Zero-Inflated and Hurdle Models for Insurance Data
Model excess zeros in insurance claim counts using zero-inflated and hurdle models for Exam MAS-I.
Zero-Inflated Models
Insurance data often has more zeros (no claims) than a Poisson or negative binomial model predicts. Zero-inflated models combine a point mass at zero with a count distribution. The zero-inflated Poisson (ZIP) has P(Y=0) = pi + (1-pi)*exp(-lambda) and P(Y=k) = (1-pi)*exp(-lambda)*lambda^k/k! for k >= 1. The mixing weight pi represents "structural zeros" from a subpopulation that cannot generate claims (e.g., fraud-free policies in a fraud model). Both pi and lambda can depend on covariates through logistic and log link functions, respectively.
Hurdle Models
Hurdle models separate the process into two stages: (1) a binary model for whether any event occurs (zero vs. positive), and (2) a truncated-at-zero count model for the positive counts. P(Y=0) = 1-p and P(Y=k|Y>0) is a truncated Poisson or negative binomial. Unlike ZIP, hurdle models treat all zeros as coming from one process. Hurdle models are easier to interpret: the binary component models "propensity to claim" and the count component models "claim intensity given at least one claim." Exam MAS-I tests both model types, their differences, and their application to insurance count data.