Bayesian Statistics in Actuarial Science: with Emphasis on Credibility

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Springer Science & Business Media, Nov 30, 1991 - Business & Economics - 238 pages
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The debate between the proponents of "classical" and "Bayesian" statistica} methods continues unabated. It is not the purpose of the text to resolve those issues but rather to demonstrate that within the realm of actuarial science there are a number of problems that are particularly suited for Bayesian analysis. This has been apparent to actuaries for a long time, but the lack of adequate computing power and appropriate algorithms had led to the use of various approximations. The two greatest advantages to the actuary of the Bayesian approach are that the method is independent of the model and that interval estimates are as easy to obtain as point estimates. The former attribute means that once one learns how to analyze one problem, the solution to similar, but more complex, problems will be no more difficult. The second one takes on added significance as the actuary of today is expected to provide evidence concerning the quality of any estimates. While the examples are all actuarial in nature, the methods discussed are applicable to any structured estimation problem. In particular, statisticians will recognize that the basic credibility problem has the same setting as the random effects model from analysis of variance.
 

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Contents

1 INTRODUCTION
1
2 BAYESIAN STATISTICAL ANALYSIS
5
B AN EXAMPLE
7
C PRIOR DISTRIBUTIONS
10
D MODEL SELECTION AND EVALUATION
13
1 Graphical Analysis
14
2 A Selection Criterion
15
3 COMPUTATIONAL ASPECTS OF BAYESIAN ANALYSIS
17
5 Graduation
89
2 The Analysis
90
D EXAMPLES ANALYSIS
97
1 Oneway
98
2 Twoway
100
3 Linear Trend
107
4 Kalman Filter
109
E PRIOR DISTRIBUTIONS
110

B NUMERICAL INTEGRATION
19
1 Adaptive Gaussian Integration
20
2 GaussHermite Integration
21
3 Estimating the Mean and Covariance
24
4 An Example
25
C MONTE CARLO INTEGRATION
29
D ADJUSTMENTS TO THE POSTERIOR MODE
32
E EMPIRICAL BAYES STYLE APPROXIMATIONS
33
F SUMMARY
35
4 PREDICTION WITH PARAMETER UNCERTAINTY
37
B A LIFE INSURANCE EXAMPLE
38
C A CASUALTY INSURANCE EXAMPLE
42
D THE KALMAN FILTER
46
E RETURN OF THE CASUALTY INSURANCE EXAMPLE
50
5 THE CREDIBILITY PROBLEM
57
A A SIMPLE MODEL
58
C CREDIBILITY ISSUES
62
6 THE HIERARCHICAL BAYESIAN APPROACH
65
B AN EXAMPLE
67
C THE GENERAL HIERARCHICAL MODEL
71
D SIMPLIFYING ASSUMPTIONS
76
2 Linearity
78
7 THE HIERARCHICAL NORMAL LINEAR MODEL
81
B EXAMPLES DESCRIPTION
82
2 Twoway
83
3 Linear Trend
85
4 Kalman Filter
86
F MODEL SELECTION AND EVALUATION
111
8 EXAMPLES
115
B ANALYSES
118
2 One Way Model Data Set 2
125
3 Empirical Bayes Style Approaches
129
4 An Iterative Approach
131
5 Other Priors
133
6 Diagnostics
135
7 Twoway Model Data Set 4
143
8 Linear Trend Model Data Set 3
145
9 Kalman Filter Data Set 3
146
10 Graduation
148
9 MODIFICATIONS TO THE HIERARCHICAL NORMAL LINEAR MODEL
151
B POISSON
152
C NONNORMAL MODELS BASED ON PARAMETER ESTIMATES
154
APPENDIX ALGORITHMS PROGRAMS AND DATA SETS
159
B ADAPTIVE GAUSSIAN INTEGRATION
163
C GAUSSHERMITE INTEGRATION
165
D POLAR METHOD FOR GENERATING NORMAL DEVIATES
166
2 Adaptive Gaussian Integration
170
3 GaussHermite Integration
173
4 Monte Carlo Integration
179
5 TierneyKadane Integration
182
F DATA SETS
184
BIBLIOGRAPHY
229
INDEX
235
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Page 233 - Proceedings of the Casualty Actuarial Society, 71, 96-121. Meyers, G. (1985), "An Analysis of Experience Rating," Proceedings of the Casualty Actuarial Society, 72, 278-317. Miller, R. and Fortney, W. (1984), "Industry-wide Expense Standards Using Random Coefficient Regression," Insurance: Mathematics and Economics, 3, 19-33.
Page 234 - Structured Credibility in Applications — Hierarchical, Multidimensional, and Multivariate Models,

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