zero-modified distribution
Many real claim-count datasets have a strange feature: the number of policies with zero claims does not match what a standard Poisson or negative binomial predicts. Often there are far more zeros than expected (most people never claim at all), occasionally fewer or none (in a setting where at least one event is guaranteed). A zero-modified distribution is a clean way to keep a familiar count model while resetting the probability of zero to match reality.
It starts from a parent distribution (Poisson, binomial, or negative binomial) and replaces P(N=0) with a chosen value, then proportionally rescales all the positive-count probabilities so everything still sums to one. If the chosen zero probability is higher than the parent's, you get a zero-inflated model (extra zeros); if you set it to exactly zero, you get the special case called zero-truncated (used when no event is impossible to observe, e.g. data on claimants only). The relative shape of the counts 1, 2, 3, ... is inherited from the parent; only the weight on zero, and the overall scaling, changes. This makes the zero-modified distribution a member of the (a,b,1) class.
Zero-modification matters wherever the zero count is large and important — dental insurance, extended warranties, and many low-frequency lines where the overwhelming majority never claim, so getting P(N=0) right dominates the fit. Practically, it lets you keep the convenient Poisson or negative binomial machinery (including the Panjer recursion) while honestly fitting the heap of zeros. The honest caveat: adjusting the zero is a targeted fix, not a cure-all. If the data are wrong in the tail as well as at zero, zero-modifying a misspecified parent simply layers one correction over an uncorrected error; you should still check the full fit, not just the zero cell.
A dental book's parent Poisson (lambda = 0.5) predicts P(N=0) = e^(-0.5) = 0.607, but the data show 75% of members claim nothing. A zero-modified Poisson sets P(N=0) = 0.75 and rescales the probabilities of 1, 2, 3, ... claims down to share the remaining 25% in the same relative proportions the Poisson gave — a much better match to the observed heap of zeros.
Reset the zero probability to match the data; keep the parent's shape for the positive counts.
Zero-modification corrects only the zero cell. If the parent model is also wrong in the tail, fixing the zeros just disguises that; always verify the fit across all counts, not just the probability of no claim.