Abstract
In the Bayesian framework, the usual choice of prior in the prediction of homogeneous Poisson processes with random effects is the gamma one. Here, we propose the use of higher order maximum entropy priors. Their advantage is illustrated in a simulation study and the choice of the best order is established by two goodness-of-fit criteria: Kullback–Leibler divergence and a discrepancy measure. This procedure is illustrated on a warranty data set from the automobile industry.
Highlights
In the study of the prediction problem for homogeneous Poisson processes (HPP), used in various fields including biomedicine [1], marketing [2] and reliability [3], the recurrent events often display extra-Poisson variation
We note that the posterior distribution (4) will not have a known closed form, but includes rather complicated high dimensional densities, rendering direct inference almost impossible because of the high dimensional integration, necessary to obtain the normalizing constant, which is not mathematically tractable. We generate from this posterior distribution a large number of samples using Markov chain Monte Carlo (MCMC) implemented in WinBUGS [21], and, from these samples, we can obtain appropriate parameter estimates such as the posterior mean of λ|( N (t1 ); α), where α is estimated by the methods described
When the value of the coefficient of variation is ≤1 and the random effects are neither generated by the gamma or a MaxEnt prior, we note that the Poisson–MaxEnt model with the 2-moment prior and the negative binomial (NB) model are similar in terms of performance where each of their predictive densities are closest to the true predictive density f ( N (t1, t2 )| N (t1 ))
Summary
In the study of the prediction problem for homogeneous Poisson processes (HPP), used in various fields including biomedicine [1], marketing [2] and reliability [3], the recurrent events often display extra-Poisson variation. We use several different prediction models with k-moment entropy priors for different values of k We study their performance using the absolute error discrepancy equal to the absolute difference between point predictors. We describe the maximum entropy principle, introduce the homogeneous Poisson process with random effects (HPPr) and define our general Poisson–MaxEnt model
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