Minimum Penalized ϕ-Divergence Estimation under Model Misspecification.

M Alba-Fernández,M Jiménez-Gamero,F Ariza-López

doi:10.3390/e20050329

M Alba-Fernández, M Jiménez-Gamero + Show 1 more

Open Access

https://doi.org/10.3390/e20050329

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Journal: Entropy	Publication Date: Apr 30, 2018
Citations: 3	License type: CC BY 4.0

Affiliation: Universidad de Sevilla, University of Jaén

Abstract

This paper focuses on the consequences of assuming a wrong model for multinomial data when using minimum penalized -divergence, also known as minimum penalized disparity estimators, to estimate the model parameters. These estimators are shown to converge to a well-defined limit. An application of the results obtained shows that a parametric bootstrap consistently estimates the null distribution of a certain class of test statistics for model misspecification detection. An illustrative application to the accuracy assessment of the thematic quality in a global land cover map is included.

Highlights

In many practical settings, individuals are classified into a finite number of unique nonoverlapping categories, and the experimenter collects the number of observations falling in each of such categories.In statistics, that sort data is called multinomial data
To give an assessment of the thematic accuracy, a comparison is needed between the label considered as true of a feature and the label assigned to the same feature after a classification
By using the fact that the minimum penalized φ-divergence estimator (MPφE) always converges to a well-defined limit, whether the model in H0 is true or not, we prove that the bootstrap consistently estimates the null distribution of these test statistics

Summary

Introduction

Individuals are classified into a finite number of unique nonoverlapping categories, and the experimenter collects the number of observations falling in each of such categories. They are strongly consistent and, conveniently normalized, asymptotically normal To derive these asymptotic properties, it is assumed that the probability model is correctly specified, that is to say, that we are sure about π ∈ P. / P, using as a test statistic a penalized φ1 -divergence between a nonparametric estimator of π, the relative frequencies, and a parametric estimator of π, obtained by assuming that the null hypothesis is true, P(θ ), θbeing an MPφ2 E. By using the fact that the MPφE always converges to a well-defined limit, whether the model in H0 is true or not, we prove that the bootstrap consistently estimates the null distribution of these test statistics.

Some Asymptotic Properties of MPφEs

Application to Bootstrapping Goodness-Of-Fit Tests

Proofs