Current status data are usually obtained with a failure time variable T which is diffcult observed but can be determined to lie below or above a random monitoring time or inspection time t. In this work we consider bivariate current status data ${t,delta_1,delta_2}$ and assume we have some prior information of the bivariate failure time variables T1 and T2. Our main goal is to find an optimal inspection time for testing the relationship between T1 and T2.

This thesis presents several statistical analysis on a cooperative project with Dr. Pallas and Yuting Mao from Biology Department of Georgia State University. This research concludes the impact of cauterize size of animals’ midbrain on auditory and visual response in brains. Besides some already commonly used statistical analysis method, such as MANOVA and Frequency Test, a unique combination of Permutation Test, Kolmogorov-Smirnov Test and Wilcoxon Rank Sum Test is applied to our non-parametric data. Some simulation results show the Permutation Test we used has very good powers, and fits the need for this study. The result confirms part of the Biology Department’s hypothesis statistically and enhances more complete understanding of the experiments and the potential impact of helping patients with Acquired Brain Injury.

In certain areas such as Pharmacokinetic(PK) and Pharmacodynamic(PD), the hazard rate function, denoted by ??, plays a central role in modeling the instantaneous risk of failure time data. In the context of assessing the appropriateness of a given parametric hazard rate model, Huh and Hutmacher [22] showed that their hazard-based visual predictive check is as good as a visual predictive check based on the survival function. Even though Huh and Hutmacher’s visual method is simple to implement and interpret, the final decision reached there depends on the personal experience of the user. In this thesis, our primary aim is to develop nonparametric goodness-ofit tests for hazard rate functions to help bring objectivity in hazard rate model selections or to augment subjective procedures like Huh and Hutmacher’s visual predictive check. Toward that aim two nonparametric goodnessofit (g-o) test statistics are proposed and they are referred to as chi-square g-o test and kernel-based nonparametric goodness-ofit test for hazard rate functions, respectively. On one hand, the asymptotic distribution of the chi-square goodness-ofit test for hazard rate functions is derived under the null hypothesis ??0 : ??(??) = ??0(??) ??? ? R + as well as under the fixed alternative hypothesis ??1 : ??(??) = ??1(??) ??? ? R +. The results as expected are asymptotically similar to those of the usual Pearson chi-square test. That is, under the null hypothesis the proposed test converges to a chi-square distribution and under the fixed alternative hypothesis it converges to a non-central chi-square distribution. On the other hand, we showed that the power properties of the kernel-based nonparametric goodness-ofit test for hazard rate functions are equivalent to those of the Bickel and Rosenblatt test, meaning the proposed kernel-based nonparametric goodness-ofit test can detect alternatives converging to the null at the rate of ???? , ?? < 1/2, where ?? is the sample size. Unlike the latter, the convergence rate of the kernel-base nonparametric g-o test is much greater; that is, one does not need a very large sample size for able to use the asymptotic distribution of the test in practice.