Mean likelihood estimators

TL;DR

This paper explores the derivation and comparison of mean likelihood estimators using Mathematica, showing they outperform maximum likelihood and Bayesian estimators in a specific time series model based on error criteria.

Contribution

It introduces a method for deriving mean likelihood estimators with Mathematica and compares their performance to other estimators in a moving-average model.

Findings

Mean likelihood estimator outperforms MLE and Bayesian estimators in the studied model.

Mathematica facilitates symbolic, numeric, and graphical analysis of estimators.

Supplementary Mathematica notebook enables reproducibility and extension of results.

Abstract

The use of {\it Mathematica} in deriving mean likelihood estimators is discussed. Comparisons between the maximum likelihood estimator, the mean likelihood estimator and the Bayes estimate based on a Jeffrey's noninformative prior using the criteria mean-square error and Pitman measure of closeness. Based on these criteria we find that for the first-order moving-average time series model, the mean likelihood estimator outperforms the maximum likelihood estimator and the Bayes estimator with a Jeffrey's noninformative prior. {\it Mathematica} was used for symbolic and numeric computations as well as for the graphical display of results. A {\it Mathematica} notebook is available which provides supplementary derivations and code from http://www.stats.uwo.ca/mcleod/epubs/mele The interested reader can easily reproduce or extend any of the results in this paper using this supplement.