Subjective Probability Models for Lifetimes (Monographs on Statistics and Applied Probability 91)

✍ Scribed by Fabio Spizzichino

• Publisher: Chapman and Hall/CRC
• Year: 2001
• Tongue: English
• Leaves: 260
• Series: Monographs on Statistics & Applied Probability 91
• Edition: 1
• Category: Library

✦ Synopsis

Bayesian methods in reliability cannot be fully utilized and understood without full comprehension of the essential differences that exist between frequentist probability and subjective probability. Switching from the frequentist to the subjective approach requires that some fundamental concepts be rethought and suitably redefined. Subjective Probability Models for Lifetimes details those differences and clarifies aspects of subjective probability that have a direct influence on modeling and drawing inference from failure and survival data. In particular, within a framework of Bayesian theory, the author considers the effects of different levels of information in the analysis of the phenomena of positive and negative aging.The author coherently reviews and compares the various definitions and results concerning stochastic ordering, statistical dependence, reliability, and decision theory. He offers a detailed but accessible mathematical treatment of different aspects of probability distributions for exchangeable vectors of lifetimes that imparts a clear understanding of what the "probabilistic description of aging" really is, and why it is important to analyzing survival and failure data.

✦ Table of Contents

Cover
......Page 1
Subjective Probability Models for Lifetimes......Page 2
ISBN 1584880600......Page 6
Contents......Page 8
Essential Bibliography......Page 11
Preface......Page 12
Notation and Acronyms......Page 16
1.1 Introduction......Page 20
1.2 Families of exchangeable events......Page 25
1.2.1 Extendibility and de Finetti s theorem......Page 31
1.2.2 The problem of prediction......Page 34
1.2.3 More on in nitely extendible families......Page 37
1.3 Exchangeable random quantities......Page 40
1.3.1 Extendibility and de Finetti s theorem for exchangeable random variables......Page 42
1.3.2 The problem of prediction......Page 48
1.4 de Finetti type theorems and parametric models......Page 51
1.4.1 Parametric models and prediction su ciency......Page 54
1.5 Exercises......Page 58
1.6 Bibliography......Page 61
2.1 Introduction......Page 66
2.2 Positive exchangeable random quantities......Page 71
2.3 Multivariate conditional hazard rates......Page 89
2.4.1 On the use of the m.c.h.r. functions......Page 99
2.4.2 Dynamic histories total time on test statistic and total hazard transform......Page 102
2.4.3 M.c.h.r. functions and dynamic su ciency......Page 109
2.5 Exercises......Page 111
2.6 Bibliography......Page 114
3.1 Introduction......Page 117
3.1.1 One dimensional stochastic orderings......Page 119
3.1.2 Stochastic monotonicity and orderings for conditional distributions......Page 123
3.2.1 Usual multivariate stochastic ordering......Page 125
3.2.2 Multivariate likelihood ratio ordering......Page 127
3.2.3 Multivariate hazard rate and cumulative hazard rate orderings......Page 128
3.2.4 Some properties of multivariate stochastic orderings and examples......Page 129
3.3.1 Positive dependence......Page 133
3.3.2 Negative dependence......Page 138
3.3.3 Simpson type paradoxes and aspects of dependence in Bayesian analysis......Page 140
3.3.4 Likelihood ratio comparisons between posterior distributions......Page 142
3.4.1 One dimensional notions of aging......Page 146
3.4.2 Dynamic multivariate notions of aging......Page 152
3.4.3 The case of exchangeable lifetimes......Page 154
3.5 Exercises......Page 158
3.6 Bibliography......Page 161
4.1 Introduction......Page 165
4.2 Schur survival functions......Page 169
4.2.1 Basic background about majorization......Page 170
4.2.2 Schur properties of survival functions and multivariate aging......Page 174
4.2.3 Examples of Schur survival functions......Page 178
4.2.4 Schur survival functions and dependence......Page 181
4.3 Schur density functions......Page 182
4.3.1 Schur-constant densities......Page 184
4.3.2 Examples of Schur densities......Page 187
4.3.3 Properties of Schur densities......Page 190
4.4.1 Schur densities and TTT plots......Page 194
4.4.2 Some other notions of Bayesian aging......Page 197
4.4.3 Heterogeneity and multivariate negative aging......Page 198
4.4.5 Extensions to non-exchangeable cases......Page 201
4.5 Exercises......Page 202
4.6 Bibliography......Page 205
5.1 Introduction......Page 208
5.1.1 Statistical decision problems......Page 213
5.1.2 Statistical decision problems and sufficiency......Page 220
5.1.3 Some technical aspects......Page 221
5.2 Stochastic orderings and orderings of decisions......Page 223
5.3 Orderings of residual lifetimes and majorization......Page 229
5.3.1 The case of observations containing failure data......Page 235
5.4 Burn-in problems for exchangeable lifetimes......Page 241
5.4.1 The case of i.i.d. lifetimes......Page 242
5.4.2 Dependence and optimal adaptive burn-in procedures......Page 245
5.4.3 Burn-in......Page 0
5.4.4 Stochastic orderings and open-loop optimal adaptive burn-in procedures......Page 250
5.5 Exercises......Page 257
5.6 Bibliography......Page 258