Inference on Quantiles under Equality Restrictions

Abstract

The thesis deals with the inference on quantiles of some probabilistic models under equality restrictions on parameters. In Chapter 1, we discuss the background of the problem and give a comprehensive literature on the related problems which are undertaken in the thesis. Specifically, the inference problems under equality restrictions have been discussed. In Chapter 2, we provide some basic definitions, fundamental notations and methodologies for the point estimation, interval estimation and hypothesis testing problems that help to frame the rest of the chapters. In Chapter 3, the problems of interval estimation of, and testing a hypothesis on the quantile have been considered when independent random samples are available from two and more normal populations with a common mean and possibly unknown and unequal variances. For two populations, the asymptotic confidence interval, parametric bootstrap intervals, and generalized confidence intervals have been obtained. For hypothesis testing, several tests, such as the one based on the Computational Approach Test, the likelihood ratio test, a test using an estimator of quantile, and tests based on a generalized p􀀀value approach, have been proposed. For several populations, in addition to the intervals proposed for two populations, we have proposed three new methods for confidence intervals, such as the two highest posterior density intervals and a confidence interval using the method of variance estimate recovery. In the case of hypothesis testing, we have also proposed three new test procedures such as the standardized likelihood ratio test, the modified likelihood ratio test and the parametric bootstrap likelihood ratio test. Chapter 4 discusses interval estimation and hypothesis testing of the quantile when samples are available from several normal populations with a common standard deviation and unequal means. The asymptotic confidence interval, an interval using the method of variance estimate recovery, bootstrap-p, bootstrap-t, highest posterior density intervals and generalized confidence interval for the quantile have been derived. We discuss several test methods for hypothesis testing, such as the one based on the computational approach test, the asymptotic likelihood ratio test, the parametric bootstrap likelihood ratio test, and tests based on the generalized p􀀀value approach. In Chapter 5, we obtain the confidence intervals and construct test procedures for the quantiles when the observations are from a bivariate normal population with a common mean. We derive several confidence intervals, such as the asymptotic confidence interval, a classical confidence interval using the method of variance estimate recovery, bootstrap-p, bootstrap-t and the highest posterior density. Furthermore, two generalized confidence intervals are obtained using some of the estimators of the common mean. In the case of hypothesis testing, several tests, such as the likelihood ratio test, tests based on the computational approach, and tests based on the generalized variable method, are derived. Chapter 6 deals with the problems of interval estimation and hypothesis testing of the quantile for several exponential populations with a common location and possibly different scale parameters. Several interval estimators for the quantile, such as the confidence intervals based on the generalized variable approach, parametric bootstrap approach and Bayesian intervals using the Markov chain Monte Carlo method, have been proposed. Several test procedures have been proposed, such as tests using the generalized variable approach, tests based on the parametric bootstrap method, and tests using a computational approach. In Chapter 7, we consider the problems of point estimation, interval estimation and hypothesis testing for the quantile of k( 2) exponential populations with a common location and different scale parameters under a progressive type-II censoring scheme. In the case of point estimation, we derive the maximum likelihood estimator (MLE), a modification to it and the uniformly minimum variance unbiased estimator (UMVUE) of the quantile. An estimator dominating the UMVUE is derived. Further, a class of affine equivariant estimators is derived, and an inadmissibility result is proved. Consequently, improved estimators dominating the UMVUE are derived. In the case of interval estimation, several confidence intervals, such as generalized confidence interval, bootstrap confidence interval, and the highest posterior density confidence interval, are obtained for the quantile. In the case of hypothesis testing, we propose generalized variable tests, a parametric bootstrap likelihood ratio test and the computational approach test. In Chapter 8, we consider the problem of comparing the quantiles for several (k 2) logistic populations. The conventional likelihood ratio test is applied, and the cut-off point is determined based on its asymptotic property. Further, we propose two modifications of the likelihood ratio test- a standardized likelihood ratio test and a parametric bootstrap likelihood ratio test. Furthermore, we suggest a computational approach test to test the equality of quantiles by leveraging technology. In all the cases, numerical comparisons of all the proposed methods are considered. The confidence intervals have been compared through average length, coverage probability, and a new measure called - the probability coverage density (wherever applicable). Also, all the proposed tests’ sizes (powers) have been computed using the Monte Carlo simulation procedure. The point estimators are compared using risk function. Real-life examples are discussed for application purposes.