A Course in Credibility Theory and its ApplicationsSpringer Science & Business Media, 2005 M08 30 - 338 pages The topic of credibility theory has been for many years — and still is — one of our major interests. This interest has led us not only to many publications, but also has been the motivation for teaching many courses on this topic over more than 20 years. These courses have undergone considerable changes over time. What we present here, “A Course in Credibility Theory and its Applications”, is the ?nal product of this evolution. Credibility theory can be seen as the basic paradigm underlying the pricing of insurance products. It resides on the two fundamental concepts “individual risk” and “collective” and solves in a rigorous way the problem of how to analyse the information obtained from these sources to arrive at the “insurance premium”. The expression “credibility” was originally coined for the weight given to the experience from the “individual risk”. Credibility theory as a mathematical discipline borrows its methods from 2 many ?elds of mathematics, e. g. Bayesian statistics, L Hilbert space te- niques, least squares, and state space modelling to mention only the most important ones. However, credibility theory remains a lifeless topic if it is not linked closely with its applications. Only through these applications has cr- ibility won its status in insurance thinking. The present book aims to convey this dual aspect of credibility and to transmit the ?avour of the insurance applications also to those readers who are not directly involved in insurance activities. |
Contents
I | 1 |
II | 7 |
III | 8 |
IV | 9 |
V | 11 |
VI | 14 |
VII | 15 |
IX | 16 |
LV | 159 |
LVI | 162 |
LVII | 165 |
LVIII | 167 |
LIX | 169 |
LXI | 170 |
LXII | 173 |
LXIII | 174 |
X | 18 |
XI | 21 |
XIII | 31 |
XIV | 34 |
XV | 36 |
XVI | 38 |
XVII | 39 |
XVIII | 46 |
XIX | 47 |
XX | 49 |
XXI | 50 |
XXII | 55 |
XXIII | 56 |
XXIV | 58 |
XXV | 59 |
XXVI | 60 |
XXVII | 64 |
XXVIII | 67 |
XXIX | 71 |
XXX | 74 |
XXXI | 77 |
XXXIII | 79 |
XXXIV | 81 |
XXXV | 84 |
XXXVI | 86 |
XXXVIII | 91 |
XXXIX | 93 |
XL | 95 |
XLI | 97 |
XLII | 106 |
XLIII | 110 |
XLIV | 111 |
XLV | 113 |
XLVI | 117 |
XLVII | 125 |
XLVIII | 130 |
XLIX | 135 |
L | 136 |
LI | 143 |
LII | 145 |
LIII | 146 |
LIV | 148 |
LXIV | 177 |
LXV | 178 |
LXVI | 180 |
LXVII | 185 |
LXIX | 187 |
LXX | 189 |
LXXI | 193 |
LXXII | 199 |
LXXIII | 201 |
LXXIV | 202 |
LXXV | 205 |
LXXVI | 208 |
LXXVIII | 217 |
LXXIX | 219 |
LXXX | 220 |
LXXXI | 223 |
LXXXII | 226 |
LXXXIII | 230 |
LXXXIV | 238 |
LXXXV | 239 |
LXXXVI | 251 |
LXXXVII | 252 |
LXXXVIII | 253 |
LXXXIX | 255 |
XC | 262 |
XCI | 264 |
XCII | 275 |
XCIII | 276 |
XCIV | 277 |
XCV | 278 |
XCVI | 280 |
XCVII | 283 |
XCVIII | 287 |
XCIX | 293 |
C | 296 |
CI | 305 |
CII | 311 |
CIII | 314 |
CIV | 318 |
CV | 323 |
329 | |
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Common terms and phrases
affine subspace aggregate claim amount assume Assumptions 4.1 average claim amount B₁ Bayes estimator Bayes premium Bayesian statistics Bühlmann-Straub model calculation Chapter claim experience claim frequency claim number claim sizes claims ratio coefficient of variation collective components conditional conditionally contracts correct individual premium CoVa covariance covariance matrix cred credibility formula credibility model credibility premium credibility theory credibility weights data compression defined Example expected value FBayes Gamma distribution given hierarchical Hilbert space homogeneous credibility estimator independent individual claim Kalman Filter large claims linear loss ratio Model Assumptions 4.7 multidimensional credibility notation number of claims observation vector On+1 orthogonal Pareto distributed PBayes pcoll pcred Poisson Poisson distributed Proof of Theorem pure risk premium quadratic loss random variables recursive Remarks risk groups Section simple Bühlmann model statistics structural parameters subspace unbiased estimator X₁ μ Θ μχ