Subjective Probability Models for Lifetimes

✍ Scribed by Fabio Spizzichino F. Spizzichino

• Publisher: Chapman & Hall /CRC
• Year: 2001
• Tongue: English
• Leaves: 259
• 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

Contents......Page 0
Subjective Probability Models for Lifetimes......Page 1
Contents......Page 7
Essential Bibliography......Page 10
Preface......Page 11
Notation and Acronyms......Page 15
1.1 Introduction......Page 19
1.2 Families of exchangeable events......Page 24
1.2.1 Extendibility and de Finetti s theorem......Page 30
1.2.2 The problem of prediction......Page 33
1.2.3 More on in nitely extendible families......Page 36
1.3 Exchangeable random quantities......Page 39
1.3.1 Extendibility and de Finetti s theorem for exchangeable random variables......Page 41
1.3.2 The problem of prediction......Page 47
1.4 de Finetti type theorems and parametric models......Page 50
1.4.1 Parametric models and prediction su ciency......Page 53
1.5 Exercises......Page 57
1.6 Bibliography......Page 60
2.1 Introduction......Page 65
2.2 Positive exchangeable random quantities......Page 70
2.3 Multivariate conditional hazard rates......Page 88
2.4.1 On the use of the m.c.h.r. functions......Page 98
2.4.2 Dynamic histories total time on test statistic and total hazard transform......Page 101
2.4.3 M.c.h.r. functions and dynamic su ciency......Page 108
2.5 Exercises......Page 110
2.6 Bibliography......Page 113
3.1 Introduction......Page 116
3.1.1 One dimensional stochastic orderings......Page 118
3.1.2 Stochastic monotonicity and orderings for conditional distributions......Page 122
3.2.1 Usual multivariate stochastic ordering......Page 124
3.2.2 Multivariate likelihood ratio ordering......Page 126
3.2.3 Multivariate hazard rate and cumulative hazard rate orderings......Page 127
3.2.4 Some properties of multivariate stochastic orderings and examples......Page 128
3.3.1 Positive dependence......Page 132
3.3.2 Negative dependence......Page 137
3.3.3 Simpson type paradoxes and aspects of dependence in Bayesian analysis......Page 139
3.3.4 Likelihood ratio comparisons between posterior distributions......Page 141
3.4.1 One dimensional notions of aging......Page 145
3.4.2 Dynamic multivariate notions of aging......Page 151
3.4.3 The case of exchangeable lifetimes......Page 153
3.5 Exercises......Page 157
3.6 Bibliography......Page 160
4.1 Introduction......Page 164
4.2 Schur survival functions......Page 168
4.2.1 Basic background about majorization......Page 169
4.2.2 Schur properties of survival functions and multivariate aging......Page 173
4.2.3 Examples of Schur survival functions......Page 177
4.2.4 Schur survival functions and dependence......Page 180
4.3 Schur density functions......Page 181
4.3.1 Schur-constant densities......Page 183
4.3.2 Examples of Schur densities......Page 186
4.3.3 Properties of Schur densities......Page 189
4.4.1 Schur densities and TTT plots......Page 193
4.4.2 Some other notions of Bayesian aging......Page 196
4.4.3 Heterogeneity and multivariate negative aging......Page 197
4.4.5 Extensions to non-exchangeable cases......Page 200
4.5 Exercises......Page 201
4.6 Bibliography......Page 204
5.1 Introduction......Page 207
5.1.1 Statistical decision problems......Page 212
5.1.2 Statistical decision problems and sufficiency......Page 219
5.1.3 Some technical aspects......Page 220
5.2 Stochastic orderings and orderings of decisions......Page 222
5.3 Orderings of residual lifetimes and majorization......Page 228
5.3.1 The case of observations containing failure data......Page 234
5.4 Burn-in problems for exchangeable lifetimes......Page 240
5.4.1 The case of i.i.d. lifetimes......Page 241
5.4.2 Dependence and optimal adaptive burn-in procedures......Page 244
5.4.3 Burn-in, optimal stopping, monotonicity, and Markovianity......Page 247
5.4.4 Stochastic orderings and open-loop optimal adaptive burn-in procedures......Page 249
5.5 Exercises......Page 256
5.6 Bibliography......Page 257