Open Journal of Statistics, 2013, 3, 45-51 Published Online December 2013 (http://www.scirp.org/journal/ojs) http://dx.doi.org/10.4236/ojs.2013.36A005 Open Access OJS Inference for the Normal Mean with Known Coefficient of Variation Yuejiao Fu, Hangjing Wang, Augustine Wong Department of Mathematics and Statistics, York University, Toronto, Canada Email: yuejiao@mathstat.yorku.ca, hangjing@mathstat.yorku.ca, august@yorku.ca Received November 1, 2013; revised December 1, 2013; accepted December 8, 2013 Copyright © 2013 Yuejiao Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor- dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Yuejiao Fu et al. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. ABSTRACT Inference for the mean of a normal distribution with known coefficient of variation is of special theoretical interest be- cause the model belongs to the curved exponential family with a scalar parameter of interest and a two-dimensional minimal sufficient statistic. Therefore, standard inferential methods cannot be directly applied to this problem. It is also of practical interest because this problem arises naturally in many environmental and agriculture studies. In this paper we proposed a modified signed log likelihood ratio method to obtain inference for the normal mean with known coeffi- cient of variation. Simulation studies show the remarkable accuracy of the proposed method even for sample size as small as 2. Moreover, a new point estimator for the mean can be derived from the proposed method. Simulation studies show that new point estimator is more efficient than most of the existing estimators. Keywords: Canonical Parameter; Coverage Probability; Curved Exponential Family; Modified Signed Log Likelihood Ratio Statistic 1. Introduction Normal distribution is one of the most widely known and commonly used distributions in statistics. Even in the introductory statistics courses, we discussed inference about the mean of a normal distribution. Usually we as- sume that the population mean and the population stan- dard deviation are unrelated parameters. However, in many physical and biological applications the population standard deviation is often found to be proportional to the mean. That is, the mean and standard deviation are re- lated. The ratio of the standard deviation to the mean is defined as the coefficient of variation (CV) in statistics. The focus of this paper is to make inference on the nor- mal mean using the extra information on the CV. In practice, this problem arises more frequently than we might anticipate. For example, in environmental stud- ies, inference about the mean of the pollutant is of special interest. And in those studies, the standard deviation of a pollutant is often assumed to be directly related to the mean of the pollutant (Niwitpong [1]). In agricultural studies, it is customary to conduct multi-location trials. From the results of a few locations, the CV can be calcu- lated and subsequently used as a known value for study- ing the mean of the experiment conducted in a new loca- tion (Bhat and Rao [2]). Brazauskas and Ghorai [3] also give examples of this problem emerging from biological and medical experiments. From the theoretical point of view, estimating a normal mean with known CV is also an interesting problem because it has a scalar parameter but a two-dimensional minimal sufficient statistic. In other words, we have a curved exponential family model, and standard inferential methods cannot be directly ap- plied (see Efron [4]). In literature, many authors have studied point estima- tion of a normal mean with known CV. For example, a consistent estimator was obtained by Searls [5] based on truncation of extreme observations. Khan [6] derived the best unbiased estimator with minimum variance. Gleser and Healy [7] obtained the uniformly minimum risk es- timator when the loss function is the squared error. Sen [8] proposed a simple and consistent estimator but the proposed estimator is biased. Guo and Pal [9] worked out an estimator based on the scaled quadratic loss function. Chaturvedi and Tomer [10] extended the method in Singh [11] and proposed a three-stage procedure and an