Stochastic Orders

herausgegeben von: Moshe Shaked, J. George Shanthikumar

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Stochastic ordering is a fundamental guide for decision making under uncertainty. It is also an essential tool in the study of structural properties of complex stochastic systems. This reference text presents a comprehensive coverage of the various notions of stochastic orderings, their closure properties, and their applications. Some of these orderings are routinely used in many applications in economics, finance, insurance, management science, operations research, statistics, and various other fields of study. And the value of the other notions of stochastic orderings still needs to be explored further.

This book is an ideal reference for anyone interested in decision making under uncertainty and interested in the analysis of complex stochastic systems. It is suitable as a text for advanced graduate course on stochastic ordering and applications.

Inhaltsverzeichnis

Frontmatter

1. Univariate Stochastic Orders

Abstract

In this chapter we study stochastic orders that compare the “location” or the “magnitude” of random variables. The most important and common orders that are considered in this chapter are the usual stochastic order ≤_st, the hazard rate order ≤hr, and the likelihood ratio order ≤_lr. Some variations of these orders, and some related orders, are also examined in this chapter.

2. Mean Residual Life Orders

Abstract

In this chapter we study two orders that are based on comparisons of functionals of mean residual lives. Like the orders in Chapter 1, the purpose of the orders here is to compare the “location” or the “magnitude” of random variables. Among other things, the relationship between the orders of Chapter 1 and the orders in this chapter will be analyzed.

3. Univariate Variability Orders

Abstract

In this chapter we study stochastic orders that compare the “variability” or the “dispersion” of random variables. The most important and common orders that are studied in this chapter are the convex and the dispersive orders. We also study in this chapter the excess wealth order (which is also called the right spread order) which is found to be useful in an increasing number of applications. Various related orders are also examined in this chapter.

4. Univariate Monotone Convex and Related Orders

Abstract

In Chapter 1 we studied orders that compare random variables according to their “magnitude”. In Chapter 3 the studied orders compare random variables according to their “variability”. The orders that are discussed in this chapter compare random variables according to both their “location” and their “spread”. The most important and common orders that are studied in this chapter are the increasing convex and the increasing concave orders. Also the transform orders that are studied here, that is, the convex, the star, and the superadditive orders, are of interest in many theoretical and practical applications. In addition, some other related orders are investigated in this chapter as well.

5. The Laplace Transform and Related Orders

Abstract

The most important common order that is studied in this chapter is the Laplace transform order. Like the orders that were discussed in Chapter 4, the Laplace transform order compares random variables according to both their “location” and their “spread”. Two other useful orders, based on ratios of Laplace transforms, are also discussed in this chapter. In addition, some other related orders are investigated in this chapter as well.

6. Multivariate Stochastic Orders

Abstract

In this chapter we describe various extensions, of the univariate stochastic orders in Chapters 1 and 2, to the multivariate case. The most important common orders that are studied in this chapter are the multivariate stochastic orders ≤_st and ≤_lr. Multivariate extensions of the orders ≤_hr and ≤_mrl are also studied in this chapter. Also, we review here further analogs of the univariate order ≤_st, such as the upper and lower orthants orders. In addition, some other related orders are investigated in this chapter as well.

7. Multivariate Variability and Related Orders

Abstract

In this chapter we describe various extensions, of the univariate variability orders in Chapters 3 and 4, to the multivariate case. The most important common orders that are studied in this chapter are the increasing and the directional convex and concave orders. Multivariate extensions of the order ≤_disp are also studied in this chapter. Some multivariate extensions of the transform orders, and of the Laplace transform order, are investigated in this chapter as well.

8. Stochastic Convexity and Concavity

Abstract

In this chapter we study stochastic monotonicities of parametric families of distributions with respect to various stochastic orders. We have already encountered stochastic monotonicities earlier in this book. For example, condition (1.A.13) in Theorem 1.A.6, condition (3.A.47) in Theorem 3.A.21, and condition (4.A.17) in Theorem 4.A.18 describe such monotonicities. In this chapter a systematic study of such stochastic monotonicities is given. Various notions of stochastic convexity and concavity are reviewed. A multivariate extension of the notion of stochastic convexity, namely, stochastic directional convexity, is investigated in this chapter as well.

9. Positive Dependence Orders

Abstract

Notions of positive dependence of two random variables X1 and X2 have been introduced in the literature in an effort to mathematically describe the property that “large (respectively, small) values of X1 tend to go together with large (respectively, small) values of X2.” Many of the notions of positive dependence are defined by means of some comparison of the joint distribution of X1 and X2 with their distribution under the theoretical assumption that X1 and X2 are independent. Often such a comparison can be extended to general pairs of bivariate distributions with given marginals. This fact led researchers to introduce various notions of positive dependence orders. These orders are designed to compare the strength of the positive dependence of the two underlying bivariate distributions. In this chapter we describe some such notions.

Backmatter

Titel: Stochastic Orders
herausgegeben von: Moshe Shaked
J. George Shanthikumar
Verlag: Springer New York
Electronic ISBN: 978-0-387-34675-5
Print ISBN: 978-0-387-32915-4
DOI: https://doi.org/10.1007/978-0-387-34675-5