On the Estimation of a Univariate Gaussian Distribution: A Comparative Approach

Abstract

Estimation of the unknown mean, μ and variance, σ 2 of a univariate Gaussian distribution given a single study variable x is considered. We propose an approach that does not require initialization of the sufficient unknown distribution parameters. The approach is motivated by linearizing the Gaussian distribution through differential techniques, and estimating, μ and σ 2 as regression coefficients using the ordinary least squares method. Two simulated datasets on hereditary traits and morphometric analysis of housefly strains are used to evaluate the proposed method (PM), the maximum likelihood estimation (MLE), and the method of moments (MM). The methods are evaluated by re-estimating the required Gaussian parameters on both large and small samples. The root mean squared error (RMSE), mean error (ME), and the standard deviation (SD) are used to assess the accuracy of the PM and MLE; confidence intervals (CIs) are also constructed for the ME estimate. The PM compares well with both the MLE and MM approaches as they all produce estimates whose errors have good asymptotic properties, also small CIs are observed for the ME using the PM and MLE. The PM can be used symbiotically with the MLE to provide initial approximations at the expectation maximization step.