Loss Models: From Data to Decisions, 4th Edition

APPENDIX D

THE RECURSIVE FORMULA

The recursive formula is (where the frequency distribution is a member of the (a, b, 1) class),

where f_S (x) = Pr (S = x). x = 0, 1, 2,…, f_X(x) = Pr(X = x), x = 0, 1, 2,…, p₀ = Pr(N = 0), and p₁ = Pr(N = 1). Note that the severity distribution (X) must place probability on nonnegative integers. The formula must be initialized with the value of f_S(0). These values are given in Table D.1. It should be noted that, if N is a member of the (a, b, 0) class. p₁ − (a + b)p₀ = 0, and so the first term will vanish. If N is a member of the compound class, the recursion must be run twice. The first pass uses the secondary distribution for p₀, p₁, a, and b. The second pass uses the output from the first pass as f_X(x) and uses the primary distribution for p₀, p₁, a, and b.

Table D.1 Starting values [ f_S(0)] for recursions.