Estimators Based on Order Statistics of Small Samples from a Three-Parameter Lognormal Distribution

Abstract

Abstract The expectations of order statistics of a logarithmic-normal distribution are expressed as functions which are linear in a location and a scale parameter but nonlinear in a shape parameter of the distribution. The regression of the observed small-sample order statistics on approximations to their expectations is considered and an iterative procedure is proposed which leads to approximate weighted least squares estimates of all three parameters. The performance of the iterative procedure as well as the properties of the estimators are studied by means of Monte Carlo simulation experiments involving samples of size 10, 15 and 20 from standardized distributions (zero median, unit geometric mean) with shape parameter (natural logarithmic variance) in the range zero to unity. It is shown that the iterative procedure always converges and that almost unbiased estimates may be obtained. The variances of the location and scale parameter estimates compare favorably with those of the well-known order-statistic estimates of a standard normal distribution. The parameterization of the lognormal distribution which is adopted leads naturally to use of the latter estimates in the limiting case of samples which exhibit symmetry or even negative skewness. While only complete samples are considered explicitly, the simple extension required to take care of censored samples is indicated.