Interval and Point Estimators for the Location Parameter of the Three-Parameter Lognormal Distribution

The three-parameter lognormal distribution is the extension of the two-parameter lognormal distribution to meet the need of the biological, sociological, and other fields. Numerous research papers have been published for the parameter estimation problems for the lognormal distributions. The inclusion of the location parameter brings in some technical difficulties for the parameter estimation problems, especially for the interval estimation. This paper proposes a method for constructing exact confidence intervals and exact upper confidence limits for the location parameter of the three-parameter lognormal distribution. The point estimation problem is discussed as well. The performance of the point estimator is compared with the maximum likelihood estimator, which is widely used in practice. Simulation result shows that the proposed method is less biased in estimating the location parameter. The large sample size case is discussed in the paper. 1. Introduction The two-parameter lognormal distribution and the three-parameter lognormal distribution have been used in many areas such as reliability, economics, ecology, biology, and atmospheric sciences. In the past twenty years, many research papers have been published on the parameter estimation problems for the lognormal distributions. See, for example, Kanefuji and Iwase [1], Sweet [2], and Crow and Shimizu [3]. The three-parameter lognormal distribution is the extension of the two-parameter lognormal distribution to meet the need of the biological and sociological science, and other fields. Some papers can be found in the literature for the parameter estimation problems for this distribution. See, for example, Komori and Hirose [4], Singh et al. [5], Eastham et al. [6], Cohen et al. [7], Chieppa and Amato [8], Griffiths [9], and Cohen and Whitten [10]. Chen [11] analyzed an application data set containing 49 plastic laminate strength measurements using the locally maximum likelihood estimation method. When the locally maximum likelihood estimation method is used, people are not using the criterion of searching the value of the parameter, which is being estimated, such that the likelihood function is maximized. This is particularly true when the location parameter of the three-parameter lognormal distribution is estimated. This is because the likelihood function goes to infinity when the value of the location parameter approaches to the smallest order statistic. The point estimation will be discussed in Section 3. The same data set is analyzed using the method presented in this paper. It should be noted