Zero-Inflated Negative Binomial Distribution
X ~ ZINB(π, r, p)
Excess-Zeros / Mixture Family • Published July 22, 2026
Two overdispersion mechanisms stacked in one model: the already-published Zero-Inflated Poisson's structural-zero mixture applied on top of the already-published Negative Binomial instead of the Poisson, reached for when a fitted Zero-Inflated Poisson still under-predicts the tail, with a variance formula derived via the law of total variance on a Bernoulli-times-NegBinom product, a no-closed-form maximum-likelihood system verified against an independent method-of-moments fit, and simulation via the mixture's own defining construction.
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About This Reference Sheet
Two overdispersion mechanisms stacked in one model: the already-published Zero-Inflated Poisson's structural-zero mixture applied on top of the already-published Negative Binomial instead of the Poisson, reached for when a fitted Zero-Inflated Poisson still under-predicts the tail, with a variance formula derived via the law of total variance on a Bernoulli-times-NegBinom product, a no-closed-form maximum-likelihood system verified against an independent method-of-moments fit, and simulation via the mixture's own defining construction.
Support
x ∈ {0, 1, 2, ...}
Parameters
π ∈ [0,1) (structural-zero probability), r > 0 (target successes), p ∈ (0,1] (success probability)