Loss Models: From Data to Decisions, 4th Edition

B.4 The compound class

Members of this class are obtained by compounding one distribution with another. That is, let N be a discrete distribution, called the primary distribution, and let M₁, M₂,… be i.i.d. with another discrete distribution, called the secondary distribution. The compound distribution is S = M₁ + ··· + M_N. The probabilities for the compound distributions are found from

for n = 1, 2,…, where a and b are the usual values for the primary distribution [which must be a member of the (a, b, 0) class] and f_y is p_y for the secondary distribution. The only two primary distributions used here are Poisson (for which p₀ = exp[-λ(1 − f₀)]) and geometric [for which p₀ = 1/(1 + β − β f₀)]. Because this information completely describes these distributions, only the names and starting values are given in the following sections.

The moments can be found from the moments of the individual distributions:

The pgf is P(z) = P_primary [P_secondary (z)].

In the following list, the primary distribution is always named first. For the first, second, and fourth distributions, the secondary distribution is the (a, b, 0) class member with that name. For the third and the last three distributions (the Poisson-ETNB and its two special cases), the secondary distribution is the zero-truncated version. ...