On the correct use of the chi-square goodness-of-fit tes

In statistical investigations concerning the law governing the claim number distribution of a collective of insured it is important to examine whether a theoretical model of the claim number process N(t) adequately describes observed empirical data. This subject is treated in a number of articles on risk theory (Bichsel, 1964; Corlier et at, 1978; Delaporte, 1960; Derron 1962; Hofmann, 1955; Lundberg, 1964; Muff, 1972; Philipson, 1960; Seal, 1969; Thyrion, 1960). In all those investigations the chi-square test is used to examine whether a theoretical counting distribution (i.e. P(N(t)=n)) differs significantly from empirical data. As all those papers mentioned contain the same incorrectnesses concerning the use of the chi-square test (this will be discussed in detail later on), it seems necessary to us to state here precisely the conditions for the chi-square statistic to be asymptotically chi-square distributed. Further we want to make some statements on how the difficulties encountered in the use of the chi-square test can be circumvented or removed in some situations important to risk theory.