Olkin Life Distributions

Structure of Nonparametric, Semiparametric, and Parametric Families
1. Auflage 2007
ISBN: 978-0-387-68477-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 785 Seiten

For over 200 years, practitioners have been developing parametric families of probability distributions for data analysis. More recently, an active development of nonparametric and semiparametric families has occurred. This book includes an extensive discussion of a wide variety of distribution families—nonparametric, semiparametric and parametric—some well known and some not. An all-encompassing view is taken for the purpose of identifying relationships, origins and structures of the various families. A unified methodological approach for the introduction of parameters into families is developed, and the properties that the parameters imbue a distribution are clarified. These results provide essential tools for intelligent choice of models for data analysis. Many of the results given are new and have not previously appeared in print. This book provides a comprehensive reference for anyone working with nonnegative data.

Autoren/Hrsg.

Olkin, Ingram

Inhaltsverzeichnis

1;Preface;7
1.1;Suggestions for Using this Book;8
2;Acknowledgements;10
3;Contents;12
4;Basic Notation and Terminology;18
4.1;Notation;18
4.2;Section and Equation Numbering;19
5;Basics;20
5.1;Preliminaries;21
5.1.1;A. Introduction;21
5.1.2;B. Probabilistic Descriptions;25
5.1.3;C. Moments and Other Expectations;40
5.1.4;D. Families of Distributions;43
5.1.5;E. Mixtures of Distributions: Introduction;44
5.1.6;F. Parametric Families: Basic Examples;46
5.1.7;G. Nonparametric Families: Basic Examples;48
5.1.8;H. Functions of Random Variables;50
5.1.9;I. Inverse Distributions: The Lorenz Curve and the Total Time on Test Transform;53
5.2;Ordering Distributions: Descriptive Statistics;64
5.2.1;A. Magnitude;66
5.2.2;B. Dispersion;78
5.2.3;C. Shape;84
5.2.4;D. Cone Orders;93
5.3;Mixtures;95
5.3.1;A. Basic Ideas;96
5.3.2;B. The Conditional Mixing Distribution;99
5.3.3;C. Limiting Hazard Rates;102
5.3.4;D. Hazard Transforms of Mixtures;104
5.3.5;E. Mixtures and Minima;108
5.3.6;F. Preservation of Orders Under Mixtures;110
6;Nonparametric Families;111
6.1;Nonparametric Families: Densities and Hazard Rates;112
6.1.1;A. Introduction;112
6.1.2;B. Log-Concave and Log-Convex Densities;113
6.1.3;C. Monotone Hazard Rates;118
6.1.4;D. Bathtub Hazard Rates;135
6.1.5;E. Determination of Hazard Rate Shape;148
6.2;Nonparametric Families: Origins in Reliability Theory;152
6.2.1;A. Coherent Systems;152
6.2.2;B. Monotone Hazard Rate Averages;166
6.2.3;C. New Better (Worse) Than Used Distributions;176
6.2.4;D. Decreasing Mean Residual Life Distributions;184
6.2.5;E. New Better (Worse) Than Used in Expectation Distributions;188
6.2.6;F. Additional Nonparametric Families of Distributions;192
6.2.7;G. Summary of Relationships and Closure Properties;195
6.2.8;H. Shock Models;197
6.2.9;I. Replacement Policies: Renewal Theory;202
6.2.10;J. Some Additional Families;207
6.3;Nonparametric Families: Inequalities for Moments and Survival Functions;209
6.3.1;A. Results Concerning Moments;209
6.3.2;B. Bounds for Survival Functions;212
7;Semiparametric Families;229
7.1;Semiparametric Families;230
7.1.1;A. Introduction;230
7.1.2;B. Location Parameters;233
7.1.3;C. Scale Parameters;237
7.1.4;D. Power Parameters;241
7.1.5;E. Frailty and Resilience Parameters: Proportional Hazards and Reverse Hazards;245
7.1.6;F. Tilt Parameters: Proportional Odds Ratios, Extreme Stable Families;255
7.1.7;G. Hazard Power Parameters;269
7.1.8;H. Moment Parameters;271
7.1.9;I. Laplace Transform Parameters;273
7.1.10;J. Convolution Parameters;274
7.1.11;K. Age Parameters: Residual Life Families;277
7.1.12;L. Successive Additions of Parameters;278
7.1.13;M. Mixing Semiparametric Families;280
7.1.14;N. Summary of Order Properties;296
7.1.15;O. Additional Semiparametric Families;297
7.1.16;P. Distributions not Admitting Parameters;298
8;Parametric Families;301
8.1;The Exponential Distribution;302
8.1.1;A. Defining Functions;303
8.1.2;B. Characterizations of the Exponential Distribution;307
8.1.3;C. Some Basic Properties of Exponential Distributions;313
8.2;Parametric Extensions of the Exponential Distribution;319
8.2.1;A. The Gamma Distribution;320
8.2.2;B. The Weibull Distribution;331
8.2.3;C. Exponential Distributions with a Resilience Parameter;343
8.2.4;D. Exponential Distributions with a Tilt Parameter;348
8.2.5;E. Generalized Gamma ( Gamma– Weibull) Distribution;358
8.2.6;F. Weibull Distribution with a Resilience Parameter;363
8.2.7;G. Residual Life of the Weibull Distribution;365
8.2.8;H. Weibull Distribution with a Tilt Parameter;365
8.2.9;I. Generalized Gamma Convolutions;369
8.2.10;J. Summary of Distributions and Hazard Rates;370
8.3;Gompertz and Gompertz–Makeham Distributions;372
8.3.1;A. The Gompertz Distribution;373
8.3.2;B. The Extensions of Makeham;384
8.3.3;C. Further Extensions of the Gompertz Distribution;399
8.3.4;D. Summary of Distributions and Hazard Rates;405
8.4;The Pareto and F Distributions and Their Parametric Extensions;408
8.4.1;A. Introduction;408
8.4.2;B. Pareto Distributions;409
8.4.3;C. Generalized F Distribution;420
8.4.4;D. The F Distribution;427
8.4.5;E. Ordering Pareto and F Distributions;432
8.4.6;F. Another Generalization of the Pareto Distribution;433
8.5;Logarithmic Distributions;435
8.5.1;A. Introduction;435
8.5.2;B. The Lognormal Distribution;439
8.5.3;C. Log Logistic Distributions;449
8.5.4;D. Log Extreme Value Distributions;450
8.5.5;E. The Log Cauchy Distribution;451
8.5.6;F. The Log Student’s t Distribution;453
8.5.7;G. Alternatives for the Logarithm Function;453
8.6;The Inverse Gaussian Distribution;458
8.6.1;A. The Inverse Gaussian Distribution;459
8.6.2;B. The Generalized Inverse Gaussian Distribution;466
8.6.3;C. The Birnbaum–Saunders Distribution;473
8.7;Distributions with Bounded Support;479
8.7.1;A. Introduction;479
8.7.2;B. The Uniform Distribution and One- Parameter Extensions;481
8.7.3;C. The Beta Distribution;485
8.7.4;D. Additional Two-Parameter Extensions of the Uniform Distribution;495
8.7.5;E. Introduction of a Scale Parameter;499
8.7.6;F. Algebraic Structure of the Distributions on [0, 1];500
8.8;Additional Parametric Families;502
8.8.1;A. Noncentral Chi-Square Distributions;502
8.8.2;B. Noncentral F Distributions;506
8.8.3;C. A Noncentral Beta Distribution and the Noncentral Squared Multiple Correlation Distribution;509
8.8.4;D. Log Distributions from Nonnegative Random Variables;514
8.8.5;E. Another Extension of the Exponential Distribution;523
8.8.6;F. Weibull–Pareto–Beta Distribution;525
8.8.7;G. Composite Distributions;528
8.8.8;H. Stable Distributions;534
9;Models Involving Several Variables;536
9.1;Covariate Models;537
9.1.1;A. Introduction;537
9.1.2;B. Some Regression Models;540
9.1.3;C. Regression Models for Other Parameters;544
9.2;Several Types of Failure: Competing Risks;545
9.2.1;A. Definitions and Notation;546
9.2.2;B. The Problem of Identifiability;551
9.2.3;C. Assumption of Independence;553
9.2.4;D. Verifiability of Independence;558
9.2.5;E. Known Copula;559
9.2.6;F. Positively Dependent Latent Variables;561
10;More About Semi-parametric Families;564
10.1;Characterizations Through Coincidences of Semiparametric Families;565
10.1.1;A. Introduction;566
10.1.2;B. Coincidences Leading to Continuous Distributions;570
10.1.3;C. Coincidences Leading to Discrete Distributions;598
10.1.4;D. Unresolved Coincidences;609
10.2;More About Semiparametric Families;612
10.2.1;A. Introduction: Stability Criteria;612
10.2.2;B. Classification of Parameters;613
10.2.3;C. Derivation of Families;620
10.2.4;D. Orderings Generated by Semiparametric Families;627
10.2.5;E. Related Stronger Orders;631
11;Complementary Topics;633
11.1;Some Topics from Probability Theory;634
11.1.1;A. Foundations;634
11.1.2;B. Moments;643
11.1.3;C. Convergence;649
11.1.4;D. Laplace Transforms and Infinite Divisibility;652
11.1.5;E. Some Discrete Distributions;657
11.1.6;F. Poisson and P´ olya Processes: Renewal Theory;662
11.1.7;G. Extreme-Value Distributions;668
11.1.8;H. Chebyshev’s Covariance Inequality;672
11.1.9;I. Multivariate Basics;673
11.2;Convexity and Total Positivity;686
11.2.1;A. Convex Functions;686
11.2.2;B. Total Positivity;693
11.3;Some Functional Equations;700
11.3.1;A. Cauchy’s Equations;700
11.3.2;B. Variants of Cauchy’s Equations;703
11.3.3;C. Some Additional Functional Equations;711
11.4;Gamma and Beta Functions;715
11.4.1;A. The Gamma Function;715
11.4.2;B. The Beta Function;720
11.5;Some Topics from Analysis;726
11.5.1;A. Basic Results from Calculus;726
11.5.2;B. Some Results Concerning Lebesgue Integrals;728
12;References;730
13;Author Index;760
14;Subject Index;768