Distribution of the Number of Claims in Motor Insurance according to the Lag of Settlement

Let Sn be the set of motor claims Si, i = 1, 2, …, N occurred during a given year n, namely Sn represents the set of claims relevant to the generation (or cohort) n.If Tn is a subset (even empty) of claims resulting without payement (that is the set of zero-claims), the set Pn = Sn − Tn shall denote the set of claims that should be settled.For every si ∈ Pn, we can define the r.v. Xi which represents the period of time required for its settlement (namely the lag of settlement).It is not sensible to deem that the r.v. Xi are equally distributed: as a matter of fact we know that the larger is the claim, the longer the lag of payement.However, we can assume that in a subset U of Pn, the r.v. Xi(U) have the same distribution function, which will be denoted by FU(x) or in short F(x).As F(x) represents the probability that a claim sj ∈ U is settled within a period o − x, the function 1 − F(x) = l(x) denotes the probability that the claim results unsettled after a lag x, that is the survival function of the claim.In this study we intend to find an analytical expression of the function 1(x) on the basis of particular assumptions about the behaviour of the adjuster with regard to the settlement of claims.The assumptions will be tested by fitting the function to some observed data.