Parametric Statistical Inference in the Zone of Moderate Deviation Probabilities

TL;DR

This paper develops a parametric statistical inference theory for the moderate deviation probability zone, introducing new proof techniques and establishing large deviation principles for estimators.

Contribution

It introduces a novel approach using Taylor series expansion of the likelihood ratio based on Hellinger distance for moderate deviations.

Findings

Proves the Large Deviation Principle for Bayesian and maximum likelihood estimators in the moderate deviation zone.

Provides a uniform approximation of the likelihood ratio logarithm.

Establishes a theorem on the concentration of the posterior Bayesian measure.

Abstract

A parametric theory of statistical inference is developed for the moderate deviation probability zone. The new approach to the proofs is based on the Taylor series expansion of the logarithm of the likelihood ratio based on the Hellinger distance. The Large Deviation Principle in the moderate deviation probability zone is proven for Bayesian estimators and maximum likelihood estimators. A uniform approximation of the logarithm of the likelihood ratio and Theorem on concentration of the posterior Bayesian measure are also established for the zone of moderate deviation probabilities.