Least informative distributions in Maximum q-log-likelihood estimation

TL;DR

This paper introduces a robust parameter estimation method using Least Informative Distributions within the Maximum q-log-likelihood framework, effectively handling outliers and contamination in data.

Contribution

It proposes a novel approach combining LIDs and q-log-likelihood to improve robustness and efficiency in parameter estimation under data contamination.

Findings

LID-based estimation outperforms traditional MLE in contaminated data scenarios.

AIC and BIC criteria are adapted for q-log-likelihood and LID models.

Real data tests demonstrate improved fitting performance with the proposed methods.

Abstract

We use the Maximum $q$-log-likelihood estimation for Least informative distributions (LID) in order to estimate the parameters in probability density functions (PDFs) efficiently and robustly when data include outlier(s). LIDs are derived by using convex combinations of two PDFs, $f_\epsilon=(1-\epsilon)f_0+\epsilon f_1$. A convex combination of two PDFs is considered as a contamination $f_1$ as outlier(s) to underlying $f_0$ distributions and $f_\epsilon$ is a contaminated distribution. The optimal criterion is obtained by minimizing the change of Maximum q-log-likelihood function when the data have slightly more contamination. In this paper, we make a comparison among ordinary Maximum likelihood, Maximum q-likelihood estimations, LIDs based on $\log_q$ and Huber M-estimation. Akaike and Bayesian information criterions (AIC and BIC) based on $\log_q$ and LID are proposed to assess the fitting performance of functions. Real data sets are applied to test the fitting performance of estimating functions that include shape, scale and location parameters.