Model selection based on penalized \(\phi \)-divergences for multinomial data
In applications there are usually several different models for describing a population from a given sample of observations and thus the problem of choosing the best fitting distribution among a list of candidates for a given set of data is an important subject. On a basis of a random sample from an unknown population and two possibly misspecified parametric models, which may be separate, overlapping or nested, the hypothesis model selection testing consists of testing if the two competing models are equally close to the true population against the hypothesis that one model is closer than the other. \textit{M. D. Jiménez-Gamero} et al. [Comput. Stat. Data Anal. 55, No. 12, 3365--3378 (2011; Zbl 1464.62099)] have studied the model selection test problem for multinomial data by using \(\phi\)-divergences as a discrepancy measure between two populations. However, other discrepancy measures could be used to measure the closeness between each competing model and the true population model. This is special appealing in cases where there are zero cell frequencies. In such cases it was illustrated by \textit{A. Mandal} et al. [Sankhyā, Ser. A 72, No. 2, 376--406 (2010; Zbl 1213.62042)] that the use of penalized \(\phi\)-divergences instead of non-penalized ones can improve inferences. For this reason, this paper deals with the model selection test problem for multinomial data by using penalized \(\phi\)-divergences. In this context, the test statistic proposed is a sample version of the difference of the distances between the true population and each competing model. Based on this test statistic and the theory developed, decision rules are given for the model selection testing problem for separate, nested or overlapping models. A simulation study evaluate the performance of the proposed methods and compare them with the ones based on non-penalized divergences. From the results, it seems that for separate models the penalization gives better results in the sense of getting a greater percentage of selecting the true population. Finally, the usefulness of the proposed methods are illustrated in a real data set modeling social preferences.
scientific article; zbMATH DE number 7444601