Stochastic Comparisons with Applications: In Order Statistics and Spacings

✍ Scribed by Subhash C. Kochar

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 280
• Category: Library

✦ Synopsis

This book emphasizes the use of stochastic orders as motivational tools for developing new statistical procedures. Stochastic orders have found useful applications in many disciplines, including reliability theory, survival analysis, risk theory, finance, nonparametric methods, economics and actuarial science.  Written by a statistician, this volume clarifies the connection between stochastic orders and nonparametric methods.

The importance of order statistics and spacings is well recognized. Classically, they mainly focus on the case when the observations are independent and identically distributed, however, several new developments have extended the comparison of order statistics to the case of non-identically distributed or non-independent observations. In addition to giving a detailed discussion of various topics in the general area of stochastic orders, a substantial part of the book is devoted to recent research on stochastic comparisons of order statistics and spacings, including a long chapter on dependence among them.

The book will be useful for graduate students and researchers in statistics, economics, actuarial science and other related disciplines.  In particular, with close to 300 references, it will be a valuable resource for reliability theorists, applied probabilists and statisticians. Readers are expected to have taken a first-year graduate level course in mathematical statistics or in applied probability. 

✦ Table of Contents

Preface
References
Acknowledgements
Contents
1 Introduction and Preliminaries
1.1 Introduction
1.2 Notations and Definitions
1.3 Order Statistics, Record Values, and Generalized Order Statistics
1.3.1 Record Values
1.3.2 Generalized Order Statistics
1.4 Multivariate Distributions and Copulas
1.5 Majorization and Related Orderings
References
2 Magnitude Orders
2.1 Introduction
2.2 (Usual) Stochastic Order
2.3 Hazard Rate and Reverse Hazard Rate Orders
2.4 Likelihood Ratio Order
2.5 Mean Residual Life and Some Other Stochastic Orders
2.6 Relations Between Magnitude Stochastic Orders
2.7 Multivariate Orders
2.8 Stochastic Comparisons Within a Random Vector
2.9 P-P Plots
2.10 Closure and Preservation Properties
2.11 P-P Order
2.12 Inference for the (Usual) Stochastic Ordering
2.13 Inference for the Hazard Rate Ordering
2.14 Inference for the Crossing Point of Two SurvivalFunctions
2.15 Testing for the Joint Stochastic and the Joint Likelihood Ratio Orders
References
3 Variability Orders
3.1 Introduction
3.2 The Dispersive Order
3.3 The Right Spread Function
3.4 The Right Spread Order (Excess Wealth Order)
3.5 Peakedness Order
3.6 Connections with Other Variability Orders
3.7 Nonparametric Inference for Variability (Spread)
References
4 Skewness and Relative Aging Orders
4.1 Introduction
4.2 Skewness Orders
4.3 Connections Between Skewness Orders and Variability Orders
4.4 Statistical Inference for Skewness and Relative Aging
References
5 Dependence Orders
5.1 Introduction
5.2 Notions of Monotone Dependence
5.3 Monotone Dependence Orders
5.4 Measures of Dependence
5.5 Dependence Between N and SN and a Positive Dependence Paradox
5.6 Nonparametric Tests for Independence
References
6 Stochastic Comparisons of Order Statistics
6.1 Introduction
6.2 Stochastic Comparisons of Order Statistics in the One-Sample Problem
6.2.1 Likelihood Ratio Ordering Among OrderStatistics
6.2.2 Hazard Rate Ordering Among Order Statistics
6.2.3 Variability Orders Among Order Statistics
6.2.4 Skewness Comparisons of Order Statistics
6.3 Stochastic Comparisons of Order Statistics in the Two-Sample Problem
6.4 Stochastic Comparisons of Order Statistics in the Proportional Hazard Rate Model
6.5 Comparing the Order Statistics of Heterogeneous Samples with Those of Comparable HomogeneousSamples
6.6 Stochastic Comparisons of Order Statistics in the Two-Sample Scale Problem
References
7 Stochastic Comparisons of Sample Spacings
7.1 Introduction
7.2 Stochastic Orders Between Spacings from Restricted Families of Distributions
7.3 Distributions of Normalized Spacings in the Case of Heterogeneous Exponential Random Variables
7.4 Stochastic Comparisons of Sample Spacings of Two Distributions
7.5 Comparing Spacings of Two Heterogeneous Exponential Samples
7.6 Stochastic Comparisons of Sample Ranges of Heterogeneous and Homogeneous Exponential Samples
7.7 Stochastic Comparisons of Relative Spacings
References
8 Dependence Among Order Statistics and Spacings
8.1 Introduction
8.2 Dependence Between Two Order Statistics
8.3 Dependence Orders Between Two Pairs of Order Statistics Based on Random Samples
8.4 Dependence of the ith Order Statistic on the First Order Statistic in the PHR Model
8.5 Order Statistics from Exchangeable BivariateDistributions
8.6 Dependence Among Spacings
References
9 Stochastic Comparisons of Weighted Sums of RandomVariables
9.1 Introduction
9.2 Peakedness Ordering of Linear Functions of Symmetric Random Variables
9.3 Stochastic Comparisons of Weighted Sums of Non-negative Random Variables
9.3.1 Comparing Weighted Sums of Non-negative Random Variables According to Magnitude and Dispersive Orders
9.3.2 Magnitude and Variability Comparisons for Weighted Sums of Gamma Random Variables
9.3.3 Skewness Orders Between Weighted Sums of Gamma Random Variables
9.3.4 Right Spread Order Between Weighted Sums of Gamma Random Variables
9.4 Stochastic Comparisons of Linear Combinations of Uniform Random Variables
9.5 Stochastic Comparisons of Convolutions of Discrete Distributions
9.5.1 Comparing Binomial and Poisson-Binomial Distributions
9.5.2 Comparing Two Poisson-Binomial Distributions
9.5.3 Comparing Two Geometric Distributions
9.6 Weighted Sums of Dependent Random Variables
References
10 Stochastic Comparisons of Mixtures of Distributions
10.1 Introduction
10.2 Aging Properties of Mixture Distributions
10.3 Stochastic Comparisons of Mixtures
References
Index