Moment Monotonicity of Weibull, Gamma and Log-normal Distributions

TL;DR

This paper provides complete proofs of the moment monotonicity for Weibull, Gamma, and Log-normal distributions, revealing parameter cancellation properties that enhance understanding and estimation in various applied fields.

Contribution

It offers the first complete mathematical proofs of moment monotonicity for these distributions and uncovers parameter cancellation properties that aid in parameter estimation.

Findings

Proved moment monotonicity for Weibull, Gamma, and Log-normal distributions.

Identified parameter cancellation properties in these distributions.

Enhanced understanding of distribution behaviors for practical applications.

Abstract

This paper investigates the moment monotonicity property of Weibull, Gamma, and Log-normal distributions. We provide the first complete mathematical proofs for the monotonicity of the function $E(X^n)^{\frac{1}{n}}$ specific to these distributions. Through the derivations, we identify a key property: in many cases, one of the two parameters defining each distribution can be effectively canceled out. This finding opens up opportunities for improved parameter estimation of these random variables. Our results contribute to a deeper understanding of the behavior of these widely used distributions and offer valuable insights for applications in fields such as reliability engineering, econometrics, and machine learning.