Bias-Corrected and Variance-Corrected MLE for the New Median Based Unit Weibull Distribution (MBUW)

TL;DR

This paper introduces bias correction and variance correction methods for maximum likelihood estimators of the new median-based Weibull distribution, addressing issues of bias and high parameter correlation in small to moderate samples.

Contribution

It proposes novel bias and variance correction techniques for MLEs of the MBUW distribution, improving estimation accuracy and parameter inference.

Findings

Bias correction reduces estimator bias in small samples.

Variance correction alleviates high parameter correlation.

Methods are demonstrated on real data.

Abstract

As the maximum likelihood method is the most commonly used method for parameters estimation being unbiased, consistent, efficient, and asymptotically normal, MLE is used to fit the new distribution (MBUW). But in small to moderate sample size, this MLE estimator is biased unlike the MLE estimators obtained from large sample sizes. In this paper, the Bias-corrected approach for this distribution is discussed and applied to real data analysis. The MLE estimators of MBUW obtained from some optimization techniques like derivative free Nelder Mead algorithm suffers from significant high correlation that is reflected on high covariance between the parameters. Also this association between the parameters affects the variances which may be inflated enough to approach infinity hampering construction of confidence intervals for each parameter. This problem may arise with any optimization technique which necessitates remedies trying to fix it. The author also elaborates a variance correction approach heavily relaying on re-parameterizing the negative log likelihood.