Stochastic Comparisons with Applications: In Order Statistics and Spacings

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