Marginalized Maximum Likelihood Estimation Method for the Three-parameter Lognormal Distribution

This paper proposes an adjustment method to overcome the difficulties encountered by the maximum likelihood method in the case of the three-parameter lognormal distribution. Endeed, when the threshold parameter is close to the smallest order statistic, the standard likelihood function is no longer bounded. In this case, maximum likelihood estimators are no longer accessible. Our strategy is twofold: first, we construct adaptive bounds intended to contain the location and shape parameters with a probability close to one as the sample size increases. Second, we construct a marginal likelihood function, which we maximize using an optimization method available in the R software through the ”nlnimb” package. This likelihood function is based on the (n-1) largest order statistics. Finally, Monte Carlo simulation studies are used to analyze the asymptotic behavior of the constructed intervals and to study the asymptotic properties of the proposed estimators through bias and the Root Mean-Squared Error(RMSE).