Moment Monotonicity of Weibull, Gamma and Log-normal Distributions

TL;DR

The paper addresses whether the moment sequence $E(X^n)^{1/n}$ is monotone for Weibull, Gamma, and Log-normal distributions, providing the first complete distribution-specific proofs. It employs ratio analyses $R=E(X^n)^m / E(X^m)^n$, leveraging the digamma function and Stirling’s approximation to show monotonicity and to identify parameter-cancellation patterns that can simplify estimation. The main results show $E(X^n)^{1/n}$ is nondecreasing in $n$ for all three distributions, with concise proofs for each via closed-form moments. These findings deepen theoretical understanding of moment behavior and offer practical implications for parameter estimation in reliability, econometrics, and machine learning contexts.

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.

Figures (3)

• Figure 1: Probability density function of Weibull distribution.
• Figure 2: Probability density function of Gamma distribution.
• Figure 3: Probability density function of Log-normal distribution.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof