Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 3, 277-287

ON A META-POISSON PROCESS FOR A COUNT DATA ANALYSIS

By

CARLO FERRERI, *University of Bologna, Italy*

SUMMARY. In this paper, a simple mixture Poisson process $\{X(t),t>0\}$, denoted by meta-Poisson process (MPP), based on a mixing density function ({\it m.d.f.}) having $(0,1]$ as support is indicated for analysing count data. It depends, just as the Poisson process (PP) does, only on a parameter- function $\vartheta $(t). Whereas the time-function of PP denotes the mean $\mu $(t)=E[X(t)], the function $% \vartheta $(t) of MPP expresses the quadratic mean function\ $\ {\cal M}%_{2}(t)=\surd \{E[X^{2}(t)]\}$. The behaviour of $\rho _{2}$(t), the ratio of variance to $\mu _{02}$(t), helps to elucidate the role of the MPP as a model accounting for the overdispersion of count data especially when $\rho_{1}$(t), the variance to mean ratio, is not much greater than one or is decreasing to one. Useful interpretations, especially for analysing spatial count data, are stressed and the estimation problem is dealt with. Then, an example aiming at illustrating the MPP heuristic possibilities is given

*AMS (1980) subject classification.* 62M99, 60G55

*Key words and phrases*. Meta-Poisson process, compound Poisson process, count data analysis, overdispersion, spatial analysis