Bayesian Statistics in Actuarial Science: with Emphasis on Credibility

242 Pages·1992·9.32 MB·English

by  Stuart A. Klugman (auth.)

Bayesian Statistics in
Actuarial Science
Huebner International Series on
Risk, lnsurance, and Economic
Security
J. David Cummins, Editor
The Wharton School
University of Pennsylvania
Philadelphia, Pennsylvania, USA
Series Advisors:
Dr. Phelim P. Boyle
University of Waterloo, Canada
Dr. Jean Lemaire
University of Pennsylvania, USA
Professor Akihiko Tsuboi
Kagawa University, Japan
Dr. Richard Zeckhauser
Harvard University, USA
Other books in the series:
Cummins, J. David; Smith, Barry D.; Vance, R. Neil;
VanDerhei, Jack L.: Risk Classification in Life
lnsurance
Mintal, Judith: lnsurance Rate Litigation
Cummins, J. David: Strategic Planning and
Mode/ing in Property-Liability lnsurance
Lemaire, Jean: Automobile lnsurance: Actuarial
Models
Rushing, William.: Social Functions and Economic
Aspects of Health lnsurance
Cummins, J. David and Harrington, Scott E.: Fair
Rate of Return in Property-Liability lnsurance
Appel, David and Borba, Philip 5.: Workers
Compensation lnsurance Pricing
Cummins, J. David and Derrig, Richard A.: Classical
lnsurance Solvency Theory
Barba, Philip S. and Appel, David: Benefits, Costs,
and Cycles in Workers Compensation
Cummins, J. David and Derrig, Richard A.: Financial
Models of lnsurance Solvency
Williams, C. Arthur: An International Comparison of
Workers' Compensation
Cummins, J. David and Derrig, Richard A.: Managing
the lnsolvency Risk of lnsurance Companies
Dionne, Georges: Contributions to lnsurance
Economics
Dionne, Georges: Foundations of lnsurance
Economics-Readings in Economics and Finance
The objecllve of the serles ls to publlsh original research and advanced
textbooks deallng wlth ali major aspects of rlsk bearlng and economic securlty.
The emphasls ls on books that wlll be of Interes! to an International audlence.
lnterdlsclpllnary toplcs as well as those !rom traditional dlsclpllnes such as
economlcs, rlsk and lnsurance, and actuarlal sclence are wlthln the scope of
the serles. The goal ls to provlde an outlet for Imaginativa approaches to
probleme In both the theory and practice of rlsk and economic securlty.
Bayesian Statistics in
Actuarial Science
with Emphasis on Credibility
Stuart A. Klugman
Drake University
'~·'
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Klugman, Stuart A., 1949-
Bayesian statistics in actuarial science : with emphasis on
credibility 1 Stuart A. Klugman.
p. cm. - (Huebner international series on risk, insurance,
and economic security)
Includes bibliographical references and index.
ISBN 978-90-481-5790-7 ISBN 978-94-017-0845-6 (eBook)
DOI 10.1007/978-94-017-0845-6
1. Insurance-Statistical methods. 2. Bayesian statistica!
decision theory. 1. Title. II. Series.
HG878l.K58 1991
368 '.01-dc20 91-25856
CIP
Copyright © 1992 by Springer Science+Business Media New York. Second Printing 1999.
Originally published by Kluwer Academic Publishers in 1992
AII rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, mechanical, photocopying, recording,
or otherwise, without the prior written permission of the publisher,
Springer Science+Business Media, LLC.
For Marie
TABLE OF CONTENTS
Preface xi
1. Introduction .................................................................................. 1
2. Bayesian Statistica! Analysis .......................................................... 5
A. The Bayesian Paradigm ............. ....................... ......... ......... 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
A. An Example ....... ... ... . .. .. .. ... ... ... ....... ... . ............ ....... ........... 17
B. Numerica} Integration ......................................................... 19
1. Adaptive Gaussian Integration................................ 20
2. Gauss-Hermite 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. Empirica} Bayes Style Approximations................................ 33
F. Summary . .................................. .......... ............ ...... ............ 35
4. Prediction with Parameter Uncertainty............................................. 37
A. The Model ...... ... . ... ... ... ....... ...... .... ... . .. ......... .... ... ... . ........ ... 37
B. A Life Insurance Example. .. . ... ... . ............ .......... ... ....... ........ 38
C. A Casualty Insurance Example............................................ 42
D. The Kalman Filter.............................................................. 46
E. Re turn of the Casualty lnsurance Example........ ...... ......... ... 50
5. The Credibility Problem ................................................................. 57
A. A Simple Model .. ....................... ........... ... ........ ... ........ ....... 58
B. Estimating the Class Mean.......... ... ........... ........ ... ........ ....... 58
C. Credibility Issues .............................................................. 62
6. The Hierarchical Bayesian Approach ................................................ 65
A. What It Is .. .... .... ....................... ... ....... .... .. ...... ... ........... .. .. 65
B. An Example .... ... ....................... ............ .. .. ......... .... ... ........ 67
C. The General Hierarchical Model........................................... 71
D. Simplifying Assumptions .................................................... 76
1. Normality .............................................................. 76
2. Linearity ............. .... ....... ........ ... . .......... .... ........ ..... 78
3. V arian ce Independent of the Mean... .... ... ......... ...... .. 78
7. The Hierarchical Normal Linear Model............................................. 81
A. The Model ......................................................................... 81
B. Examples - Description ...................................................... 82
1. One-Way ............................................................... 82
2. Two-Way ............................................................... 83
3. Linear Trend .. ..... . ... . ... ......... ............. ....... ......... .... 85
4. Kalman Filter ..... ... ....... ......... .. .. ......... ....... ........ ..... 86
5. Graduation ........ .......... ............. ......... ........... ..... .. . . 89
C. Analysis of the Model ........... .... ............. ......... .................... 89
1. Preliminaries .. .. . .. .. .. . .. . .. .. .. . .. .. .. .. . .. .... .. .. .. . . .. . . .. .. . . . . 89
2. The Analysis ..... .. . ... .... ......... ............. ...... ... ....... .. .. 90
D. Examples - Analysis ........ ..... .............. ........... ...... ......... ..... 97
1. One-Way ............................................................... 98
2. Two-Way ............................................................... 100
3. Linear Trend ......................................................... 107
4. Kalman Filter ......................................................... 109
5. Graduation ............................................................ 109
E. Prior Distributions ............................................................. 110
F. Model Selection and Evaluation. .......................................... 111
8. Examples ...................................................................................... 115
A. Data Sets .......................................................................... 115
B. Analyses ........................................................................... 118
1. One-way Model, Data Set !..................................... 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. Two-way Model, Data Set 4 .................................... 143
8. Linear Trend Model, Data Set 3 .............................. 145
9. Kalman Filter, Data Set 3 ....................................... 146
10. Graduation .......................................................... 148
VIU
9. Modifications to the Hierarchical Normal Linear Model .................... 151
A. Lognormal ......................................................................... 151
B. Poisson .......................................................... ................... 152
C. Non-normal Models Based on Parameter Estimates ............. 154
Appendix. Algorithms, Programs, and Data Sets .................................. 159
A. The Simplex Method of Function Maximization.................. 159
B. Adaptive Gaussian Integration ............................................ 163
C. Gauss-Hermite Integration .................................................. 165
D. Polar Method for Generating Normal Deviates .................... 166
E. GAUSS Programs ............................................................... 166
1. Simplex Maximization............................................. 166
2. Adaptive Gaussian Integration .................... .... .. ...... 170
3. Gauss-Hermite Integration ...................................... 173
4. Monte Carlo Integration.......................................... 179
5. Tierney-Kadane Integration..................................... 182
F. Data Sets .......................................................................... 184
1. Data Set 1 ............................................................. 184
2. Data Sets 2-4 ...................................................... 184
Bibliography 229
Index 235
IX
PREFACE
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.
As the technological problem is the most difficult, this monograph
spends a great deal of time on computational issues. The algorithms are
presented in the text and computer programs are listed in the Appendix. In
addition, each model is illustrated with an example that demonstrates that
these algorithms can be applied to realistic data sets. The data sets are also
included in the Appendix so that the interested reader can duplicate the
results presented here as well as try other models.
This work had its origins when I spent the Fali of 1984 at the
National Council on Compensation Insurance in New York City. I learned