A revisit to maximum likelihood estimation of Weibull model parameters

TL;DR

This paper revisits the maximum likelihood estimation of Weibull distribution parameters, providing analytical proofs of existence and uniqueness, and explores their sampling distributions through theoretical and computational methods.

Contribution

It offers new analytical insights into the existence, uniqueness, and distributional properties of Weibull MLEs, supported by simulation results.

Findings

MLEs exist and are unique for Weibull parameters.

Sampling distributions of MLEs can be approximated by Weibull distributions.

Simulation confirms bias and MSE expressions for MLEs.

Abstract

In this work, we revisit the estimation of the model parameters of a Weibull distribution based on iid observations, using the maximum likelihood estimation (MLE) method which does not yield closed expressions of the estimators. Among other results, it has been shown analytically that the MLEs obtained by solving the highly non-linear equations do exist (i.e., finite), and are unique. We then proceed to study the sampling distributions of the MLEs through both theoretical as well as computational means. It has been shown that the sampling distributions of the two model parameters' MLEs can be approximated fairly well by suitable Weibull distributions too. Results of our comprehensive simulation study corroborate some recent results on the first-order bias and first-order mean squared error (MSE) expressions of the MLEs.