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

Yuan-Quan Wang, Ying-Ying Zhang and Jia-Lei Liu

Communications in Statistics - Theory and Methods, 2023, vol. 52, issue 17, 6018-6036

Abstract: We provide a novel method to analytically calculate the high-order origin moments of a hypergeometric distribution, that is, the expectation identity method. First, the expectation identity of the hypergeometric distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments of the hypergeometric distribution by using the expectation identity. Furthermore, we analytically calculate the general kth (k=1,2,…) origin moment of the hypergeometric distribution by using the expectation identity, and the results are summarized in a theorem. Moreover, we use the general kth origin moment to validate the first four origin moments of the hypergeometric distribution. Next, the coefficients of the first ten origin moments of the hypergeometric distribution are summarized in a table containing Stirling numbers of the second kind. Moreover, the general kth origin moment of the hypergeometric distribution by using the expectation identity is restated by another theorem involving Stirling numbers of the second kind. Finally, we provide some numerical and theoretical results.

Date: 2023
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DOI: 10.1080/03610926.2021.2024235

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