Parameter Estimation in Credibility Theory

The problem of distribution-free parameter estimation in recent credibility theory is discussed in the papers [I], [3] and [4] of the bibliography. Here, we consider a multiclass model with regression assumption. In that case, already treated by Charles Hachemelster, [3], this author obtains an unsymmetrical matrix as an estimator of a covariance matrix. Of course, for practical use, this matrix is symmetrized in the obvious way. We show that this procedure can be avoided and that a lot of symmetrical unbiased estimators can be obtained at once. By particularisations to the I-rank model, we find the estimators given by Buhlmann and Straub, [I], [4]. In the multirank case, a generalization of the minimum variance principle (minimumization of the trace of the covariance matrix) leads to an optimal estimator of the mean regression vector. A noteworthy conclusion of our discussion is that there is no difference at all between the various credibility formulae (the inhomogeneous formula, the homogeneous formula, the mean-free formula) if the mean regression vector is estimated optimally. Finally we show that it must not be hoped to find, in a large set of unbiased estimators of the covariance matrix, one estimator furnishing, always, a semi definite positive estimate