x 13 = 0.3 S 33 = 76 d 133 = 20 d 233 = 4 Mathematically, the indicator ijk is then defined as ijk = 0 if min(t 1ijk, T 2ijk ) > IT i, 1 if T 1ijk < min(t 2ijk, IT i ), 2 if T 2ijk < min(t 1ijk, IT i ), (1.1) and then δ ijk will be used to denote the realization of ijk. An example of data observed from such a setting is presented in Table 1.1. Sometimes, the causes of failures cannot be diagnosed or determined. In this case, the cause is said to be masked. The indicator ijk is usually set to be 1 in this situation. There are a lot of literatures discussing the masked causes of failures:usher and Hodgson (1988) developed a maximum likelihood inference on the component reliability with masked system life test data; Flehinger et al. (1998) studied the survival function with masked causes of failures under proportional hazards assumption; Sen et al. (2010) proposed a Bayesian approach to handle masked cause of death. 8

35 1.6 EM Algorithm Maximum likelihood method is commonly used in practice for estimating the model parameters due to its well-known desirable properties. When there exist missing data, the likelihood function would become complicated and the MLEs are therefore not usually in a simple closed-from. In such a case, EM algorithm is a natural way to handle the missing data problem; see Casella and Berger (2002) and McLachlan and Krishnan (2008) for more details. The EM algorithm consists of two parts: the expectation step (E-step) followed by the maximization step (M-step). In the E-step, the expectation of complete log-likelihood, conditional on the observed data and current estimate of the model parameter is calculated. In the M-step, the estimation is updated though maximizing the expected log-likelihood function obtained from the E-step. By repeating the E-step and M-step iteratively until convergence to the desired level of accuracy, the MLEs will be obtained. EM algorithm has been widely adopted in statistical literature for estimating the model parameters when any form of censoring is present in the data. For example, Ng et al. (2002) developed EM algorithm for estimating the parameters of Weibull and lognormal distributions from progressively censored data. Nandi and Dewan (2010) estimated the parameters of the bivariate Weibull distribution under random censoring. Pal (2014) applied the EM algorithm for estimating the parameters of a cure rate model. Park (2005) and Craiu and Duchesne (2004) developed EM algorithm when these are masked cases in a competing risk setup. In the case of one-shot devices, the missing data are the exact lifetimes of the components of the devices, T, and the observed data are the status of the devices, 9

36 δ. Let θ be the vector of model parameters. The complete log-likelihood function is denoted by l c (θ, δ, T ). The EM algorithm then seeks the MLE of θ by iterating these two steps: 1. calculte the expectation E T (l c (θ, δ, T ) θ (m), δ) ; 2. update the parameter estimation, θ (m+1), by θ (m+1) = arg max θ (E T (l c (θ, δ, T ) θ, δ)). 1.7 Semi-Parametric Model In parametric modeling, the life distribution is assumed to be a specific distribution. The inferential methods are then based on this specific model assumption. However, the inference may be biased if the assumed model is incorrect. For this reason, a semi-parametric model, that combines parametric and non-parametric components, becomes a good candidate for model fitting purposes. In our case, we assume the hazard rates are proportional to each other at different stress levels and no other assumption on the underlying lifetime distribution. This is commonly referred to as the Cox model or proportional hazards model. Proportional hazards model was first proposed by Cox (1972), and since then it has become a prominent model in many applications. For example, Finkelstein and Wolfe (1985) proposed a semi-parametric method for analyzing interval censored data in an animal tumorigenicity study. Cheng and Wei (2000) provided inference for a semi-parametric model with AIDS panel data. In this thesis, we use such a proportional hazards model in the context of one-shot device testing data analysis in 10

37 order to put forward a flexible semi-parametric model for this purpose. 1.8 Bayesian Approach The Bayesian approach provides an alternative statistical inference from a perspective different from the likelihood approach. In the case of one-shot devices with high reliability, the failure observations may be very rare in which case MLEs may not always exist. The Bayesian approach incorporates prior information and so it can provide useful inference even in this situation. Such available information can be converted in the form of a prior distribution for the unknown parameters. A more accurate prior information means a stronger prior distribution for the parameters. After data collection, the joint posterior distribution is then derived and the Bayesian estimates are obtained by the use of Metropolis-Hastings algorithm, which is one of the popular Markov Chain Monte Carlo methods. In reliability theory, Bayesian estimation is a popular approach. Martza and Wailera (1990) developed a reliability analysis for a complex system of independent binomial serial or parallel components. Coolen and Newby (1994) provided a Bayesian method which combines the prior information from several experts in a consistent way. Philippe and Lionel (2006) modelled the reliability of a complex system with Dynamic Object Oriented Bayesian Networks. Recently, Fan et al. (2009) adopted a Bayesian approach for analyzing highly reliable one-shot device testing data. 11

38 1.9 Scope of the Thesis The rest of this thesis proceeds as follows. In Chapters 2 and 3, we focus on the development of EM algorithms for one-shot device testing with two competing risks under exponential and Weibull lifetime distributions, respectively. In Chapter 4, a semi-parametric model is developed for the one-shot devices by using a proportional hazards model. In Chapter 5, a Bayesian approach is developed for one-shot device testing with two competing risks under exponential lifetime distribution. In Chapter 6, a one-shot device testing data under Weibull lifetime distribution is analyzed by the Bayesian methodology. Finally, some concluding remarks and possible future directions are mentioned in Chapter 7. All the detailed proofs and derivations are relegated to the Appendix for conciseness in the presentation of the main text of the thesis. 12

39 Chapter 2 EM Algorithm for One-Shot Device Testing with Two Competing Risks under Exponential Distribution 2.1 Introduction This Chapter provides an extension of the work of Balakrishnan and Ling (2012a) by introducing a competing risk model into one-shot device testing analysis under an accelerated life test setting. As mention before, one shot devices with multiple components are very common. For example, a fire extinguisher contains a cylinder, a valve and chemicals inside; an automobile air bag contains a crash sensor, an inflator and an air bag. However, the reliability of one-shot devices has not been studied 13

40 by considering a competing risks setup. Balakrishnan and Ling (2012a) developed an EM algorithm for the estimation of parameters of one-shot device model under exponential lifetime distribution with single stress factor, and Balakrishnan and Ling (2012b) further extended it to multiple stress factors. In this Chapter, we introduce the one-shot device testing model with competing risks. The EM algorithm is then developed for finding the MLEs of the model parameters. We show that the proposed method is quite robust to the choice of initial values and yield accurate estimates as compared to linearly constrained least-square estimates. Then, the reliability and the mean lifetime under normal stress condition are also estimated. For convenience, we will confine our attention throughout this discussion to the case of two competing risks for the failure of device. The results presented in this chapter have been published in the article N. Balakrishnan, H.Y. So, and M.H. Ling, EM algorithm for one-shot device testing with competing risks under exponential distribution, Reliability Engineering & System Safety, vol. 137, pp , c 2015 with permission from Elsevier. 2.2 Model Description Consider the testing of electro-explosive devices. The structure of an ordinary electroexplosive device is shown in Figure 1.1. Let us now assume that there are only two potential causes responsible for the failure of detonation, say burnout of resistance wires (part 20 in Figure 1.1) as Cause 1 and leakage of organic fuel (part 6 in Figure 1.1) as Cause 2. An accelerated life test for such one-shot devices is set up as follows: 14

41 1. the tests are only checked at inspection times IT i, for i = 1,..., I; 2. the devices are tested under different temperatures (as stress levels) x j, for j = 1,..., J; 3. there are K ij devices tested at IT i and x j ; 4. the number of devices failed due to the r-th cause at IT i and x j is denoted by D rij, for r = 1,, R, and d rij is the realization of it; 5. the number of devices that survive (successfully detonated) at IT i and x j is denoted by S ij = K ij R r=1 D rij, and s ij is the realization of that. Let us denote the random variable for the failure time due to Causes 1 and 2 as T rijk, for r = 1, 2, i = 1,, I, j = 1,, J and k = 1,, K ij, respectively. We now assume that T rijk follows Exponential distribution with rate parameter λ rj with p.d.f. f rj (t) = λ rj e λ rjt, r = 1, 2, = 1,, J, where λ rj is the failure rate of the r-th component in the device under temperature x j. Of course, t rijk will be used to denote the realization of the r.v. T rijk. The relationship between λ rj and x j is assumed to be a log-link function of the form λ rj (α) = α r0 exp(α r1x j ), α r0, α r1, x j 0. (2.1) We define ijk to be the indicator of the k-th item under temperature x j and inspection time IT i as stated in (1.1). We also denote p 0ij, p 1ij and p 2ij for the survival probability, failure probability due to Cause 1 and failure probability due to Cause 2, respectively, which are as 15

42 follows: p 0ij = (1 F 1 (IT i x j ))(1 F 2 (IT i x j )) = exp( (λ 1j (α) + λ 2j (α))it i ), (2.2) ( ) λ 1j (α) p 1ij = (1 exp( (λ 1j (α) + λ 2j (α))it i )), (2.3) λ 1j (α) + λ 2j (α) ( ) λ 2j (α) p 2ij = (1 exp( (λ 1j (α) + λ 2j (α))it i )). (2.4) λ 1j (α) + λ 2j (α) Now the data collected at temperatures x = {x j, j = 1, 2..., J} and inspection times IT = {IT i, i = 1,..., I} are the numbers of devices with the indicator values δ ijk = 0, δ ijk = 1 and δ ijk = 2, which are denoted by s ij, d 1ij and d 2ij, respectively. Then, the likelihood function of α = {α 10, α 11, α 20, α 21 } is given by where K ij = s ij + d 1ij + d 2ij. L(α δ, IT, w) = I J i=1 j=1 p s ij 0ij pd 1ij 1ij pd 2ij 2ij, (2.5) 2.3 EM Algorithm It is known that the EM algorithm is an efficient way to determine the Maximum Likelihood Estimates (MLEs) of the model parameters in the presence of missing data which may occur due to censoring and masking effects; see Casella and Berger (2002) and McLachlan and Krishnan (2008) for details. It involves approximating the missing data by their expectation given the observed data and the current estimates of the parameters (E-step) and then maximizing the corresponding likelihood function to obtain the updated parameter estimates (M-step). If we repeat the E-step and M-step iteratively, by the monotonicity of EM algorithm, the numerical values of the 16

43 MLEs will be attained to the desired level of accuracy. In the example of electro-explosive device, the parameters of interest are α r0, α r1, r = 1, 2, and the data that are not observable are the true lifetimes of the devices. Let us denote that lifetime by T ( ijk) rijk defined by T ( ijk) rijk = T rijk min(t 1ijk, T 2ijk ) > IT i, when ijk = 0, T rijk T 1ijk < min(t 2ijk, IT i ), when ijk = 1, T rijk T 2ijk < min(t 1ijk, IT i ), when ijk = 2. (2.6) Let us denote α = (α 10, α 11, α 20, α 21) and α for the current estimate of α. Then, the complete data log-likelihood is given by l complete (α) = = I i=1 I i=1 K J ij ( ) ( ) log f 1 (T (δ ik) 1ijk ) + log f 2 (T (δ ik) 2ijk ) j=1 k=1 J j=1 l complete ij (α), where l complete ij (α) = K ij k= E-step log(λ 1j (α)) λ 1j (α)t (δ ik) 1ijk + log(λ 2j(α)) λ 2j (α)t (δ ik) 2ijk K ij = K ij (log(λ 1j (α)) + log(λ 2j (α))) λ 1j (α) k=1 T (δ ik) 1ijk K ij λ 2j(α) k=1 T (δ ik) 2ijk. In the E-step of the EM algorithm, we shall take the expected value of the missing data, given the observed data and the current parameter estimates, to approximate 17