Maximum Likelihood Estimation in the Three-Parameter Lognormal Distribution

Abstract

Summary Several authors (Hill, 1963; Box and Cox, 1964; Harter and Moore, 1966) have discussed the problem of estimating α, μ and σ2 where observations xi, i = 1, 2, ..., n, are available such that y = log (x + α) has a normal distribution with mean μ and variance σ2. In this paper it is shown that the difficulties encountered because the “likelihood” becomes unbounded are avoided if one takes into account the grouping error inherent in the data and uses the “correct” likelihood (Kempthorne and Folks, 1971; Copas, 1972). The behaviour of the maximum likelihood estimators is investigated using asymptotic theory and Monte Carlo simulations.