## Sponsor

Portland State University. Department of Mathematics and Statistics

## First Advisor

Subhash Kochar

## Term of Graduation

2010

## Date of Publication

1-1-2010

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

## Department

Mathematics and Statistics

## Language

English

## Subjects

Statistics, Stochastic orders

## DOI

10.15760/etd.391

## Physical Description

1 online resource (xi, 132 pages)

## Abstract

The statistics literature has mostly focused on the case when the data available is in the form of a random sample. In many cases, the observations are not identically distributed. Such samples are called heterogeneous samples. The study of heterogeneous samples is of great interest in many areas, such as statistics, econometrics, reliability engineering, operation research and risk analysis.

Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc.

In this dissertation, most of the work is devoted to investigating the properties of statistics based on heterogeneous samples with the aid of stochastic orders. We will see the effect of the change in the stochastic properties of various functions of observations as their parameters change. The theory of majorization will be used for this purpose.

First, order statistics from heterogeneous samples will be investigated. Order statistics appear everywhere in statistics and related areas. The *k*-out-of-*n* systems are building blocks of a coherent system. The lifetime of such a system is the same as that of the (*n*-*k*+1)th order statistic in a sample size of *n*. Stochastic comparisons between order statistics have been studied extensively in the literature in case the parent observations are independent and identically distributed. However, in practice this assumption is often violated as different components in a system may not have the same distribution. Comparatively less work has been done in the case of heterogeneous random variables, mainly because of the reason that their distribution theory is very complicated. Some open problems in the literature have been solved in the dissertation. Some new problems associated with order statistics have been investigated in the thesis.

Next, stochastic properties of spacings based on heterogeneous observations are studied. Spacings are of great interest in many areas of statistics, in particular, in the characterizations of distributions, goodness-of-fit tests, life testing and reliability models. In particular, the stochastic properties of the sample range are investigated in detail. Applications in reliability theory are highlighted.

The relative dependence between extreme order statistics will be investigated in Chapter 4. In particular, the open problem discussed in *Dolati, et al.* (2008) is solved in this Chapter.

In the last Chapter, convolutions of random variables from heterogeneous samples will be investigated. Convolutions have been widely used in many areas to model many practical situations. For example, in reliability theory, it arises as the lifetime of a redundant standby system; in queuing theory, it is used to model the total service time by an agent in a system; in insurance, it is used to model total claims on a number of policies in the individual risk model. I will compare the dispersion and skewness properties of convolutions of different heterogeneous samples. The tail behavior of convolutions are investigated as well.

The work in this dissertation has significant applications in many diverse areas of applied probability and statistics. For example, statistics based on order statistics and spacings from heterogeneous samples arise in studying the robust properties of statistical procedures; the work on order statistics will also provide a better estimation of lifetime of a coherent system in reliability engineering; convolution results will be of great interest in insurance and actuarial science for evaluating risks.

## Rights

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## Persistent Identifier

http://archives.pdx.edu/ds/psu/4754

## Recommended Citation

Xu, Maochao, "Stochastic Orders in Heterogeneous Samples with Applications" (2010). *Dissertations and Theses.* Paper 391.

https://doi.org/10.15760/etd.391