Bias-Corrected and Variance-Corrected MLE for the New Median Based Unit Weibull Distribution (MBUW)
Iman M. Attia
*
Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt.
*Author to whom correspondence should be addressed.
Abstract
In this paper, the author discusses the MLE of a new unit distribution called Median-Based unit Weibull (MBUW), previously outlined in another article. The distribution has two parameters, (α) and (β), and the author focuses on the bias corrected MLE, particularly useful for small to moderate samples. The author elaborates on the well-known corrective approach that other researchers have applied for estimating the bias-corrected MLE in various distributions. Additionally, the author presents a new re-parameterization for the negative Log- Likelihood because the two parameters of the MBUW distribution are highly correlated. This correlation, which varies depending on the data, can be either linear or nonlinear and may be positive or negative. It is identified as the primary cause of high variance or even infinite variance, complicating the construction of confidence intervals. The author refers to this method as the variance-corrected MLE. To address this correlation, the author defines one parameter (β) in terms of the other parameter (α) leading to re-parametrization of the negative Log-Likelihood. Consequently, the author estimates the parameter (α) and then substitutes it into the relationship to get the other parameter (β) . The Nelder-Mead algorithm is utilized for the MLE process. Furthermore, the author employs the expected Fisher information matrix to estimate the variance of \((\hat{α}),\) then uses the delta method to derive the variance of \((\hat{β}).\) The author also applies the bias-corrected MLE and compares the estimated parameters to those obtained through the variance-corrected approach. The results show that the parameters are nearly identical. The bias of the estimated \((\hat{α})\) is far less than the bias of the estimated \((\hat{β}).\) Other statistical indices remain nearly equal before and after deploying the bias-corrected MLE, highlighting the effective performance of the variance-corrected MLE in estimating both parameters and their variances. Thus, this new approach helps mitigate the correlation between the two parameters. The MBUW distribution fits the datasets used in the paper, although some well-known unit distributions may slightly outperform MBUW regarding goodness-of-fit (GoF) tests such as the KS-test, CVM test, and AD test, along with indices like AIC, CAIC, BIC, HQIC, and nLL. However, due to the greater dependency between its two parameters compared to other well-known unit distributions like the Beta and Kumaraswamy distributions, the author felt compelled to explore solutions to mitigate the correlation and compare this solution to the established corrective approach of the bias-corrected MLE. This new variance-corrected approach yields minimal bias.
Keywords: Cox and Snell bias-correction, median based unit Weibull (MBUW), maximum likelihood estimators, variance-corrected MLE, Bias-corrected MLE