## Bayesian Statistics in Actuarial Science: with Emphasis on CredibilityThe 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 |

229 | |

235 | |

### Other editions - View all

Bayesian Statistics in Actuarial Science: with Emphasis on Credibility Stuart A. Klugman Limited preview - 2013 |

Bayesian Statistics in Actuarial Science: with Emphasis on Credibility Stuart A. Klugman No preview available - 2014 |

Bayesian Statistics in Actuarial Science: with Emphasis on Credibility Stuart A. Klugman No preview available - 2010 |

### Common terms and phrases

actuarial adaptive Gaussian integration argument Bayes estimate Bayesian analysis Bayesian approach Cl Yr Payroll Class Year Payroll constant covariance matrix credibility Data Set diagonal elements dimension distribution with parameter empirical Bayes style ENDIF ENDO evaluated example expected value exposure factor gamma distribution Gauss-Hermite integration hierarchical model HNLM indicates inserted Insurance integral with respect integrand interval inverse iterations Kalman filter Klugman linear trend model logarithm Loss Cl Yr maximum likelihood estimates mean and covariance Monte Carlo Integration multivariate normal multivariate normal distribution natural logarithm noninformative prior normal distribution number of claims obtain Occ Yr Exp one-way model parameter estimates parameter values Payroll Loss Cl plot Poisson distribution posterior density posterior distribution posterior mean posterior mode predictive distribution prior density prior distribution problem pure premium quantities relative error residuals Section Set 2 Class Set 3 Cl simplex method solution transformation Weibull Yr Payroll Loss zero

### Popular passages

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,