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Life distributions: structure of nonparametric, semiparametric, and parametric families

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Bibliographic Details
Main Authors:	Marshall, Albert W. (Author), Olkin, Ingram 1924-2016 (Author)
Format:	Book
Language:	English
Published:	New York Springer 2007
Series:	Springer series in statistics
Subjects:	Distribution (Théorie des probabilités) Distribution (Probability theory) Wahrscheinlichkeitsverteilung
Online Access:	Inhaltsverzeichnis
Item Description:	Hier auch später erschienene, unveränderte Nachdrucke
Physical Description:	XX, 782 S. graph. Darst.
ISBN:	0387203338 9780387203331

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adam_text	Contents Preface ............................................................................ vii Acknowledgements ............................................................ xi Basic Notation and Terminology ........................................ xix Part I. Basics 1. Preliminaries..... .......................................................... 3 A. Introduction ............................................................... 3 B. Probabilistic Descriptions .............................................. 7 C. Moments and Other Expectations .................................... 22 D. Families of Distributions ............................................... 25 E. Mixtures of Distributions: Introduction ............................. 26 F. Parametric Families: Basic Examples ................................ 28 G. Nonparametric Families: Basic Examples ........................... 30 H. Functions of Random Variables ....................................... 32 I. Inverse Distributions: The Lorenz Curve and the Total Time on Test Transform ....................................... 35 2. Ordering Distributions: Descriptive Statistics ................ 47 A. Magnitude ................................................................. 49 B. Dispersion ................................................................. 61 С Shape ....................................................................... 67 D. Cone Orders ............................................................... 76 3. Mixtures ..................................................................... 79 A. Basic Ideas ................................................................ 80 B. The Conditional Mixing Distribution ................................ 83 C. Limiting Hazard Rates .................................................. 86 :iv Contents D. Hazard Transforms of Mixtures ....................................... 88 E. Mixtures and Minima ................................................... 92 F. Preservation of Orders Under Mixtures ............................. 94 Part II. Nonparametric Families 4. Nonparametric Families: Densities and Hazard Rates .............................................................. 97 A. Introduction ............................................................... 97 B. Log-Concave and Log-Convex Densities ............................. 98 C. Monotone Hazard Rates ................................................ 103 D. Bathtub Hazard Rates .................................................. 120 E. Determination of Hazard Rate Shape ................................ 133 5. Nonparametric Families: Origins in Reliability Theory ....................................................... 137 A. Coherent Systems ........................................................ 137 B. Monotone Hazard Rate Averages ..................................... 151 C. New Better (Worse) Than Used Distributions ..................... 161 D. Decreasing Mean Residual Life Distributions ...................... 169 E. New Better (Worse) Than Used in Expectation Distributions .............................................................. 173 F. Additional Nonparametric Families of Distributions ............. 177 G. Summary of Relationships and Closure Properties ................ 180 H. Shock Models ............................................................. 182 I. Replacement Policies: Renewal Theory .............................. 187 J. Some Additional Families .............................................. 192 6. Nonparametric Families: Inequalities for Moments and Survival Functions ................................................. 195 A. Results Concerning Moments .......................................... 195 B. Bounds for Survival Functions ........................................ 198 Part III. Semiparametric Families 7. Semiparametric Families .............................................. 217 A. Introduction ............................................................... 217 B. Location Parameters .................................................... 220 C. Scale Parameters ......................................................... 224 D. Power Parameters ........................................................ 228 Contents xv E. Frailty and Resilience Parameters: Proportional Hazards and Reverse Hazards .................................................... 232 F. Tilt Parameters: Proportional Odds Ratios, Extreme Stable Families ................................................ 242 G. Hazard Power Parameters .............................................. 256 H. Moment Parameters ..................................................... 258 I. Laplace Transform Parameters ........................................ 260 J. Convolution Parameters ................................................ 261 K. Age Parameters: Residual Life Families ............................. 264 L. Successive Additions of Parameters .................................. 265 M. Mixing Semiparametric Families ...................................... 267 N. Summary of Order Properties ......................................... 283 0. Additional Semiparametric Families ................................. 284 P. Distributions not Admitting Parameters ............................ 285 Part IV. Parametric Families 8. The Exponential Distribution ....................................... 291 A. Defining Functions ...................................................... 292 B . Characterizations of the Exponential Distribution ................ 296 C. Some Basic Properties of Exponential Distributions ............. 302 9. Parametric Extensions of the Exponential Distribution ................................................................ 309 A. The Gamma Distribution .............................................. 310 B. The Weibull Distribution .............................................. 321 C. Exponential Distributions with a Resilience Parameter ......... 333 D . Exponential Distributions with a Tilt Parameter ................. 338 E. Generalized Gamma (Gamma-Weibull) Distribution ............ 348 F. Weibull Distribution with a Resilience Parameter ................ 353 G. Residual Life of the Weibull Distribution ........................... 355 H. Weibull Distribution with a Tilt Parameter ....................... 355 1. Generalized Gamma Convolutions ................................... 359 J. Summary of Distributions and Hazard Rates ...................... 360 10. Gompertz and Gompertz-Makeham Distributions .......... 363 A. The Gompertz Distribution ........................................... 364 B. The Extensions of Makeham .......................................... 375 С Further Extensions of the Gompertz Distribution ................ 390 D. Summary of Distributions and Hazard Rates ...................... 396 xvi Contents 11. Pareto and F Distributions and Their Parametric Extensions ................................................. 399 A. Introduction .............................................................. 399 B. Pareto Distributions .................................................... 400 C. Generalized F Distribution ............................................ 411 D. The F Distribution ...................................................... 418 E. Ordering Pareto and F Distributions ................................ 423 F . Another Generalization of the Pareto Distribution ............... 424 12. Logarithmic Distributions ............................................ 427 A. Introduction .............................................................. 427 B. The Lognormal Distribution .......................................... 431 С Log Logistic Distributions ............................................. 441 D. Log Extreme Value Distributions .................................... 442 E. The Log Cauchy Distribution ......................................... 443 F. The Log Student s í Distribution ..................................... 445 G. Alternatives for the Logarithm Function ........................... 445 13. The Inverse Gaussian Distribution ................................ 451 A. The Inverse Gaussian Distribution ................................... 452 B. The Generalized Inverse Gaussian Distribution ................... 459 C. The Birnbaum-Saunders Distribution ............................... 466 14. Distributions with Bounded Support ............................. 473 A. Introduction .............................................................. 473 B. The Uniform Distribution and One-Parameter Extensions ................................................................ 475 C. The Beta Distribution .................................................. 479 D. Additional Two-Parameter Extensions of the Uniform Distribution ................................................... 489 E. Introduction of a Scale Parameter ................................... 493 F. Algebraic Structure of the Distributions on [0, 1]................. 494 15. Additional Parametric Families ..................................... 497 A. Noncentral Chi-Square Distributions ................................ 497 B. Noncentral F Distributions ............................................ 501 C. A Noncentral Beta Distribution and the Noncentral Squared Multiple Correlation Distribution ......................... 504 D . Log Distributions from Nonnegative Random Variables ......... 509 E. Another Extension of the Exponential Distribution .............. 518 F. Weibull-Pareto-Beta Distribution ................................... 521 Contents xvii G. Composite Distributions ............................................... 523 H. Stable Distributions ..................................................... 529 Part V. Models Involving Several Variables 16. Covariate Models ......................................................... 533 A. Introduction .............................................................. 533 B. Some Regression Models ............................................... 536 C. Regression Models for Other Parameters ........................... 540 17. Several Types of Failure: Competing Risks .................... 541 A. Definitions and Notation ............................................... 542 B. The Problem of Identifiability ........................................ 547 C. Assumption of Independence .......................................... 549 D. Verifiability of Independence .......................................... 554 E. Known Copula ........................................................... 555 F. Positively Dependent Latent Variables .............................. 557 Part VI. More About Semi-parametric Families 18. Characterizations Through Coincidences of Semiparametric Families .............................................. 563 A. Introduction .............................................................. 564 B. Coincidences Leading to Continuous Distributions ............... 568 C. Coincidences Leading to Discrete Distributions ................... 596 D. Unresolved Coincidences ............................................... 607 19. More About Semiparametric Families ........................... 611 A. Introduction: Stability Criteria ....................................... 611 B . Classification of Parameters ........................................... 612 C. Derivation of Families .................................................. 619 D. Orderings Generated by Semiparametric Families ................ 626 E. Related Stronger Orders ............................................... 630 Part VII. Complementary Topics 20. Some Topics from Probability Theory ........................... 635 A. Foundations ............................................................... 635 B. Moments .................................................................. 644 C. Convergence .............................................................. 650 xviii Contents D . Laplace Transforms and Infinite Divisibility ....................... 653 E. Some Discrete Distributions ........................................... 658 F. Poisson and Pólya Processes: Renewal Theory .................... 663 G. Extreme-Value Distributions .......................................... 669 H. Chebyshev s Covariance Inequality .................................. 673 I. Multivariate Basics ...................................................... 674 21. Convexity and Total Positivity ...................................... 687 A. Convex Functions ........................................................ 687 B. Total Positivity .......................................................... 694 22. Some Functional Equations ........................................... 701 A. Cauchy s Equations ..................................................... 701 B. Variants of Cauchy s Equations ....................................... 704 C. Some Additional Functional Equations ............................. 712 23. Gamma and Beta Functions ......................................... 717 A. The Gamma Function .................................................. 717 B. The Beta Function ...................................................... 722 24. Some Topics from Analysis.. ......................................... 729 A. Basic Results from Calculus ........................................... 729 B. Some Results Concerning Lebesgue Integrals ...................... 731 References ........................................................................ 733 Author Index ................................................................... 763 Subject Index ................................................................... 771
adam_txt	Contents Preface . vii Acknowledgements . xi Basic Notation and Terminology . xix Part I. Basics 1. Preliminaries. . 3 A. Introduction . 3 B. Probabilistic Descriptions . 7 C. Moments and Other Expectations . 22 D. Families of Distributions . 25 E. Mixtures of Distributions: Introduction . 26 F. Parametric Families: Basic Examples . 28 G. Nonparametric Families: Basic Examples . 30 H. Functions of Random Variables . 32 I. Inverse Distributions: The Lorenz Curve and the Total Time on Test Transform . 35 2. Ordering Distributions: Descriptive Statistics . 47 A. Magnitude . 49 B. Dispersion . 61 С Shape . 67 D. Cone Orders . 76 3. Mixtures . 79 A. Basic Ideas . 80 B. The Conditional Mixing Distribution . 83 C. Limiting Hazard Rates . 86 :iv Contents D. Hazard Transforms of Mixtures . 88 E. Mixtures and Minima . 92 F. Preservation of Orders Under Mixtures . 94 Part II. Nonparametric Families 4. Nonparametric Families: Densities and Hazard Rates . 97 A. Introduction . 97 B. Log-Concave and Log-Convex Densities . 98 C. Monotone Hazard Rates . 103 D. Bathtub Hazard Rates . 120 E. Determination of Hazard Rate Shape . 133 5. Nonparametric Families: Origins in Reliability Theory . 137 A. Coherent Systems . 137 B. Monotone Hazard Rate Averages . 151 C. New Better (Worse) Than Used Distributions . 161 D. Decreasing Mean Residual Life Distributions . 169 E. New Better (Worse) Than Used in Expectation Distributions . 173 F. Additional Nonparametric Families of Distributions . 177 G. Summary of Relationships and Closure Properties . 180 H. Shock Models . 182 I. Replacement Policies: Renewal Theory . 187 J. Some Additional Families . 192 6. Nonparametric Families: Inequalities for Moments and Survival Functions . 195 A. Results Concerning Moments . 195 B. Bounds for Survival Functions . 198 Part III. Semiparametric Families 7. Semiparametric Families . 217 A. Introduction . 217 B. Location Parameters . 220 C. Scale Parameters . 224 D. Power Parameters . 228 Contents xv E. Frailty and Resilience Parameters: Proportional Hazards and Reverse Hazards . 232 F. Tilt Parameters: Proportional Odds Ratios, Extreme Stable Families . 242 G. Hazard Power Parameters . 256 H. Moment Parameters . 258 I. Laplace Transform Parameters . 260 J. Convolution Parameters . 261 K. Age Parameters: Residual Life Families . 264 L. Successive Additions of Parameters . 265 M. Mixing Semiparametric Families . 267 N. Summary of Order Properties . 283 0. Additional Semiparametric Families . 284 P. Distributions not Admitting Parameters . 285 Part IV. Parametric Families 8. The Exponential Distribution . 291 A. Defining Functions . 292 B . Characterizations of the Exponential Distribution . 296 C. Some Basic Properties of Exponential Distributions . 302 9. Parametric Extensions of the Exponential Distribution . 309 A. The Gamma Distribution . 310 B. The Weibull Distribution . 321 C. Exponential Distributions with a Resilience Parameter . 333 D . Exponential Distributions with a Tilt Parameter . 338 E. Generalized Gamma (Gamma-Weibull) Distribution . 348 F. Weibull Distribution with a Resilience Parameter . 353 G. Residual Life of the Weibull Distribution . 355 H. Weibull Distribution with a Tilt Parameter . 355 1. Generalized Gamma Convolutions . 359 J. Summary of Distributions and Hazard Rates . 360 10. Gompertz and Gompertz-Makeham Distributions . 363 A. The Gompertz Distribution . 364 B. The Extensions of Makeham . 375 С Further Extensions of the Gompertz Distribution . 390 D. Summary of Distributions and Hazard Rates . 396 xvi Contents 11. Pareto and F Distributions and Their Parametric Extensions . 399 A. Introduction . 399 B. Pareto Distributions . 400 C. Generalized F Distribution . 411 D. The F Distribution . 418 E. Ordering Pareto and F Distributions . 423 F . Another Generalization of the Pareto Distribution . 424 12. Logarithmic Distributions . 427 A. Introduction . 427 B. The Lognormal Distribution . 431 С Log Logistic Distributions . 441 D. Log Extreme Value Distributions . 442 E. The Log Cauchy Distribution . 443 F. The Log Student's í Distribution . 445 G. Alternatives for the Logarithm Function . 445 13. The Inverse Gaussian Distribution . 451 A. The Inverse Gaussian Distribution . 452 B. The Generalized Inverse Gaussian Distribution . 459 C. The Birnbaum-Saunders Distribution . 466 14. Distributions with Bounded Support . 473 A. Introduction . 473 B. The Uniform Distribution and One-Parameter Extensions . 475 C. The Beta Distribution . 479 D. Additional Two-Parameter Extensions of the Uniform Distribution . 489 E. Introduction of a Scale Parameter . 493 F. Algebraic Structure of the Distributions on [0, 1]. 494 15. Additional Parametric Families . 497 A. Noncentral Chi-Square Distributions . 497 B. Noncentral F Distributions . 501 C. A Noncentral Beta Distribution and the Noncentral Squared Multiple Correlation Distribution . 504 D . Log Distributions from Nonnegative Random Variables . 509 E. Another Extension of the Exponential Distribution . 518 F. Weibull-Pareto-Beta Distribution . 521 Contents xvii G. Composite Distributions . 523 H. Stable Distributions . 529 Part V. Models Involving Several Variables 16. Covariate Models . 533 A. Introduction . 533 B. Some Regression Models . 536 C. Regression Models for Other Parameters . 540 17. Several Types of Failure: Competing Risks . 541 A. Definitions and Notation . 542 B. The Problem of Identifiability . 547 C. Assumption of Independence . 549 D. Verifiability of Independence . 554 E. Known Copula . 555 F. Positively Dependent Latent Variables . 557 Part VI. More About Semi-parametric Families 18. Characterizations Through Coincidences of Semiparametric Families . 563 A. Introduction . 564 B. Coincidences Leading to Continuous Distributions . 568 C. Coincidences Leading to Discrete Distributions . 596 D. Unresolved Coincidences . 607 19. More About Semiparametric Families . 611 A. Introduction: Stability Criteria . 611 B . Classification of Parameters . 612 C. Derivation of Families . 619 D. Orderings Generated by Semiparametric Families . 626 E. Related Stronger Orders . 630 Part VII. Complementary Topics 20. Some Topics from Probability Theory . 635 A. Foundations . 635 B. Moments . 644 C. Convergence . 650 xviii Contents D . Laplace Transforms and Infinite Divisibility . 653 E. Some Discrete Distributions . 658 F. Poisson and Pólya Processes: Renewal Theory . 663 G. Extreme-Value Distributions . 669 H. Chebyshev's Covariance Inequality . 673 I. Multivariate Basics . 674 21. Convexity and Total Positivity . 687 A. Convex Functions . 687 B. Total Positivity . 694 22. Some Functional Equations . 701 A. Cauchy's Equations . 701 B. Variants of Cauchy's Equations . 704 C. Some Additional Functional Equations . 712 23. Gamma and Beta Functions . 717 A. The Gamma Function . 717 B. The Beta Function . 722 24. Some Topics from Analysis. . 729 A. Basic Results from Calculus . 729 B. Some Results Concerning Lebesgue Integrals . 731 References . 733 Author Index . 763 Subject Index . 771
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spelling	Marshall, Albert W. Verfasser (DE-588)1053449909 aut Life distributions structure of nonparametric, semiparametric, and parametric families Albert W. Marshall ; Ingram Olkin New York Springer 2007 XX, 782 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Hier auch später erschienene, unveränderte Nachdrucke Distribution (Théorie des probabilités) Distribution (Probability theory) Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s DE-604 Olkin, Ingram 1924-2016 Verfasser (DE-588)118889036 aut Erscheint auch als Online-Ausgabe 978-0-387-68477-2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015511708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
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title	Life distributions structure of nonparametric, semiparametric, and parametric families
title_auth	Life distributions structure of nonparametric, semiparametric, and parametric families
title_exact_search	Life distributions structure of nonparametric, semiparametric, and parametric families
title_exact_search_txtP	Life distributions structure of nonparametric, semiparametric, and parametric families
title_full	Life distributions structure of nonparametric, semiparametric, and parametric families Albert W. Marshall ; Ingram Olkin
title_fullStr	Life distributions structure of nonparametric, semiparametric, and parametric families Albert W. Marshall ; Ingram Olkin
title_full_unstemmed	Life distributions structure of nonparametric, semiparametric, and parametric families Albert W. Marshall ; Ingram Olkin
title_short	Life distributions
title_sort	life distributions structure of nonparametric semiparametric and parametric families
title_sub	structure of nonparametric, semiparametric, and parametric families
topic	Distribution (Théorie des probabilités) Distribution (Probability theory) Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd
topic_facet	Distribution (Théorie des probabilités) Distribution (Probability theory) Wahrscheinlichkeitsverteilung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015511708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT marshallalbertw lifedistributionsstructureofnonparametricsemiparametricandparametricfamilies AT olkiningram lifedistributionsstructureofnonparametricsemiparametricandparametricfamilies

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