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

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

von: Albert W. Marshall, Ingram Olkin

Springer-Verlag, 2007

ISBN: 9780387684772 , 788 Seiten

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 192,59 EUR

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


 

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