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<p>In this thesis, inferential methods for one-shot device testing data from accelerated life-test are developed. Due to constraints on time and budget, accelerated life-tests are commonly used to induce more failures within a reasonable amount of test-time for obtaining more lifetime information that will be especially useful in reliability analysis. One-shot devices, which can be used only once as they get destroyed immediately after testing, yield observations only on their condition and not on their real lifetimes. So, only binary response data are observed from an one-shot device testing experiment. Since no failure times of units are observed, we use the EM algorithm for determining the maximum likelihood estimates of the model parameters. Also, inference for the reliability at a mission time and the mean lifetime at normal operating conditions are also developed.</p> <p>The thesis proceeds as follows. Chapter 2 considers the exponential distribution with single-stress relationship and develops inferential methods for the model parameters, the reliability and the mean lifetime. The results obtained by the EM algorithm are compared with those obtained from the Bayesian approach. A one-shot device testing data is analyzed by the proposed method and presented as an illustrative example. Next, in Chapter 3, the exponential distribution with multiple-stress relationship is considered and corresponding inferential results are developed. Jackknife technique is described for the bias reduction in the developed estimates. Interval estimation for the reliability and the mean lifetime are also discussed based on observed information matrix, jackknife technique, parametric bootstrap method, and transformation technique. Again, we present an example to illustrate all the inferential methods developed in this chapter. Chapter 4 considers the point and interval estimation for the one-shot device testing data under the Weibull distribution with multiple-stress relationship and illustrates the application of the proposed methods in a study involving the development of tumors in mice with respect to risk factors such as sex, strain of offspring, and dose effects of benzidine dihydrochloride. A Monte Carlo simulation study is also carried out to evaluate the performance of the EM estimates for different levels of reliability and different sample sizes. Chapter 5 describes a general algorithm for the determination of the optimal design of an accelerated life-test plan for one-shot device testing experiment. It is based on the asymptotic variance of the estimated reliability at a specific mission time. A numerical example is presented to illustrate the application of the algorithm. Finally, Chapter 6 presents some concluding remarks and some additional research problems that would be of interest for further study.</p> / Doctor of Philosophy (PhD)

<p>Left truncation arises because in many situations, failure of a unit is observed only if it fails after a certain period. In many situations, the units under study may not be followed until all of them fail and the experimenter may have to stop at a certain time when some of the units may still be working. This introduces right censoring into the data. Some commonly used lifetime distributions are lognormal, Weibull and gamma, all of which are special cases of the flexible generalized gamma family. Likelihood inference via the Expectation Maximization (EM) algorithm is used to estimate the model parameters of lognormal, Weibull, gamma and generalized gamma distributions, based on left truncated and right censored data. The asymptotic variance-covariance matrices of the maximum likelihood estimates (MLEs) are derived using the missing information principle. By using the asymptotic variances and the asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. For comparison purpose, Newton-Raphson (NR) method is also used for the parameter estimation, and asymptotic confidence intervals corresponding to the NR method and parametric bootstrap are also obtained. Through Monte Carlo simulations, the performance of all these methods of inference are studied. With regard to prediction analysis, the probability that a right censored unit will be working until a future year is estimated, and an asymptotic confidence interval for the probability is then derived by the delta-method. All the methods of inference developed here are illustrated with some numerical examples.</p> / Doctor of Philosophy (PhD)

<p>Recently, the Conway-Maxwell Poisson (COM-Poisson) cure rate model has been proposed which includes as special cases some of the well-known cure rate models discussed in the literature. Data obtained from cancer clinical trials are often right censored and the expectation maximization (EM) algorithm can be efficiently used for the determination of the maximum likelihood estimates (MLEs) of the model parameters based on right censored data.</p> <p>By assuming the lifetime distribution to be exponential, lognormal, Weibull, and gamma, the necessary steps of the EM algorithm are developed for the COM-Poisson cure rate model and some of its special cases. The inferential method is examined by means of an extensive simulation study. Model discrimination within the COM-Poisson family is carried out by likelihood ratio test as well as by information-based criteria. Finally, the proposed method is illustrated with a cutaneous melanoma data on cancer recurrence. As the lifetime distributions considered are not nested, it is not possible to carry out a formal statistical test to determine which among these provides an adequate fit to the data. For this reason, the wider class of generalized gamma distributions is considered which contains all of the above mentioned lifetime distributions as special cases. The steps of the EM algorithm are then developed for this general class of distributions and a simulation study is carried out to evaluate the performance of the proposed estimation method. Model discrimination within the generalized gamma family is carried out by likelihood ratio test and information-based criteria. Finally, for the considered cutaneous melanoma data, the two-way flexibility of the COM-Poisson family and the generalized gamma family is utilized to carry out a two-way model discrimination to select a parsimonious competing cause distribution along with a suitable choice of a lifetime distribution that provides the best fit to the data.</p> / Doctor of Philosophy (PhD)

Researchers conduct life testing on objects of interest in an attempt to determine their life distribution as a means of studying their reliability (or survivability). Determining the life distribution of the objects under study helps manufacturers to identify potential faults, and to improve quality. Researchers sometimes conduct accelerated life tests (ALTs) to ensure that failure among the tested units is earlier than what could result under normal operating (or environmental) conditions. Moreover, such experiments allow the experimenters to examine the effects of high levels of one or more stress factors on the lifetimes of experimental units. Examples of stress factors include, but not limited to, cycling rate, dosage, humidity, load, pressure, temperature, vibration, voltage, etc. A special class of ALT is step-stress accelerated life testing. In this type of experiments, the study sample is tested at initial stresses for a given period of time. Afterwards, the levels of the stress factors are increased in agreement with prefixed points of time called stress-change times. In practice, time and resources are limited; thus, any experiment is expected to be constrained to a deadline which is called a termination time. Hence, the observed information may be subjected to Type-I censoring. This study discusses maximum likelihood inferential methods for the parameters of multiple step-stress models from a generalized Birnbaum-Saunders distribution under time constraint alongside other inference-related problems. A couple of general inference frameworks are studied; namely, the observed likelihood (OL) framework, and the expectation-maximization (EM) framework. The last-mentioned framework is considered since there is a possibility that Type-I censored data are obtained. In the first framework, the scoring algorithm is used to get the maximum likelihood estimators (MLEs) for the model parameters. In the second framework, EM-based algorithms are utilized to determine the required MLEs. Obtaining observed information matrices under both frameworks is also discussed. Accordingly, asymptotic and bootstrap-based interval estimators for the model parameters are derived. Model discrimination within the considered generalized Birnbaum-Saunders distribution is carried out by likelihood ratio test as well as by information-based criteria. The discussed step-stress models are illustrated by analyzing three real-life datasets. Accordingly, establishing optimal multiple step-stress test plans based on cost considerations and three optimality criteria is discussed. Since maximum likelihood estimators are obtained by numerical optimization that involves maximizing some objective functions, optimization methods used, and their software implementations in R are discussed. Because of the computational aspects are in focus in this study, the benefits of parallel computing in R, as a high-performance computational approach, are briefly addressed. Numerical examples and Monte Carlo simulations are used to illustrate and to evaluate the methods presented in this thesis. / Thesis / Doctor of Science (PhD)

本文研究美國退休福利保險公司(PBGC)保險價值的計算,延伸 Marcus (1987)模型,提出狀態轉換過程保險價值模型計算,也就是將市場分為兩種情況,正成長率視為正常狀態,負成長率為衰退狀態,利用狀態轉換過程評價 PBGC 契約在經濟困難而終止和介入終止下合理的保險價值。在參數估計方面,本文以 S&P500股價指數和一年期國庫券資料參數估計值及Marcus(1987)和Pennacchi and Lewis(1994)的方式給定參數,以 EM-PSO-Gradient 延伸 EM-Gradient 方法並以最大概似函數值、AIC 準則和 BIC 準則比較估計結果。最後固定其他參數, 探討狀態轉換過程保險價值模型對參數調整後保險價值的影響之敏感度分析。 / In this paper, we evaluate Pension Benefit Guaranty Corporation insurance values through regime switching models, which is the extension of the models of Marcus (1987). That is, we can separate periods of economy with faster growth from those with slower growth when observing long-term trends in economy and calculate the reasonable PBGC insurance values under distress termination and intervention termination by regime switching processes. We set parameters by estimating S&P 500 index and 1-year treasury bills by EM-PSO-Gradient, which is the extensive method of EM-Gradient and refer the methods of setting parameters from Marcus (1987) and Pennacchi and Lewis (1994). After that, we compare the maximum likelihood estimates, AIC and BIC of the estimative results. Finally, we do sensitivity analysis through given the other parameters and look into what would impact on our models of insurance values when adjusting one parameter.

我們提供一個新的可靠度模型，DwACM，並提供一個模式選擇準則CCP，我們利用DwACM和CCP來選擇衰變量。 / We propose a new reliability model, DwACM (Dichotomous-data with Auxiliary Continuous Measurements model) to describe a data set which consists of classical dichotomous response (Go or No Go) associated with a set of continuous auxiliary measurement. In this model, the lifetime of each individual is considered as a latent variable. Given the value of the latent variable, the dichotomous response is either 0 or 1 depending on if it fails or not at the measuring time. The continuous measurement can be regarded as observations of an underlying possible degradation candidate of which descending process is a function of the lifetime. Under the assumption that the failure of products is defined as the time at which the continuous measurement reaches a threshold, these two measurements can be linked in the proposed model. Statistical inference under this model are both in frequentist and Bayesian frameworks. To evaluate the continuous measurements, we provide a criterion, CCP (correct classification probability), to select the best degradation measurement. We also report our simulation studies of the performances of parameters estimators and CCP.

本文主要著重在處理條件評估法下，「不知道」受訪者的回應。當「不知道」受訪者的產生機制並未符合完全隨機時，考量他們的真實意向就顯得極為重要。 文中使用中央研究院生醫所在其研究計畫「竹東及朴子地區心臟血管疾病長期追蹤研究」（CardioVascular Disease risk FACtor Two-township Study，簡稱CVDFACTS）第五循環中的研究調查資料。 　　由於以往的文獻對於「不知道」受訪者的處理，皆有不足之處。如Wang (1997)所提出的方法，就只能針對某種特定的「不知道」受訪者來做處理；而Caudill and Groothuis (2005)所提的方法，由於將「不知道」受訪者的差補與願付價格的估計分開，亦使其估計結果不具備一些好的性質。在本文中，我們提出一個能同時處理「不知道」受訪者且估計願付價格的方法。除了使得統計上較有效率外，也保有EM演算法的一個特性：願付價格模型中的估計參數為最大概似估計值。此外，在加入三要素混合模型(Tsai (2005))後，我們也可避免用到極端受訪者的訊息去差補那些「不知道」受訪者的意向。 　　在分析願付價格的過程中，我們發現此筆資料的「不知道」受訪者，其產生的機制為隨機，而非為完全隨機，這意謂著不考量「不知道」受訪者的分析結果，必定會產生偏差。而在比較有考量「不知道」受訪者與沒有的情況後，其結果確實應證了我們的想法：只要「不知道」受訪者不是完全隨機產生的，那麼不考量他們必定會產生某種程度的偏差。 / This paper investigates how to deal with “Don’t Know” (DK) responses in contingent valuation surveys, which must be taken into consideration when they are not completely at random. The data we use is collected from the fifth cycle of the Cardiovascular Disease Risk Factor Two-township Study (CVDFACTS), which is a series of long-term surveys conducted by the Institute of Biomedical Sciences, Academia Sinica. Previous methods used in dealing with DK responses have not been satisfactory because they only focus on some types of DK respondents (Wang (1997)), or separate the imputation of DK responses from the WTP estimation (Caudill and Groothuis (2005)). However, in this paper, we introduce an integrated method to cope with the incomplete data caused by DK responses. Besides being more efficient, the single-step method guarantees maximum likelihood estimates of the WTP model to be obtained due to the good property that the EM algorithm possesses. Furthermore, by adding the concept of the three-component mixture model (Tsai (2005)), some extreme information are drawn out when imputing the DK inclinations. In this hypertension data, the mechanism of the DK responses is “Don’t know at random”, which means the analysis of DK-dropped results in a bias. By using our method, the difference between DK-dropped and DK-included is actually revealed, which proves our suspicion that a DK-dropped analysis is accompanied by a biased result when DK is not completely at random.

Les modèles à sur-représentation de zéros discrets et continus ont une large gamme d'applications et leurs propriétés sont bien connues. Bien qu'il existe des travaux portant sur les modèles discrets à sous-représentation de zéro et modifiés à zéro, la formulation usuelle des modèles continus à sur-représentation -- un mélange entre une densité continue et une masse de Dirac -- empêche de les généraliser afin de couvrir le cas de la sous-représentation de zéros. Une formulation alternative des modèles continus à sur-représentation de zéros, pouvant aisément être généralisée au cas de la sous-représentation, est présentée ici. L'estimation est d'abord abordée sous le paradigme classique, et plusieurs méthodes d'obtention des estimateurs du maximum de vraisemblance sont proposées. Le problème de l'estimation ponctuelle est également considéré du point de vue bayésien. Des tests d'hypothèses classiques et bayésiens visant à déterminer si des données sont à sur- ou sous-représentation de zéros sont présentées. Les méthodes d'estimation et de tests sont aussi évaluées au moyen d'études de simulation et appliquées à des données de précipitation agrégées. Les diverses méthodes s'accordent sur la sous-représentation de zéros des données, démontrant la pertinence du modèle proposé. Nous considérons ensuite la classification d'échantillons de données à sous-représentation de zéros. De telles données étant fortement non normales, il est possible de croire que les méthodes courantes de détermination du nombre de grappes s'avèrent peu performantes. Nous affirmons que la classification bayésienne, basée sur la distribution marginale des observations, tiendrait compte des particularités du modèle, ce qui se traduirait par une meilleure performance. Plusieurs méthodes de classification sont comparées au moyen d'une étude de simulation, et la méthode proposée est appliquée à des données de précipitation agrégées provenant de 28 stations de mesure en Colombie-Britannique. / Zero-inflated models, both discrete and continuous, have a large variety of applications and fairly well-known properties. Some work has been done on zero-deflated and zero-modified discrete models. The usual formulation of continuous zero-inflated models -- a mixture between a continuous density and a Dirac mass at zero -- precludes their extension to cover the zero-deflated case. We introduce an alternative formulation of zero-inflated continuous models, along with a natural extension to the zero-deflated case. Parameter estimation is first studied within the classical frequentist framework. Several methods for obtaining the maximum likelihood estimators are proposed. The problem of point estimation is considered from a Bayesian point of view. Hypothesis testing, aiming at determining whether data are zero-inflated, zero-deflated or not zero-modified, is also considered under both the classical and Bayesian paradigms. The proposed estimation and testing methods are assessed through simulation studies and applied to aggregated rainfall data. The data is shown to be zero-deflated, demonstrating the relevance of the proposed model. We next consider the clustering of samples of zero-deflated data. Such data present strong non-normality. Therefore, the usual methods for determining the number of clusters are expected to perform poorly. We argue that Bayesian clustering based on the marginal distribution of the observations would take into account the particularities of the model and exhibit better performance. Several clustering methods are compared using a simulation study. The proposed method is applied to aggregated rainfall data sampled from 28 measuring stations in British Columbia.

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