Some Investigations about the Properties of Maximum Likelihood Estimations Based on Lower Record Values for a Sub-Family of the Exponential Family

TL;DR

This paper investigates the properties of maximum likelihood estimators for a sub-family of the exponential family, comparing estimations based on samples and lower record values, and analyzing their asymptotic behaviors.

Contribution

It introduces new theoretical results on MLEs based on lower record values for a specific exponential sub-family, including their asymptotic properties and relations to standard MLEs.

Findings

Derived MLEs for parameters, PDFs, and CDFs based on lower record values.

Established theoretical relations between MLEs from samples and lower record values.

Proved asymptotic properties of these estimators.

Abstract

Here, in this paper it has been considered a sub family of exponential family. Maximum likelihood estimations (MLE) for the parameter of this family, probability density function, and cumulative density function based on a sample and based on lower record values have been obtained. It has been considered Mean Square Error (MSE) as a criterion for determining which is better in different situations. Additionally, it has been proved some theories about the relations between MLE based on lower record values and based on a random sample. Also, some interesting asymptotically properties for these estimations have been shown during some theories.