Loss Models: From Data to Decisions, 4th Edition

9.6 The recursive method

Suppose that the severity distribution f_X(x) is defined on 0, 1, 2, …, m representing multiples of some convenient monetary unit. The number m represents the largest possible payment and could be infinite. Further, suppose that the frequency distribution, p_k, is a member of the (a, b,1) class and therefore satisfies

Then the following result holds.

Theorem 9.8 For the (a, b, 1) class,

(9.20)

noting that x Λ m is notation for min(x, m).

Proof: This result is identical to Theorem 7.2 with appropriate substitution of notation and recognition that the argument of f_X(x) cannot exceed m.

Corollary 9.9 For the (a, b, 0) class, the result (9.20) reduces to

(9.21)

Note that when the severity distribution has no probability at zero, the denominator of (9.20) and (9.21) equals 1. Further, in the case of the Poisson distribution, (9.21) reduces to

(9.22)

The starting value of the recursive schemes (9.20) and (9.21) is f_S(0) = P_N [f_X(0)] following Theorem 7.3 with an appropriate change of notation. In the case of the Poisson distribution we have

Starting values for other ...