Modified Profile Likelihood Estimation in the Lomax Distribution

Abstract

In this paper, we consider improving maximum likelihood inference for the scale parameter of the Lomax distribution. The improvement is based on using modifications to the maximum likelihood estimator based on the Barndorff-Nielsen modification of the profile likelihood function. We apply these modifications to obtain improved estimators for the scale parameter of the Lomax distribution in the presence of a nuisance shape parameter. Due to the complicated expression for the Barndorff-Nielsen’s modification, several approximations to this modification are considered in this paper, including the modification based on the empirical covariances and the approximation based on using suitably derived approximate ancillary statistics. We obtained the approximations for the Lomax profile likelihood function and the corresponding modified maximum likelihood estimators. They are not available in simple closed forms and can be obtained numerically as roots of some complicated likelihood equations. Comparisons between maximum profile likelihood estimator and modified profile likelihood estimators in terms of their biases and mean squared errors were carried out using simulation techniques. We found that the approximation based on the empirical covariances to have the best performance according to the criteria used. Therefore we recommend to use this modified version of the maximum likelihood estimator for the Lomax scale parameter, especially for small sample sizes with heavy censoring, which is quite common in industrial life testing experiments and reliability studies. An example based on real data is given to illustrate the methods considered in this paper.