(a,b,0) class
/ ay-bee-zero class /
Actuaries noticed that the three workhorse count distributions — Poisson, binomial, and negative binomial — all share a hidden family resemblance. The probability of k claims relates to the probability of k-1 claims by a simple rule, the same shape of rule for all three. Grouping them by this shared recursion gives the (a,b,0) class: a small, tidy family of frequency distributions defined by one recursive relationship.
The defining rule is that the ratio of successive probabilities is linear in k: P(N=k) divided by P(N=k-1) equals a plus b divided by k, for k = 1, 2, 3, and onward, with two constants a and b. The label 'zero' means the recursion starts from the natural value at k=0 with no special adjustment. Different sign patterns of a and b pick out the members: a=0 gives the Poisson, a negative gives the binomial, and a between 0 and 1 gives the negative binomial (and its geometric special case). So just two numbers, a and b, encode which of these familiar distributions you have.
Why bother naming this family? Because the same recursion that defines it is exactly what makes the Panjer recursion work: any (a,b,0) frequency, combined with a discretised severity, lets you build up the aggregate-loss distribution one step at a time, avoiding a brute-force convolution. So the (a,b,0) class is not a curiosity — it is the gateway that turns frequency-severity modelling into a fast, exact computation. The honest limit: only these specific distributions belong to (a,b,0); if your data need, say, an inflated number of zeros, you must step up to the (a,b,1) class instead.
For a Poisson with lambda = 4, the recursion is P(k)/P(k-1) = 4/k, which matches a = 0 and b = 4. So P(1)/P(0) = 4, P(2)/P(1) = 2, P(3)/P(2) = 4/3 — each probability built from the last by a single multiply, the very step the Panjer recursion exploits.
Two constants a and b, via P(k)/P(k-1) = a + b/k, identify Poisson, binomial, or negative binomial.
The (a,b,0) class contains exactly three distributions (Poisson, binomial, negative binomial). It is not a free-form family — if real data show, for instance, far too many zeros, you cannot stay in (a,b,0); you move to the (a,b,1) class.