Minimum $L^q$-distance estimators for non-normalized parametric models

TL;DR

This paper introduces a new minimum $L^q$-distance estimation method for smooth density models on the positive axis, providing theoretical properties and empirical comparisons with classical estimators.

Contribution

It develops a novel $L^q$-norm based estimation approach for non-normalized models, with rigorous existence, measurability, and consistency results.

Findings

The new estimator is consistent under common asymptotic conditions.

Simulation studies show competitive performance with classical estimators.

The method effectively handles non-normalized models like exponential-polynomial families.

Abstract

We propose and investigate a new estimation method for the parameters of models consisting of smooth density functions on the positive half axis. The procedure is based on a recently introduced characterization result for the respective probability distributions, and is to be classified as a minimum distance estimator, incorporating as a distance function the $L^q$-norm. Throughout, we deal rigorously with issues of existence and measurability of these implicitly defined estimators. Moreover, we provide consistency results in a common asymptotic setting, and compare our new method with classical estimators for the exponential-, the Rayleigh-, and the Burr Type XII distribution in Monte Carlo simulation studies. We also assess the performance of different estimators for non-normalized models in the context of an exponential-polynomial family.