Abstract
Estimation of the unknown mean, μ and variance, σ2 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 σ2 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.
Highlights
The Gaussian distribution is a continuous function characterized by the mean μ and variance σ2
The mean μ and the variance σ2 are referred to as sufficient parameters in most of the statistics literature and this is due to the fact that they contain all information about the probability distribution function, see Equation (1)
The maximum likelihood estimation (MLE) is regarded as the standard approach to most of the nonlinear estimation problems as it always converges to the required minimum given “good” initial guess approximations, it requires the maximization of the log-likelihood method [20]
Summary
The Gaussian distribution is a continuous function characterized by the mean μ and variance σ2. It is regarded as the mostly applied distribution in all of the science disciplines since it can be used to approximate several other. We consider a single observation x obtained from a univariate Gaussian distribution with both the ( ) mean μ and variance, σ2, unknown, that is x ~ N μ,σ 2 , −∞ < μ < ∞. In this paper the problems of estimating the sufficient parameters of a normal distribution using the iterative methods are discussed. The mean μ and the variance σ2 are referred to as sufficient parameters in most of the statistics literature and this is due to the fact that they contain all information about the probability distribution function, see Equation (1)
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