Life Distributions (eBook)

Albert W. Marshall, Professor Emeritus of Statistics at the University of British Colombia, previously served on the faculty of the University of Rochester and on the staff of the Boeing Scientific Research Laboratories. His fundamental contributions to reliability theory have had a profound effect in furthering its development. Ingram Olkin is Professor Emeritus of Statistics and Education at Stanford University, after having served on the faculties of Michigan State University and the University of Minnesota. He has made significant contributions in multivariate analysis and in the development of statistical methods in meta-analysis, which has resulted in its use in many applications. Professors Marshall and Olkin, coauthors of papers on inequalities, multivariate distributions, and matrix analysis, are about to celebrate 50 years of collaborations. Their basic book on majorization has promoted awareness of the subject, and led to new applications in such fields as economics, combinatorics, statistics, probability, matrix theory, chemistry, and political science.

Preface 7
Suggestions for Using this Book 8
Acknowledgements 10
Contents 12
Basic Notation and Terminology 18
Notation 18
Section and Equation Numbering 19
Basics 20
Preliminaries 21
A. Introduction 21
B. Probabilistic Descriptions 25
C. Moments and Other Expectations 40
D. Families of Distributions 43
E. Mixtures of Distributions: Introduction 44
F. Parametric Families: Basic Examples 46
G. Nonparametric Families: Basic Examples 48
H. Functions of Random Variables 50
I. Inverse Distributions: The Lorenz Curve and the Total Time on Test Transform 53
Ordering Distributions: Descriptive Statistics 64
A. Magnitude 66
B. Dispersion 78
C. Shape 84
D. Cone Orders 93
Mixtures 95
A. Basic Ideas 96
B. The Conditional Mixing Distribution 99
C. Limiting Hazard Rates 102
D. Hazard Transforms of Mixtures 104
E. Mixtures and Minima 108
F. Preservation of Orders Under Mixtures 110
Nonparametric Families 111
Nonparametric Families: Densities and Hazard Rates 112
A. Introduction 112
B. Log-Concave and Log-Convex Densities 113
C. Monotone Hazard Rates 118
D. Bathtub Hazard Rates 135
E. Determination of Hazard Rate Shape 148
Nonparametric Families: Origins in Reliability Theory 152
A. Coherent Systems 152
B. Monotone Hazard Rate Averages 166
C. New Better (Worse) Than Used Distributions 176
D. Decreasing Mean Residual Life Distributions 184
E. New Better (Worse) Than Used in Expectation Distributions 188
F. Additional Nonparametric Families of Distributions 192
G. Summary of Relationships and Closure Properties 195
H. Shock Models 197
I. Replacement Policies: Renewal Theory 202
J. Some Additional Families 207
Nonparametric Families: Inequalities for Moments and Survival Functions 209
A. Results Concerning Moments 209
B. Bounds for Survival Functions 212
Semiparametric Families 229
Semiparametric Families 230
A. Introduction 230
B. Location Parameters 233
C. Scale Parameters 237
D. Power Parameters 241
E. Frailty and Resilience Parameters: Proportional Hazards and Reverse Hazards 245
F. Tilt Parameters: Proportional Odds Ratios, Extreme Stable Families 255
G. Hazard Power Parameters 269
H. Moment Parameters 271
I. Laplace Transform Parameters 273
J. Convolution Parameters 274
K. Age Parameters: Residual Life Families 277
L. Successive Additions of Parameters 278
M. Mixing Semiparametric Families 280
N. Summary of Order Properties 296
O. Additional Semiparametric Families 297
P. Distributions not Admitting Parameters 298
Parametric Families 301
The Exponential Distribution 302
A. Defining Functions 303
B. Characterizations of the Exponential Distribution 307
C. Some Basic Properties of Exponential Distributions 313
Parametric Extensions of the Exponential Distribution 319
A. The Gamma Distribution 320
B. The Weibull Distribution 331
C. Exponential Distributions with a Resilience Parameter 343
D. Exponential Distributions with a Tilt Parameter 348
E. Generalized Gamma ( Gamma– Weibull) Distribution 358
F. Weibull Distribution with a Resilience Parameter 363
G. Residual Life of the Weibull Distribution 365
H. Weibull Distribution with a Tilt Parameter 365
I. Generalized Gamma Convolutions 369
J. Summary of Distributions and Hazard Rates 370
Gompertz and Gompertz–Makeham Distributions 372
A. The Gompertz Distribution 373
B. The Extensions of Makeham 384
C. Further Extensions of the Gompertz Distribution 399
D. Summary of Distributions and Hazard Rates 405
The Pareto and F Distributions and Their Parametric Extensions 408
A. Introduction 408
B. Pareto Distributions 409
C. Generalized F Distribution 420
D. The F Distribution 427
E. Ordering Pareto and F Distributions 432
F. Another Generalization of the Pareto Distribution 433
Logarithmic Distributions 435
A. Introduction 435
B. The Lognormal Distribution 439
C. Log Logistic Distributions 449
D. Log Extreme Value Distributions 450
E. The Log Cauchy Distribution 451
F. The Log Student’s t Distribution 453
G. Alternatives for the Logarithm Function 453
The Inverse Gaussian Distribution 458
A. The Inverse Gaussian Distribution 459
B. The Generalized Inverse Gaussian Distribution 466
C. The Birnbaum–Saunders Distribution 473
Distributions with Bounded Support 479
A. Introduction 479
B. The Uniform Distribution and One- Parameter Extensions 481
C. The Beta Distribution 485
D. Additional Two-Parameter Extensions of the Uniform Distribution 495
E. Introduction of a Scale Parameter 499
F. Algebraic Structure of the Distributions on [0, 1] 500
Additional Parametric Families 502
A. Noncentral Chi-Square Distributions 502
B. Noncentral F Distributions 506
C. A Noncentral Beta Distribution and the Noncentral Squared Multiple Correlation Distribution 509
D. Log Distributions from Nonnegative Random Variables 514
E. Another Extension of the Exponential Distribution 523
F. Weibull–Pareto–Beta Distribution 525
G. Composite Distributions 528
H. Stable Distributions 534
Models Involving Several Variables 536
Covariate Models 537
A. Introduction 537
B. Some Regression Models 540
C. Regression Models for Other Parameters 544
Several Types of Failure: Competing Risks 545
A. Definitions and Notation 546
B. The Problem of Identifiability 551
C. Assumption of Independence 553
D. Verifiability of Independence 558
E. Known Copula 559
F. Positively Dependent Latent Variables 561
More About Semi-parametric Families 564
Characterizations Through Coincidences of Semiparametric Families 565
A. Introduction 566
B. Coincidences Leading to Continuous Distributions 570
C. Coincidences Leading to Discrete Distributions 598
D. Unresolved Coincidences 609
More About Semiparametric Families 612
A. Introduction: Stability Criteria 612
B. Classification of Parameters 613
C. Derivation of Families 620
D. Orderings Generated by Semiparametric Families 627
E. Related Stronger Orders 631
Complementary Topics 633
Some Topics from Probability Theory 634
A. Foundations 634
B. Moments 643
C. Convergence 649
D. Laplace Transforms and Infinite Divisibility 652
E. Some Discrete Distributions 657
F. Poisson and P´ olya Processes: Renewal Theory 662
G. Extreme-Value Distributions 668
H. Chebyshev’s Covariance Inequality 672
I. Multivariate Basics 673
Convexity and Total Positivity 686
A. Convex Functions 686
B. Total Positivity 693
Some Functional Equations 700
A. Cauchy’s Equations 700
B. Variants of Cauchy’s Equations 703
C. Some Additional Functional Equations 711
Gamma and Beta Functions 715
A. The Gamma Function 715
B. The Beta Function 720
Some Topics from Analysis 726
A. Basic Results from Calculus 726
B. Some Results Concerning Lebesgue Integrals 728
References 730
Author Index 760
Subject Index 768