Estimation and testing on independent not identically distributed observations based on R\'enyi's pseudodistances

TL;DR

This paper introduces a robust family of estimators based on Rénye's pseudodistances for independent, non-identically distributed data, extending classical methods like MLE with improved robustness and applicability to regression models.

Contribution

It develops a new class of estimators and Wald-type tests based on Rénye's pseudodistances for i.n.i.d.o., including influence function analysis and practical evaluation.

Findings

Proposed estimators include MLE as a special case.

Simulation results show improved robustness over classical methods.

Real data analysis demonstrates practical applicability.

Abstract

In real life we often deal with independent but not identically distributed observations (i.n.i.d.o), for which the most well-known statistical model is the multiple linear regression model (MLRM) without random covariates. While the classical methods are based on the maximum likelihood estimator (MLE), it is well known its lack of robustness to small deviations from the assumed conditions. In this paper, and based on the R\'enyi's pseudodistance (RP), we introduce a new family of estimators in case our information about the unknown parameter is given for i.n.i.d.o.. This family of estimators, let say minimum RP estimators (as they are obtained by minimizing the RP between the assumed distribution and the empirical distribution of the data), contains the MLE as a particular case and can be applied, among others, to the MLRM without random covariates. Based on these estimators, we introduce Wald-type tests for testing simple and composite null hypotheses, as an extension of the classical MLE-based Wald test. Influence functions for the estimators and Wald-type tests are also obtained and analysed. Finally, a simulation study is developed in order to asses the performance of the proposed methods and some real-life data are analysed for illustrative purpose.