Closed-form solutions for parameter estimation in exponential families based on maximum a posteriori equations

TL;DR

This paper introduces closed-form estimators for exponential family parameters using MAP equations, offering a computationally simpler alternative that performs comparably to traditional methods, with improved accuracy as sample size grows.

Contribution

It provides a novel closed-form solution for parameter estimation in exponential families based on MAP equations, reducing computational complexity.

Findings

Estimators' bias and MSE decrease with larger sample sizes.

Performance is comparable to traditional MAP and ML estimators.

Proposed estimators are computationally simpler, avoiding numerical optimization.

Abstract

In this paper, we derive closed-form estimators for the parameters of certain exponential family distributions through the maximum a posteriori (MAP) equations. A Monte Carlo simulation is conducted to assess the performance of the proposed estimators. The results show that, as expected, their accuracy improves with increasing sample size, with both bias and mean squared error approaching zero. Moreover, the proposed estimators exhibit performance comparable to that of traditional MAP and maximum likelihood (ML) estimators. A notable advantage of the proposed method lies in its computational simplicity, as it eliminates the need for numerical optimization required by MAP and ML estimation.