Exact expressions for the weights used in least-squares regression estimation for the log-logistic and Weibull distribution

TL;DR

This paper derives exact variance expressions for residuals in heteroscedastic regression models for the log-logistic and Weibull distributions, improving estimation accuracy in small samples by using weighted regression.

Contribution

It provides new exact variance formulas enabling weighted regression for better estimation in small samples for these distributions.

Findings

Weighted regression outperforms maximum likelihood in small samples.

Exact residual variance expressions improve estimation accuracy.

Maximum likelihood performs best in large samples.

Abstract

Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting. The equations leads to highly heteroscedastic regression. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples maximum likelihood performs best, but it is shown that in smaller samples the weighted regression outperforms maximum likelihood estimation with respect to bias and mean square error.