We show how using censored regressors leads to expansion bias, or estimated effects that are proportionally too large. We show the necessity of this effect in bivariate regression and illustrate the bias using results for normal regressors. We study the bias when there is a censored regressor among many regressors, and we note how censoring can work to undo errors-in-variables bias. We discuss several approaches to correcting expansion bias. We illustrate the concepts by considering how censored regressors can arise in the analysis of wealth effects on consumption, and on peer effects in productivity.

Cancer is a dangerous disease causing the most deaths in the world today and around 550,000 deaths in America per year (American Cancer Society). Larynx cancer data was recorded by Kardaun (1983). The data was collected at a Dutch hospital during 1970 – 1978. Ninety male adults with cancer of larynx were involved into the study. Each patient was divided into one of four groups depending on his or her illness condition. The data also recorded their age, lifetime, and year of entering the research. These are common factors as factors of other cancer data. The purpose of this thesis is to apply proportional hazard regression model, additive hazard regression model, censored quantiles regression model, and censored linear regression model to analyze the above larynx cancer data and find the best regression model of data by using each method. Comparison and suggestion for which method should be used in specific situation are also made. Some related topics are also mentioned so we can have resource for future study. Key words: right censoring, proportional risk model, additive risk model, quantiles regression model, linear regression model.

My dissertation includes three essays that examine or relax classical restrictive assumptions used in econometrics estimation methods. The first chapter proposes methods for examining how a response variable is influenced by a covariate. Rather than focusing on the conditional mean I consider a test of whether a covariate has an effect on the entire conditional distribution of the response variable given the covariate and other conditioning variables. This type of analysis is useful in situations where the econometrician or policy maker is interested in knowing whether a variable or policy would improve the distribution of the response outcomes in a stochastic dominance sense. The response variable is assumed to be continuous, while both discrete and continuous covariate cases are considered. I derive the asymptotic distribution of the test statistics and show that they have simple known asymptotic distributions under the null by using and extending conditional empirical process results given by Horvath and Yandell (1988). Monte Carlo experiments are conducted, and the tests are shown to have good small sample behavior. The tests are applied to a study on father's labor supply. The second chapter is based on previous joint work with Jason Abrevaya. It considers estimation of censored panel-data models with individual-specific slope heterogeneity. The slope heterogeneity may be random (random-slopes model) or related to covariates (correlated-random-slopes model). Maximum likelihood and censored least-absolute deviations estimators are proposed for both models. Specification tests are provided to test the slope-heterogeneity models against nested alternatives. The proposed estimators and tests are used for an empirical study of Dutch household portfolio choice. Strong evidence of correlated random slopes for the age variables is found, indicating that the age profile of portfolio adjustment varies significantly with other household characteristics. The third chapter proposes specification tests in models with endogenous covariates. In empirical studies, econometricians often have little information on the functional form of the structural model, regardless of whether covariates in model are exogenous or endogenous. In this chapter, I propose tests for restricted structural model specifications with endogenous covariates against the fully nonparametric alternative. The restricted model specifications include the nonparametric specification with a restricted set of covariates, the semiparametric single index specification and the parametric linear specification. Test statistics are “leave-one-out” type kernel U-statistic as used in Fan and Lee (1996). They are constructed using the idea of the control function approach. Monte Carlo results are provided and tests are shown to have reasonable small sample behavior. / text

以貝氏方法來處理部分區分(partially-classified)或是失去部分訊息資料的類別抽樣(categorical sampling with censored data)，大部分建立在「誠實回答」(truthful reporting)以及「無價值性失去部分訊息」(non-informative censoring)的前提下。Dr.Jiang(1995)取消以上兩個限制，提出quasi-Bayes method來近似這類問題的貝氏解。另外我們也嘗試利用Gelfand and Smith(1990)針對Gibbs sampler所提出的收斂方法來估計。本文重點在比較此兩種方法的估計值準確性，並考慮先驗參數(prior)對估計精準的影響。

在多組壽命試驗中，為了準確地估計模型的參數，我們必須找出最合適的實驗品數量，以分配給每一個應力水平。近來, Ng, Chan and Balakrishnan(2006)，在完整樣本情況下，利用「極值回歸模型」發展了找尋實驗品數量最合適的分配方法。其後，Ka, Chan, Ng and Balakrishnan (2011)在同一個回歸模型下，研究了對於「II型截尾樣本」最合適的分配方法。因為我們仍未確立對「I型截尾樣本」的最合適分配方法，所以我們將會在本篇論文中探討如何在「I型截尾壽命試驗」中找出最合適的實驗品分配方法。 / 在本論文中，我們會利用最大似然估計的方法去估計模型參數。我們也會計算出「逆費雪訊息矩陣」(「漸近方差協方差矩陣」)I⁻¹，用以量度參數估計值的準確度。以下是三個對最合適分配方法的決定準則: / 1.費雪訊息矩陣的行列式最大化, / 2. ν1估計值的方差最小化, var( ν1)(V -優化準則 ) / 3.漸近方差協方差矩陣的跡最小化, tr(⁻¹)(A-優化準則 ) / 我們也會討論在「極值回歸模型」的特例：「指數回歸模型」之下最合適的分配方法。 / In multi-group life-testing experiment, it is essential to optimize the allocation of the items under test to dierent stress levels in order to estimate the model parameter accurately. Recently Ng, Chan and Balakrishnan(2006) developed the optimal allocation for complete sample case with extreme value regression model, and Ka, Chan, Ng and Balakrishnan (2011) discussed about the optimal allocation for Type -II censoring cases with the same model. The optimal allocation for Type-I censoring scheme has not been established, so in this thesis, we are going to investigate the optimal allocation if Type-I censoring scheme is adopted in life-testing experiment. / Maximum likelihood estimation method will be adopted in this thesis for estimating model parameter. The inverted Fisher information matrix (asymptotic variance -covariance matrix),I⁻¹ , will be derived and used to measure the accuracy of the estimated parameters. The optimal allocation will be determined based on three optimal criteria: / 1. Maximizing the determinant of the expected Fisher Information matrix, / 2. Minimizing the variance of the estimator of ν1, var( ν1) (V -optimality ) / 3. Minimizing the trace of the variance-covariance matrix, tr(I⁻¹) (A-optimality ) / Optimal allocation under the exponential regression model,which is a spe¬cial case of extreme value regression model, will also be discussed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / So, Hon Yiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 46-48). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Accelerated Life Test --- p.1 / Chapter 1.2 --- Life-Stress Relationship --- p.1 / Chapter 1.3 --- Type I Censoring --- p.3 / Chapter 1.4 --- Optimal Allocation --- p.3 / Chapter 1.5 --- The Scope of the Thesis --- p.4 / Chapter 2 --- Extreme Value Regression Model --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Model and Maximum Likelihood Estimation --- p.5 / Chapter 2.3 --- Expected Fisher Information --- p.8 / Chapter 3 --- Criteria for Optimization and the Optimal Allocation --- p.12 / Chapter 3.1 --- Introduction --- p.12 / Chapter 3.2 --- Criteria for Optimization --- p.12 / Chapter 3.3 --- Numerical Illustrations and the Optimal Allocation --- p.14 / Chapter 4 --- Sensitivity Analysis --- p.17 / Chapter 4.1 --- Introduction --- p.17 / Chapter 4.2 --- Sensitivity Analysis --- p.17 / Chapter 4.3 --- Numerical Illustrations --- p.19 / Chapter 4.3.1 --- Illustration with McCool (1980) Data --- p.19 / Chapter 4.3.2 --- Further Study --- p.21 / Chapter 5 --- Exponential Regression Estimation --- p.26 / Chapter 5.1 --- Introduction --- p.26 / Chapter 5.2 --- The Model and the Likelihood Inference --- p.27 / Chapter 5.3 --- Optimal Sample Size Allocation for Estimation of Model Pa- rameters --- p.30 / Chapter 5.4 --- Numerical Illustration --- p.33 / Chapter 5.5 --- Sensitivity Analysis --- p.35 / Chapter 5.5.1 --- Parameter Misspeci cation --- p.35 / Chapter 5.5.2 --- Censoring Time --- p.38 / Chapter 5.5.3 --- Further Study --- p.40 / Chapter 6 --- Conclusion and Further Research --- p.44

Comparison of two or more failure time distributions based on interval-censored data is tested by extension of log-rank test proposed by Sun (1996, 2001, 2004). Furthermore, Chang (2004) verified that the proposed test statistics are approximately chi-cquare with degrees of freedom p-1 after constants factor adjustment which can be obtained from simulations. In this paper we approach in a different way to estimate the adjustment factor of a given interval-censored data by applying the bootstrap technique to the test statistics. Simulation results indicate that the bootstrap technique performs well on those test statistics except the one proposed in 1996. By using chi-square goodness of fit test, we found that Sun's test in 1996 is significantly far from any chi-square.

The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. We obtain efficient estimators of the parameters of type I (maximum) extreme value distribution without solving the likelihood equations. This research provides for the first time simple expressions for the elements of the information matrix for type II censoring. We construct efficient estimators of the parameters using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also perform an extensive Monte Carlo simulation study to examine the performance of the estimators for different sample sizes.

This thesis is mainly concerned with the estimation of parameters in autoregressive models with censored data. For convenience, attention is restricted to the first-order stationary autoregressive (AR(1)) model in which the response random variables are subject to right-censoring. In their present form, currently available methods of estimation in regression analysis with censored autocorrelated data, which includes the MLE, are applicable only if the errors of the AR component of the model are Gaussian. Use of these methods in AR processes with non-Gaussian errors requires, essentially, rederivations of the estimators. Hence, in this thesis, we propose new estimators which arerobust in the sense that they can be applied with minor or no modifications to AR models with non-Gaussian. We propose three estimators, two of which the form of the distribution of the errors needs to be specified. The third estimator is a distribution-free estimator. As the reference to this estimator suggests, it is free from distributional assumptions in the sense that the error distribution is calculated from the observed data. Hence, it can be used in a wide variety of applications.In the first part of the thesis, we present a summary of the various currently available estimators for the linear regression model with censored independent and identically distributed (i.i.d.) data. In our review of these estimators, we note that the linear regression model with censored i.i.d. data has been studied quite extensively. Yet, use of autoregressive models with censored data has received very little attention. Hence, the remainder of the thesis focuses on the estimation of parameters for censored autocorrelated data. First, as part of the study, we review currently available estimators in regression with censored autocorrelated data. Then we present descriptions of the new estimators for censored ++ / autocorrelated data. With the view that extensions to the AR(p), model, p > 1, and to left-censored data can be easily achieved, all the estimators, both currently available and new, are discussed in the context of the AR(1) model. Next, we establish some asymptotic results for the estimators in which specification of the form of the error distribution is necessary. This is followed by a simulation study based on Monte Carlo experiments in which we evaluate and compare the performances of the new and currently available estimators among themselves and with the least-squares estimator for the uncensored case. The performances of the asymptotic variance estimators of the parameter estimators are also evaluated.In summary, we establish that for each of the two new estimators for which the distribution of the errors is assumed known, under suitable conditions on the moments of the error distribution function, if the estimator is consistent, then it is also asymptotically normally distributed. For one of these estimators, if the errors are Gaussian and alternate observations are censored, then the estimator is consistent. Hence, for this special case, the estimator is consistent and asymptotically normal. The simulation results suggest that this estimator is comparable with the distribution-free estimator and a currently available pseudolikelihood (PL) estimator. All three estimators perform worse than the least squares estimator for the uncensored case. The MLE and another currently available PL estimator perform comparably not only with the least squares estimator for the uncensored case but also with estimators from the abovementioned group of three estimators, which includes the distribution-free estimator. The other new estimator for which the form of the error distribution is assumed known compares favourably with the least- squares estimator for the uncensored case ++ / and better than the rest of the estimators when the true value of the autoregression parameter is 0.2. When the true value of the parameter is 0.5, this estimator performs comparably with the rest of the estimators and worse when the true value of the parameter is O.S. The simulation results of the asymptotic variance estimators suggest that for each estimator and for a fixed value of the true autoregression parameter, if the error distribution is fixed and the censoring rate is constant, the asymptotic formulas lead to values which are asymptotically insensitive to the censoring pattern. Also, the estimated asymptotic variances decrease as the sample size increases and their behaviour, with respect to changes in the true value of autoregression parameter, is consistent with the behaviour of the asymptotic variance of the least-squares estimator for the uncensored case.Some suggestions for possible extensions conclude the thesis.

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Monotone function, such as growth function and cumulative distribution function, is often a study of interest in statistical literature. In this dissertation, we propose a nonparametric least-squares method for estimating monotone functions induced from stochastic processes in which the starting time of the process is subject to interval censoring. We apply this method to estimate the mean function of tumor growth with the data from either animal experiments or tumor screening programs to investigate tumor progression. In this type of application, the tumor onset time is observed within an interval. The proposed method can also be used to estimate the cumulative distribution function of the elapsed time between two related events in human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) studies, such as HIV transmission time between two partners and AIDS incubation time from HIV infection to AIDS onset. In these applications, both the initial event and the subsequent event are only known to occur within some intervals. Such data are called doubly interval-censored data. The common property of these stochastic processes is that the starting time of the process is subject to interval censoring. A unified two-step nonparametric estimation procedure is proposed for these problems. In the first step of this method, the nonparametric maximum likelihood estimate (NPMLE) of the cumulative distribution function for the starting time of the stochastic process is estimated with the framework of interval-censored data. In the second step, a specially designed least-squares objective function is constructed with the above NPMLE plugged in and the nonparametric least-squares estimate (NPLSE) of the mean function of tumor growth or the cumulative distribution function of the elapsed time is obtained by minimizing the aforementioned objective function. The theory of modern empirical process is applied to prove the consistency of the proposed NPLSE. Simulation studies are extensively carried out to provide numerical evidence for the validity of the NPLSE. The proposed estimation method is applied to two real scientific applications. For the first application, California Partners' Study, we estimate the distribution function of HIV transmission time between two partners. In the second application, the NPLSEs of the mean functions of tumor growth are estimated for tumors with different stages at diagnosis based on the data from a cancer surveillance program, the SEER program. An ad-hoc nonparametric statistic is designed to test the difference between two monotone functions under this context. In this dissertation, we also propose a numerical algorithm, the projected Newton-Raphson algorithm, to compute the non– and semi-parametric estimate for the M-estimation problems subject to linear equality or inequality constraints. By combining the Newton-Raphson algorithm and the dual method for strictly convex quadratic programming, the projected Newton-Raphson algorithm shows the desired convergence rate. Compared to the well-known iterative convex minorant algorithm, the projected Newton-Raphson algorithm achieves much quicker convergence when computing the non- and semi-parametric maximum likelihood estimate of panel count data.