Asymptotic properties of generalized closed-form maximum likelihood estimators

TL;DR

This paper introduces a generalized closed-form maximum likelihood estimator with proven asymptotic properties, enabling real-time statistical inference for models like Gamma, Nakagami, and Beta distributions, especially useful in embedded systems.

Contribution

It develops a new generalized MLE that provides closed-form solutions and retains key properties, improving computational efficiency over traditional methods.

Findings

The generalized MLE has asymptotic normality and consistency.

Closed-form estimators are derived for Gamma, Nakagami, and Beta distributions.

Extension to bivariate gamma distribution demonstrates broader applicability.

Abstract

The maximum likelihood estimator (MLE) is pivotal in statistical inference, yet its application is often hindered by the absence of closed-form solutions for many models. This poses challenges in real-time computation scenarios, particularly within embedded systems technology, where numerical methods are impractical. This study introduces a generalized form of the MLE that yields closed-form estimators under certain conditions. We derive the asymptotic properties of the proposed estimator and demonstrate that our approach retains key properties such as invariance under one-to-one transformations, strong consistency, and an asymptotic normal distribution. The effectiveness of the generalized MLE is exemplified through its application to the Gamma, Nakagami, and Beta distributions, showcasing improvements over the traditional MLE. Additionally, we extend this methodology to a bivariate gamma distribution, successfully deriving closed-form estimators. This advancement presents significant implications for real-time statistical analysis across various applications.