Statistica Sinica 23 (2013),

MULTIPLE-INFLATION POISSON MODEL WITH

$L_1$ REGULARIZATION

Xiaogang Su$^1$, Juanjuan Fan$^2$, Richard A. Levine$^2$,

Xianming Tan$^3$, and Arvind Tripathi$^1$

$^1$University of Alabama at Birmingham, $^2$San Diego State University,

and $^3$McGill University

Abstract: A multiple-inflation Poisson (MIP) model is put forward for analyzing count data that have multiple inflated values. Analogous to the zero-inflated Poisson model (ZIP; Lambert (1992)), MIP assumes a mixture distribution of Poisson and degenerate distributions, where the probabilities for the inflated values are from a cumulative logit model. We explore the properties of the proposed model, with a detailed treatment given to its maximum likelihood estimation. Moreover, we address variable selection by adopting an $L_1$ regularization scheme. Both simulation experiments and an analysis of a health care data set are provided to illustrate the multiple-inflation Poisson model.

Key words and phrases: Count data, LASSO, Poisson distribution, variable selection, zero-inflated.