Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Least squares regression based on probability plots, also called rank regression, can be used to estimate the parameters of some distributions. Regression is performed between a function of the empirical distribution function and the order statistic as the independent variable. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, the exponential and the Gumbel (extreme value type I) distributions. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators with respect to the mean square error, especially in small samples.