Inference on Powers of Scale Parameters under Common Parameter Setup

Abstract

The thesis deals with the problems of point estimation, interval estimation and hypothesis testing on the powers of scale parameters for normal and exponential model setups with a common parameter assumption. In the case of point estimation, some decision-theoretic results are derived. Further, the problems of interval estimation and hypothesis testing about the powers of scale parameters are also addressed. Chapter 1 introduces the model problem and discusses in detail the related literature. Further, the summary of the results obtained in the thesis is also discussed. In Chapter 2, we provide some basic definitions, fundamental results, and methodologies on point estimation, interval estimation, and hypothesis testing problems that help to frame the rest of the chapters. In Chapter 3, the problems of inference on the powers of scale parameters have been considered when samples are available from two normal populations with a common mean. In particular, we derived the maximum likelihood estimators (MLEs) and several plug-in estimators using some of the popular estimators of the common mean. A sufficient condition for improving affine equivariant estimators using the quadratic loss function is derived. Moreover, we propose several interval estimators, such as the asymptotic confidence interval, bootstrap confidence intervals, highest posterior density intervals, and intervals based on generalized pivot variables. Further, several test procedures are derived, such as the likelihood ratio test, a parametric bootstrap approach test, two computational approach tests, and tests based on the generalized p-value method. A simulation study has been conducted to assess and compare the performances of all the suggested intervals in terms of average length, coverage probability, and a new measure called - the probability coverage density. The test procedures are compared in terms of their sizes and powers. Real-life examples have been considered to demonstrate the potential applicability of all the suggested inferential procedures. In Chapter 4, we consider the model discussed in Chapter 3 and aim to estimate the powers of scale parameters when they follow certain simple ordering. Maximum-likelihood estimators and several plug-in type estimators using some popular estimators of the common mean have been proposed. Sufficient conditions for improving equivariant estimators for the powers of the scale parameters under order restriction have been derived. Consequently, several improved estimators have been proposed. A numerical comparison among all the proposed estimators in the special cases (c = 0.5 and c = 1) has been made in terms of risk values using the quadratic loss function, and recommendations are given for the use of the estimators. Finally, a real-life example has been considered to show the potential application of the model problem. In Chapter 5, we discuss the problems of interval estimation and hypothesis testing on the powers of the ratio of scale parameters for two normal populations with a common mean. We derive several confidence intervals, such as the asymptotic confidence interval, bootstrap confidence interval (bootstrap-p, bootstrap-t), and the generalized intervals. The generalized confidence intervals are constructed utilizing some of the existing estimators of the common mean. We discuss several test methods for hypothesis testing, such as the computational approach test, the asymptotic likelihood ratio test, the parametric bootstrap likelihood ratio test, and tests based on the generalized p-value approach. Numerical comparisons have been made to compare all the suggested confidence intervals and test procedures. Chapter 6 addresses the problems of point estimation, interval estimation, and hypothesis testing on the powers of scale parameters for two exponential populations with a common location. In the case of point estimation, various point estimators are derived, such as the maximum likelihood estimator (MLE), the uniform minimum variance unbiased estimator (UMVUE), and two plug-in estimators using modified MLE and the UMVUE of the common location parameter. An Inadmissibility result is proved for the class of affine equivariant estimators. In the realm of interval estimation, several approaches are discussed, such as approximate intervals utilizing the asymptotic normality of MLE, bootstrap intervals, and intervals using the generalized variable method. Several hypothesis testing procedures are developed, namely, the parametric bootstrap likelihood ratio test (PBLRT), computational approach tests (CAT), and tests based on the generalized p-value approach. Simulation studies are conducted to compare the performances of point estimators, interval estimators and test procedures. The chapter concludes with a real-life application. In Chapter 7, we consider the model discussed in Chapter 6 and estimate the powers of scale parameters under order restriction. The chapter introduces several classical estimators, such as the maximum likelihood estimators, plug-in type restricted maximum likelihood estimators, and uniform minimum variance unbiased estimators. Sufficient conditions for constructing improved estimators have been derived under the scale and affine group of transformations. Consequently, several improved estimators for the powers of the scale parameters under order restriction have been proposed. A simulation study has been conducted using the quadratic loss function to compare these estimators in terms of risk values, and recommendations are given for the use of the estimators based on our simulation study. Chapter 8 discusses inference about the powers of the common scale parameter when samples are available from two exponential populations with different location parameters. It mainly focuses on hypothesis testing and interval estimation procedures. A parametric likelihood ratio test is proposed using artificial bootstrap samples to test the null hypothesis against an appropriate alternative. Further, the study suggests two computational approach tests and several generalized test procedures to test the hypothesis. Moreover, the chapter presents various interval estimation techniques for the powers of the common scale parameter. These include bootstrap intervals, the highest posterior density interval, and certain generalized intervals. A comprehensive simulation study is conducted to assess and compare the performance of all the proposed tests and intervals. Finally, the practical applicability of the proposed model problem is illustrated through real-life examples.