ON THE NON-SINGULARITY ISSUE FOR THE POISSON-GAMMA MODEL
Journal Title: Jaunųjų mokslininkų darbai - Year 2014, Vol 41, Issue 1
Abstract
The problem of estimation of small probabilities in large populations (e.g., the estimation of probability of some disease, death, suicides, etc.) is considered. The number of corresponding events depends on the size of the population and the probability of the single event. It is assumed that the number of events in populations has a Poisson distribution with certain parameters. Let us have two models of distribution of unknown probabilities: the probabilities have a gamma distribution (Poisson- gamma model), or logits of the probabilities have a Gaussian distribution (Poisson-Gaussian model). In the case of the Poisson- Gaussian model it is known that if a certain non-singularity condition does not hold then empirical Bayes estimates of unknown probabilities are equal to mean relative risk estimates, and corresponding distribution of logits of the probabilities have singular distribution with zero variance. In the case of the Poisson-gamma model we have a similar non-singularity issue. In practice it means that the shape and scale parameters given by iterative procedures for finding distribution parameters converge to infinity and we do not obtain finite values of shape and scale parameters. The non-singularity condition depends only on population sizes and the number of observed events. We will consider the Poisson-gamma model for some sets of data and we will show the behaviour of iterative procedures. We will focus on the behaviour of a partial derivative of the maximum likelihood function, which is essential for the non-singularity condition for the Poisson-gamma model.
Authors and Affiliations
Gintautas Jakimauskas
Keywords
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