Two regression credibility models

In this communication we will discuss two regression credibility models from Non – Life Insurance Mathematics that can be solved by means of matrix theory. In the first regression credibility model, starting from a well-known representation formula of the inverse for a special class of matrices a risk premium will be calculated for a contract with risk parameter θ. In the next regression credibility model, we will obtain a credibility solution in the form of a linear combination of the individual estimate (based on the data of a particular state) and the collective estimate (based on aggregate USA data). To illustrate the solution with the properties mentioned above, we shall need the well-known representation theorem for a special class of matrices, the properties of the trace for a square matrix, the scalar product of two vectors, the norm with respect to a positive definite matrix given in advance and the complicated mathematical properties of conditional expectations and of conditional covariances.