Work

Expectation identity of the negative binomial distribution and its application in the calculations of high-order origin moments

Year: 2025

Type: article

Abstract: Analytically calculating high-order origin moments of a negative binomial distribution has long been a persistent and challenging problem. In this study, we employ an expectation identity method to analytically compute these moments. First, we restate the expectation identity theorem for the negative binomial distribution. Subsequently, using this method, we derive analytical expressions for the first four origin moments as well as the general kth (k=1,2,…) moment of the negative binomial distribution. Finally, we present a table displaying polynomial coefficients for the first 10 origin moments and highlight that their leading coefficients correspond to Stirling numbers of the second kind.

Source: Communication in Statistics- Theory and Methods

DOI: 10.1080/03610926.2025.2461627

Author: Ying-Ying Zhang

Institution: Yunnan University

Language: English

Topic: Statistical Distribution Estimation and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Social Science Fund of China

Award: G8300451444