Maximum L$q$-likelihood estimation

TL;DR

This paper introduces the maximum L$q$-likelihood estimator (ML$q$E), a new parameter estimation method based on nonextensive entropy, analyzing its properties and demonstrating its advantages in bias-variance tradeoff for small to moderate samples.

Contribution

The paper proposes the ML$q$E, a novel estimator based on nonextensive entropy, and studies its asymptotic properties and practical performance through analysis and simulations.

Findings

ML$q$E reduces mean squared error with proper q selection.

Asymptotic normality and efficiency are achieved when q approaches 1.

ML$q$E effectively balances bias and variance in small samples.

Abstract

In this paper, the maximum L$q$-likelihood estimator (ML$q$E), a new parameter estimator based on nonextensive entropy [Kibernetika 3 (1967) 30--35] is introduced. The properties of the ML$q$E are studied via asymptotic analysis and computer simulations. The behavior of the ML$q$E is characterized by the degree of distortion $q$ applied to the assumed model. When $q$ is properly chosen for small and moderate sample sizes, the ML$q$E can successfully trade bias for precision, resulting in a substantial reduction of the mean squared error. When the sample size is large and $q$ tends to 1, a necessary and sufficient condition to ensure a proper asymptotic normality and efficiency of ML$q$E is established.