## ON EFFICIENT ESTIMATION FOR THE LOG-NORMAL MODEL MEAN

### by Winston A. Richards,
Ashok Sahai, Robin Antoine, Letetia Addison,
and
M. Raghunadh Acharya
.

**Abstract:**
It is not uncommon to have situations where highly skewed data appear in research investigations. In many instances, transformations of such data are advocated, for the primary purpose of establishing a normal distribution required for parametric statistical analysis. Such a transformation retains its advantage in the context of ’meta-analysis’ (Joseph et. al. (2000)), as well as in the context of ‘Bayesian’ Case-Studies (Stevens et. al. (2003)). The logarithmic transformation is a commonly-employed method within the decision sciences used to establish a normal distribution of skewed data. Patterson (1966) discussed and defined the challenges involved in the estimation of the population mean following the transformation of sample data. The same is equally true for the logarithmic transformations as well. The purpose of this paper is to address the logarithmic transformation and to ultimately provide the most efficient estimation of the lognormal mean. This has been achieved by using the sample information from the resultant normal distribution, in order to ultimately estimate the mean of the non-transformed population. The aspects of the gains in efficiency of the proposed ‘Optimal Mean Estimator’ are numerically illustrated through a simulation study, comparing it with the minimum “Risk/RMSE (Relative Mean Square Error)” estimator of log-normal mean recently studied by Shen et. al. (2006).
**Key Words: **
population mean, lognormal modeling, simulation study

**Authors:**

Ashok Sahai, sahai.ashok@gmail.com

Winston A. Richards, ugu@psu.edu

rmantoine@hotmail.com M. Raghunadh Acharya, rmantoine@hotmail.com

**Editor:**
Ahmed H. Youssef,ahyoussef@hotmail.com

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