ON ORDERED RANDOM VARIABLES AND ITS CONCOMITANTS FOR SOME CONTINUOUS DISTRIBUTIONS

Abstract

Concomitants of ordered random variables have a wide variety of applications in many areas such as selection problems, prediction analysis, double sampling plans, inference problems and information theory. Their importance appears in many fields such as biological, biomedical, physical, industrial and economical disciplines. The aim of this thesis is to study the concomitants of ordered random variables arising from Morgenstern family and its extensions.

The thesis consists of seven chapters:

Chapter 1 This chapter is an introductory chapter. It contains definitions and basic concepts that are used throughout the thesis. A summary of the previous studies is introduced as follows: the different types of ordered random variables and its concomitants, Morgenstern distributions, measures of information and Bayesian analysis.

Chapter 2 In this chapter, the concomitants of case-II of GOS’s from Morgenstern distributions and recurrence relations between their moments are studied. Further, we provide the BLUE of the location and scale parameters of the concomitants of order statistics from some distributions.

Some results of this chapter are published in: Journal of Statistics Applications and Probability, Vol. 3(3), 345-353, 2015.

Chapter 3 Our goal in this chapter is to obtain and study the residual and past entropies of the Morgenstern family for concomitants of ordered random variables. We also consider the characterization results based on the entropy function for concomitants of ordered random variables of residual and past lifetime distributions. Some results of this chapter are published in: Journal of Probability and Statistics, Vol. 2015, 1-6, 2015.

Chapter 4 Shannon entropy and Fisher information for concomitants of case-I of GOS’s from subfamilies of Morgenstern family when the marginal distributions are Weibull, exponential, Pareto and power function are considered in this chapter. Also, we provide some numerical results of Shannon entropy and Fisher information for concomitants of order statistics. Some results of this chapter are published in: Arabian Journal of Mathematics, Vol. 4(3), 171-184, 2015.

Chapter 5 Concomitants of ordered random variables are studied for one and two variables about different subjects such as moments, recurrence relation, uncertainty and so on. We classify this chapter into two parts. In the first part, the joint density of the concomitants of case-I of GOS’s for Morgenstern family is proposed and study on the moments of such model is considered. Statistical inferences such as MLE, Bayesian estimation under different types of loss function and Bayesian prediction are obtained for the association parameter of Morgenstern family. Applications of these inferences are presented. In the second part, the joint densities of the concomitants of caseI of GOSs for exponential and power function subfamilies of Morgenstern family are derived. Statistical inference such as MLE and Bayesian estimation under different types of loss functions based on IP and NIP for the distribution parameters, reliability and cumulative hazard functions are obtained. In addition, Bayesian prediction bounds, Bayes predictive estimator and approximate confidence intervals of the estimators are considered. Applications of these results are given for concomitants of order statistics. Some results of this chapter are published in: