following models were proposed to model the data; zero inflated negative poisson and negative binomial. Before discussing them let’s consider poisson regression model and the zero inflated model. The response variable is the number of death that occurred in the year and is represented by y and the predictor variable is the age at which death occurred and is represented by x.

Models
Pension data consist of count variable outcome interest which might contain too many zeros. And with this count data the expected number of occurrence of death is the dependent variable and the age is the predictor variable. Different models were proposed to fit count data with too many zeros than expected: Lambert (1992) described the zero-inflated Poisson regression models with an application to defects in manufacturing; Hall (2000) also described the zero-inflated binomial regression model and incorporated random effects into ZIP and ZIB models.

Many count datasets has the joint presence of excess zero observations and long right tails features that may be accounted for by over-dispersion in the data, which are both relative to the Poisson assumption, Gurmu and Trivedi (1996). The proportion of the zeros increase whenever there zeros are too many relative to the Poisson assumption, so the negative binomial regression and zero-inflation negative binomial regression model tend to improve the fit of the data. The model selection is done using the likelihood ratio test.

Poisson regression model
Poisson regression model is used to model count data. It is a discrete probability distribution that is used to model the number of events occurring within a given time interval. The Poisson distribution models the log-odds as a linear function of the observed covariates. This gives the generalized linear model with Poisson response and ling log.
If the number of occurrence has a variable Y which has a poisson distribution with parameter μ and it takes integer values of y = 0, 1, 3, … then the probability distribution is given by
P(Y = y) = (μ^y e^(-λ))/y! ; λ > 0 3.1
where λ is the shape parameter which indicates the average number of events