A Bounded Derivative Method for the Maximum Likelihood Estimation on Weibull Parameters

TL;DR

This paper introduces a new bounded derivative method for maximum likelihood estimation of Weibull distribution parameters, ensuring unique solutions and improving computational efficiency through a novel root-finding algorithm.

Contribution

It provides a simple proof of global monotonicity, defines derivative bounds based on scale invariance, and develops an efficient, convergent root-finding algorithm for Weibull MLE.

Findings

The proposed algorithm converges reliably and efficiently.

Numerical experiments show improved performance over existing methods.

The method guarantees unique solutions for Weibull parameter estimation.

Abstract

For the basic maximum likelihood estimating function of the two parameters Weibull distribution, a simple proof on its global monotonicity is given to ensure the existence and uniqueness of its solution. The boundary of the function's first-order derivative is defined based on its scale-free property. With a bounded derivative, the possible range of the root of this function can be determined. A novel root-finding algorithm employing these established results is proposed accordingly, its convergence is proved analytically as well. Compared with other typical algorithms for this problem, the efficiency of the proposed algorithm is also demonstrated by numerical experiments.