Comments on Different Deductions of Expressions for Conditional Expectations

In a paper presented to the fifth ASTIN Colloquium (Lucerne, 1965) Buhlmann has given some propositions with regard to Experience Rating understood as a sequence of estimates of the expectation with respect to the distribution function of (equation- see paper) say, of (equation-see paper) which is the mean for fixed THETA with respect to the distribution function (equation- see paper) of the variables (equation- see paper) for each value to (equation- see paper). In such problems the estimator function for (equation- see paper) is generally chosen to be the conditional mean of (equation- see paper) for a given set of observed values of (equation- see paper). This is generally justified by the principle of least square deviation. According to Buhlmann this justification is not sufficient. Therefore, he bases the choice of this estimator function upon a postulate of equilibrium, described in the following lines.