Increased Statistical Efficiency in a Lognormal Mean Model

Grant H Skrepnek,Ashok Sahai

doi:10.1155/2014/964197

Abstract

Within the context of clinical and other scientific research, a substantial need exists for an accurate determination of the point estimate in a lognormal mean model, given that highly skewed data are often present. As such, logarithmic transformations are often advocated to achieve the assumptions of parametric statistical inference. Despite this, existing approaches that utilize only a sample’s mean and variance may not necessarily yield the most efficient estimator. The current investigation developed and tested an improved efficient point estimator for a lognormal mean by capturing more complete information via the sample’s coefficient of variation. Results of an empirical simulation study across varying sample sizes and population standard deviations indicated relative improvements in efficiency of up to 129.47 percent compared to the usual maximum likelihood estimator and up to 21.33 absolute percentage points above the efficient estimator presented by Shen and colleagues (2006). The relative efficiency of the proposed estimator increased particularly as a function of decreasing sample size and increasing population standard deviation.

Highlights

The presence of highly skewed data is commonplace across both basic and applied sciences [1, 2]
Relative efficiencies for both efficient estimators increased as a function of lower sample size, though the proposed estimator, GS(2014), was more efficient at lower population standard deviations
Findings support the proposed estimator, GS(2014), under almost all combinations of sample sizes and population standard deviations compared to the current efficient estimator and the usual maximum likelihood estimator

Summary

Introduction

The presence of highly skewed data is commonplace across both basic and applied sciences [1, 2]. The logarithmic transformation of such data may be undertaken with the primary purpose of establishing a normal distribution and improving variance, which may include removing heteroskedasticity in the process, to achieve N(μ, σ2). Shen et al [4] proposed an improved efficient minimum risk/relative mean squared error (RMSE) estimator of the lognormal mean, and numerous researchers have addressed the estimation of parameters within this distribution from both a frequentist and Bayesian context [5,6,7,8,9]. In presenting the lognormal distribution from a more fundamental perspective, if Y is a random variable with a lognormal distribution and a mean of E(Y) = ς, ln(Y) will be normally distributed with mean of μ and variance of σ2. Y may be expressed as Y ∼ ln(μ, σ2) with a mean of ς, observing that ς

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