Analysis of bivariate life-time data has important applications in various fields. Examples include data analysis of life-time studies of twins, paired organs (i.e. kidneys, eyes, ears, etc.) or successive survival events. The main interest in bivariate survival analysis is to investigate the dependent relationship between the two variables. However statistical inference in bivariate survival analysis may complicated due to the complexity of bivariate censoring mechanism. In the thesis, we consider some statistical inference problems for data that subject to bivariate censoring. The problem has been studied by various statisticians and is known as a "notoriously tough problem" (Gill, 1992). In Chapter 3, we review some nonparametric estimators to the problem and then in Section 3.3 we propose two simplified estimators under particular censoring patterns and derive their asymptotic properties. We also develop a semi-parametric estimator in Section 3.4 and discuss its asymptotic properties. In Chapter 4, we perform simulations to compare large sample performance of several bivariate survival function estimators. Chapter 5 discusses estimation of Kendall's tau for bivariate survival data. It seems that existing estimators of tau modified for censored data are not statistically reliable. We propose an estimator of tau which is expressed as a functional of the joint survival function. We apply the von Mises $\delta$-method to deduce some asymptotic properties of the proposed estimator. In Chapter 6, we discuss modeling and model selection issues for bivariate survival data. It turns out that the well-known frailty and Archimedean copulas models are special case of the semi-parameter structure discussed earlier. We propose two model selection methods suitable for censored data which can be used to choose a particular frailty and copula model. The inference problems discussed in the thesis are interrelated. Specifically nonparametric estimation of the bivariate survival function plays a key role in deducing important statistics in model selection and Kendall's tau estimation. Chapter 7 discusses issues related to time-dependent association. In Chapter 8, we summarize the main results and briefly discuss possible future work.

Suppose we collect bivariate continuous data $(X\sb{k},Y\sb{k})\ (k=1,2,\...,t)$ that are independently and identically distributed from some unknown distribution, F. We can compute the empirical (sample) quantiles (e.g., medians, tertiles, quartiles, quintiles) of the two marginal distributions and then partition the original data into r row and c column categories defined by these empirical marginal quantiles. The counts of such an $r\times c$ contingency table do not follow the multinomial distribution because the marginal totals in each row and column of the table are fixed as a result of using the random empirical marginal quantiles as cut-points to partition the data. Rather, we say that the counts in such a contingency table have the "empirical bivariate quantile-partitioned" (EBQP) distribution. Alternatively, suppose we collect bivariate continuous data from some known parametric family with distribution function $F(x, y\vert\theta)$ that depends on an unknown parameter vector $\theta.$ In this case, we can use the original data to estimate $\theta$ and then compute the estimated quantiles of the two marginal distributions. In turn, we estimate the expected counts of the $r\times c$ contingency table defined by these parametric marginal quantiles. We say that the estimated expected counts in such a contingency table have the "parametric bivariate quantile-partitioned" (PBQP) distribution. In this dissertation, we present the asymptotic distribution theory for both the EBQP and PBQP distributions and develop several important simplifications. We also show how to compute the asymptotic variances of several measures of agreement based on contingency tables, including kappa and weighted kappa, from both EBQP and PBQP tables. Our simulations confirm the accuracy of our theory for samples of moderate size, while our numerical calculations allow us to study the asymptotic efficiency of statistics calculated from EBQP tables relative to those calculated from PBQP tables. Furthermore, we note that EBQP, PBQP, and multinomial methods predict asymptotic variances that can differ substantially from each other. In addition, we apply our methods to analyze examples from nutritional epidemiology. Finally, we include examples of computer code to illustrate our algorithms and to allow interested researchers to perform their own empirical (EBQP) and parametric (PBQP) analyses.

Markers and quantitative trait loci (QTL) are ordered on a chromosome using discrete genotypic and continuous phenotypic data, respectively. Genotypic data is obtained from chromosomal alleles of the subject. Phenotypic, or QTL, data is a physical characteristic of the subject that can be measured. An additional phenotype data prediction step will be used in the QTL problem. The possible orders of the markers and QTL are dependent upon the likelihood of the data for each configuration. When all possible permutations of the markers and QTL are realized, their relative likelihoods will be compared to predict the actual order along the chromosome. Bayesian methodology may be employed to provide solutions to the problem of ordering genetic markers and QTL on a chromosome. By allowing priors, more precise answers to both problems can be found. If conjugate priors are chosen, exact integrated solutions to the marker order problem can be found. A Semi-Markov Chain Monte Carlo algorithm, SMCMC, will also be used to establish posterior probabilities on the orders. Gray code generation of the posterior probabilities will also be possible. Gray code will enumerate every order in the permutation space. The likelihood equation corresponding to each order will then be computed. These will be compared by inserting the fixed posterior means on the recombination rates into the likelihood equations. In this manner Gray code may be used to establish posterior probabilities on the orders. These three methods will be compared for the marker order problem to establish the accuracy of the SMCMC. For the more complicated QTL problem the posterior distributions will not be integrable. Thus the SMCMC and Gray code will provide the only solutions to the order problem. The SMCMC algorithm will be superior to the Gray code algorithm because the SMCMC will sample from the entire posterior distributions on the recombination rates, while the Gray code will calculate the order probabilities using only the posterior means.