Practical Credibility Theory with Emphasis on Optimal Parameter Estimation

Abstract

We develop Hachemeister's regression model 111 credlbihty theory (without proofs) and indicate how the involved structural parameters can be estimated from the observable variables (with proofs for the simple results and those not yet published).Large famlhes of unbiased estinaators are available. From the practical viewpoint this is rather a handicap because it creates the problem to decide what estimators actually to use. In order to fix optimal estimators, we adopt the small-sample critermn of minimum-variance But in the research for general solutmns three kinds of difficulties arise.(i) The calculations become too lengthy. (fi) The optimal estimators depend on some of the parameters to be estimated (Then we call them pseudo-estimators). (ill) The optimal estimators depend on new structural parameters defined in terms of fourth-order momentsOnly a compromise allows to cope with this reahty. Sltuatmn (ill) creates new estimation problems. They can only be avoided at the cost of the Introduction of special assumptions or approxiinatlons. Then problem (i) is more or less automatically solved. By an obvious method of successive approximations pseudo-estmlators can serve as true estimators. Thus (li) is no real problem. GENERAL NOTATIONS AND DEFINITIONS MatricesThe same main symbol is used to denote a matrix c and its elements c{. Here c{ is the element at the intersectmn of row i and column j. The row i is denoted by c, and the column j by d. When the number of rows is m and the number of columns n, we say that the matrix has dimensions n (m)-

Credibility Models with Time-Varying Trend Components

“…The estimates of B and Z, are biased and, furthermore, the moment estimate of the scaled covariance matrix B need not be non-negative definite. These drawbacks can be overcome, in part, by either using-the iterative estimation approach of DEVYLDER (1981DEVYLDER ( , 1984, or a true Bayes approach instead of an empirical Bayes approach. The details of the Bayes analysis can be found in KLUGMAN (1987).…”

Bayesian Methods in Actuarial Science

“…De Vylder (1982) suggested a model which is based on credibility theory. He assumed that the distribution of the random variable Xi (the vector of observations for year of origin i) depends on the structure variable Oi' The model is based on the following assumptions (superscript 0 corresponds to vectors or matrices whose elements relate to observed variables):…”

“…yO is an unknown random vector (consisting of elements corresponding to XO) and [3(fn is a scalar function of 8j • Using the Hachemeister regression model (see Section 2.2) the following credibility estimator B, for [3(8;) is obtained: Vylder (1982). Mack (1990) modified the De Vylder credibility model by replacing assumption (c) by var(XijI8j ) = YJ 0 2 (8;)/ pj and so formally stating that the variance of Xij is dependent on the development year j.…”

“…In his classical model, Biihlmann (1967) Vylder (1981) who suggested forms of unbiased estimates. Herzog (1990) examined the compatibility of the Bayesian and the Biihlmann models and showed that the Biihlmann credibility estimate is the best linear approximation to the Bayesian estimate of the pure premium.…”

“…To model trend (e.g. inflation trend) Hachemeister (1975) Vylder (1986), who considered a non-linear regression model.…”