Fitting three-parameter lognormal models by numerical global optimization — an improved algorithm

Abstract

The theoretical maximum likelihood point estimates of the parameters of the three-parameter lognormal distribution (LD) for complete, ungrouped samples are infeasible values for which the likelihood function is positively infinite. This has prompted the search for usable point estimates which are called local-maximum likelihood estimates (LMLE's) and which correspond to the largest finite local maximum of the log-likelihood function (LLF). Previously published work has shown that an effective way to obtain LMLE's is to numerically maximize the conditional LLF of the observations using sequential interior penalty and barrier function methods. These methods are robust but necessitate the numerical solution of a sequence of maximization problems. This paper describes an algorithm for computing LMLE's which involves only a single numerical maximization of a suitably transformed conditional LLF. The results of computational tests indicate that the algorithm, when implemented using computationally efficient and reliable univariate global optimization methods, is a computationally robust method for computing LMLE's of LD parameters.