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

As the maximum likelihood method is the most commonly used method for parameter estimation being unbiased, consistent, efficient, and asymptotically normal, MLE is used to fit the new distribution (MBUW). But in small to moderate sample sizes, 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 the derivative free Nelder Mead algorithm suffer from a significant high correlation reflected in high covariance between the parameters. Also, this association between the parameters affects the variances which may be inflated enough to approach infinity hampering the 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 relying on re-parameterizing the negative log-likelihood.