Life Distributions - Structure of Nonparametric, Semiparametric, and Parametric Families

Preface

Suggestions for Using this Book

Acknowledgements

Contents

Basic Notation and Terminology

Notation

Section and Equation Numbering

Basics

Preliminaries

A. Introduction

B. Probabilistic Descriptions

C. Moments and Other Expectations

D. Families of Distributions

E. Mixtures of Distributions: Introduction

F. Parametric Families: Basic Examples

G. Nonparametric Families: Basic Examples

H. Functions of Random Variables

I. Inverse Distributions: The Lorenz Curve and the Total Time on Test Transform

Ordering Distributions: Descriptive Statistics

A. Magnitude

B. Dispersion

C. Shape

D. Cone Orders

Mixtures

A. Basic Ideas

B. The Conditional Mixing Distribution

C. Limiting Hazard Rates

D. Hazard Transforms of Mixtures

E. Mixtures and Minima

F. Preservation of Orders Under Mixtures

Nonparametric Families

Nonparametric Families: Densities and Hazard Rates

A. Introduction

B. Log-Concave and Log-Convex Densities

C. Monotone Hazard Rates

D. Bathtub Hazard Rates

E. Determination of Hazard Rate Shape

Nonparametric Families: Origins in Reliability Theory

A. Coherent Systems

B. Monotone Hazard Rate Averages

C. New Better (Worse) Than Used Distributions

D. Decreasing Mean Residual Life Distributions

E. New Better (Worse) Than Used in Expectation Distributions

F. Additional Nonparametric Families of Distributions

G. Summary of Relationships and Closure Properties

H. Shock Models

I. Replacement Policies: Renewal Theory

J. Some Additional Families

Nonparametric Families: Inequalities for Moments and Survival Functions

A. Results Concerning Moments

B. Bounds for Survival Functions

Semiparametric Families

Semiparametric Families

A. Introduction

B. Location Parameters

C. Scale Parameters

D. Power Parameters

E. Frailty and Resilience Parameters: Proportional Hazards and Reverse Hazards

F. Tilt Parameters: Proportional Odds Ratios, Extreme Stable Families

G. Hazard Power Parameters

H. Moment Parameters

I. Laplace Transform Parameters

J. Convolution Parameters

K. Age Parameters: Residual Life Families

L. Successive Additions of Parameters

M. Mixing Semiparametric Families

N. Summary of Order Properties

O. Additional Semiparametric Families

P. Distributions not Admitting Parameters

Parametric Families

The Exponential Distribution

A. Defining Functions

B. Characterizations of the Exponential Distribution

C. Some Basic Properties of Exponential Distributions

Parametric Extensions of the Exponential Distribution

A. The Gamma Distribution

B. The Weibull Distribution

C. Exponential Distributions with a Resilience Parameter

D. Exponential Distributions with a Tilt Parameter

E. Generalized Gamma ( Gamma– Weibull) Distribution

F. Weibull Distribution with a Resilience Parameter

G. Residual Life of the Weibull Distribution

H. Weibull Distribution with a Tilt Parameter

I. Generalized Gamma Convolutions

J. Summary of Distributions and Hazard Rates

Gompertz and Gompertz–Makeham Distributions

A. The Gompertz Distribution

B. The Extensions of Makeham

C. Further Extensions of the Gompertz Distribution

D. Summary of Distributions and Hazard Rates

The Pareto and F Distributions and Their Parametric Extensions

A. Introduction

B. Pareto Distributions

C. Generalized F Distribution

D. The F Distribution

E. Ordering Pareto and F Distributions

F. Another Generalization of the Pareto Distribution

Logarithmic Distributions

A. Introduction

B. The Lognormal Distribution

C. Log Logistic Distributions

D. Log Extreme Value Distributions

E. The Log Cauchy Distribution

F. The Log Student’s t Distribution

G. Alternatives for the Logarithm Function

The Inverse Gaussian Distribution

A. The Inverse Gaussian Distribution

B. The Generalized Inverse Gaussian Distribution

C. The Birnbaum–Saunders Distribution

Distributions with Bounded Support

A. Introduction

B. The Uniform Distribution and One- Parameter Extensions

C. The Beta Distribution

D. Additional Two-Parameter Extensions of the Uniform Distribution

E. Introduction of a Scale Parameter

F. Algebraic Structure of the Distributions on [0, 1]

Additional Parametric Families

A. Noncentral Chi-Square Distributions

B. Noncentral F Distributions

C. A Noncentral Beta Distribution and the Noncentral Squared Multiple Correlation Distribution

D. Log Distributions from Nonnegative Random Variables

E. Another Extension of the Exponential Distribution

F. Weibull–Pareto–Beta Distribution

G. Composite Distributions

H. Stable Distributions

Models Involving Several Variables

Covariate Models

A. Introduction

B. Some Regression Models

C. Regression Models for Other Parameters

Several Types of Failure: Competing Risks

A. Definitions and Notation

B. The Problem of Identifiability

C. Assumption of Independence

D. Verifiability of Independence

E. Known Copula

F. Positively Dependent Latent Variables

More About Semi-parametric Families

Characterizations Through Coincidences of Semiparametric Families

A. Introduction

B. Coincidences Leading to Continuous Distributions

C. Coincidences Leading to Discrete Distributions

D. Unresolved Coincidences

More About Semiparametric Families

A. Introduction: Stability Criteria

B. Classification of Parameters

C. Derivation of Families

D. Orderings Generated by Semiparametric Families

E. Related Stronger Orders

Complementary Topics

Some Topics from Probability Theory

A. Foundations

B. Moments

C. Convergence

D. Laplace Transforms and Infinite Divisibility

E. Some Discrete Distributions

F. Poisson and P´ olya Processes: Renewal Theory

G. Extreme-Value Distributions

H. Chebyshev’s Covariance Inequality

I. Multivariate Basics

Convexity and Total Positivity

A. Convex Functions

B. Total Positivity

Some Functional Equations

A. Cauchy’s Equations

B. Variants of Cauchy’s Equations

C. Some Additional Functional Equations

Gamma and Beta Functions

A. The Gamma Function

B. The Beta Function

Some Topics from Analysis

A. Basic Results from Calculus

B. Some Results Concerning Lebesgue Integrals

References

Author Index

Subject Index