(a,b,1) class
/ ay-bee-one class /
Real claim data often have a peculiar number of policies with zero claims — sometimes far more zeros than a Poisson predicts (lots of people simply never claim), and occasionally a setting where zero is impossible so you need none at all. The plain (a,b,0) distributions cannot bend the zero probability freely. The (a,b,1) class fixes this by letting you set the probability of zero to whatever the data demand, while keeping a clean recursion for the rest.
Mechanically, the (a,b,1) class uses the same linear recursion P(N=k)/P(N=k-1) = a + b/k, but only from k=2 onward, leaving the probability at zero, P(N=0), as a free quantity you choose separately. The counts for k=1 and above are then rescaled to keep everything summing to one. Two important members: a zero-truncated distribution sets P(N=0) = 0 (used when at least one event is certain, e.g. you only observe claimants), and a zero-modified distribution sets P(N=0) to some chosen value, higher or lower than the parent model would give. The Poisson, binomial, and negative binomial all have such zero-truncated and zero-modified versions.
This class matters because the count of zeros is often the single most important and most distorted part of an insurance frequency table — think of dental or warranty data where the vast majority never claim. Being able to tune P(N=0) without otherwise abandoning the familiar Poisson or negative binomial shape gives a much better fit. And, crucially, the (a,b,1) members still feed the Panjer recursion (with a small modification at the first step), so aggregate-loss computation stays exact. The caveat: the freedom to fit zeros is powerful but can mask a deeper misspecification — if both the zeros and the tail are off, a richer model, not just zero-modification, may be needed.
Warranty data show 96% of products never claim, but a fitted Poisson predicts only 90% zeros. Instead of abandoning the Poisson, the actuary uses a zero-modified Poisson: set P(N=0) = 0.96 directly, then let the original Poisson recursion shape the relative probabilities of 1, 2, 3+ claims among the remaining 4% — a far better fit to the heap of zeros.
Set the zero probability freely, keep the recursion for the rest — that is the (a,b,1) idea.
Zero-modification only adjusts the probability at zero; it does not by itself fix a wrong tail. If both the count of zeros and the upper tail misfit, zero-modifying a still-wrong parent just hides one problem behind another.