Construction of Zero Inflated Poisson Mixture Distributions and Application to Fertility Data

Abstract

In applications involving count data, it is common to encounter the frequency of observed zeros significantly higher than predicted by the model based on the standard parametric family of discrete distributions. In such situations, Zero Inflated Poisson and Zero Inflated Negative Binomial distributions have been widely used in modeling the data, yet other models may be more appropriate in handling the data with excess zeros. Such situations normally result to misspecification of the statistical model leading to erroneous conclusions and bringing uncertainty into research and practice. Mixture models cover several distinct fields of the statistical science. Their broad acceptance as adequate models to describe diverse situations is evident from the plethora of their applications in the statistical literature. The univariate Poisson mixture distributions have been formed together with the structural properties of inflated power series distributions. However, the continuous mixture distributions with a Zero Inflated Poisson as the inflate model have neither been considered nor applied on fertility data. Therefore the problem was to form continuous Zero Inflated Poisson mixture distributions by using direct integration, in recursive formula, through expectation forms and by use of special functions. The mixture model formed had a Zero Inflated Poisson model as an inflate model and a prior distribution. Furthermore, the inflated mixture distributions obtained explicitly and by the method of moments had not been proved to be identical and application of ZIP mixture distribution! on fertility data. This work concentrated on the construction of continuous ZIP mixture distributions together with their properties and the proofs of identities resulting from the continuous mixture distributions. According to this study, the method that resulted in a good number of mixture Poisson distributions compared to the other methods was that of obtaining recursive relations using integration by parts. This was a clear indication that there is no restriction on what kind of method to use for a particular given mixing distribution, that is, any method can be used whenever possible. However, some mixture distributions e.g the ZIP Lomax distribution, could not be constructed by direct integration. The ZIG distribution was then fitted to fertility data. Then ZIG model was chosen because it lies in the domain of [0, 00] and it is also used in modeling of rare and discrete events, which fits the characteristics of fertility data. This clearly showed that the Zero-Inflated mixture distributions are the most appropriate in modeling of count data. The models that were derived in this work could be used by actuaries in assessing the credit worthiness of an investor and in claims compensation. The demographers can also use these models to study different components in population.