Abstract
A breakthrough is provided in the study of the existence problem for maximum likelihood estimators (MLE) in the hierarchical generalized linear model (HGLM) of Poisson-gamma type, as well as in the negative binomial regression model. Any more than the uniqueness problem associated, the existence problem of MLE for these models has not yet been studied except in the very special case of the sample. This issue is addressed here for the Poisson-gamma HGLM, and a sufficient condition is obtained to ensure the MLE existence in that case. It is also shown that this condition has the same effect in the negative binomial regression model with the index parameter considered as unknown. In the latter model, the obtained condition appears as a natural extension of the necessary and sufficient condition well known for solving the existence and uniqueness problems for the index parameter MLE in the sample case.
Highlights
AMS 2000 subject classifications: Primary 62J02; secondary 62F10
It should be noted that a Poisson-gamma hierarchical generalized linear model (HGLM) considered at a single moment follows a negative binomial regression model (Cameron and Trivedi [4], Hilbe[12]), since a mixture of Poisson distributions by a gamma distribution gives a negative binomial distribution (Johnson et al [13])
The aim of this paper is to provide a breakthrough in the study of the existence problem for maximum likelihood estimators (MLE) in Poisson-gamma HGLM
Summary
Consider a sample of n individuals kept under observation over T periods. For the individual k during the period t, a random count variable Ykt is observed and represents the dependent variable, while the J deterministic characteristics xkt = (xkt1, . . . , xktJ )′ ∈ RJ are known and represent the covariables. For the individual k during the period t, a random count variable Ykt is observed and represents the dependent variable, while the J deterministic characteristics xkt = T , and the nT × J regression matrix X = (x11x12 · · · xnT )′ In this setting, denoting by P(λ) the Poisson distribution of parameter λ, λ > 0, and, by gamma(a, b) the gamma distribution of parameters a and b, a > 0, b > 0, the Poisson-gamma HGLM requires the three following assumptions:. Observe that the non-conditional distribution of Ykt is the negative binomial distribution N B(a, λkt/(a + λkt)), since, by (H1) and (H2), for ykt ∈ N: P(Ykt = ykt) =. An extension of the Poisson-gamma HGLM, useful for instance in insurance, is to assume that individuals are not necessarily observed for the same number of periods. All results presented seem to be extended to cover this case
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