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 () 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.
Highlights
It is easy to see that our analytical formulas of the first four origin moments of the hypergeometric distribution by using the expectation identity (2) are the same as those obtained in Li and Tian (2010), in which they use the definition of expectation for the calculations
We will provide an assessment of the computation complexity (see Chapter 5 of Progri (2011)) which is given in number of multiplications of the Definition Method (DM) (1) and the Analytical Method (AM) (16)
We have provided a noval method to calculate the high-order origin moments of the hypergeometric distribution Y ∼ HG (N, M, n), that is, the expectation identity method
Summary
The hypergeometric distribution has attracted continuous interest in the literature since 2000. Childs and Balakrishnan (2000) examined some approximations to the multivariate hypergeometric distribution with applications to hypothesis testing. Kumar (2002) introduced extended generalized hypergeometric probability distributions. Hida and Akahira (2003) considered an improvement on the approximation to the generalized hypergeometric distribution. Dinwoodie et al (2004) presented two new transform methods for computing with hypergeometric distributions on lattice points. Hush and Scovel (2005) provided an improved concentration of measure theorem for the hypergeometric distribution. Kumar (2007) studied some properties of bivariate generalized hypergeometric probability distributions. Lahiri et al (2007) investigated normal approximation to the hypergeometric distribution in nonstandard cases and established a sub-Gaussian Berry-Esseen theorem. Fog (2008) developed sampling methods for Wallenius’ and Fisher’s noncentral hypergeometric distributions. Li and Tian (2010) provided a simple algorithm for the high-order origin moments of the hypergeometric distribution. Eisinga and Pelzer (2011) proposed saddlepoint approximations to the mean and variance of the extended hypergeometric distribution. Lebrun (2013) proposed an efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Plya distributions. Liu and Wang (2014) obtained the high-order origin moments, the high-order central moments, and the high-order cumulants of the hypergeometric distribution. de Klerk et al (2015) considered an error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. Danielian et al (2016) proposed a new regularly varying generalized hypergeometric distribution of the second type. Greene and Wellner (2017) established exponential bounds for the hypergeometric distribution. Mano (2017) discussed partition structure and the A-hypergeometric distribution associated with the rational normal curve. Takayama et al (2018) considered A-hypergeometric distributions and Newton polytopes. Hafid et al (2020) proposed a novel methodology-based joint hypergeometric distribution to analyze the security of sharded blockchains. Krishnamoorthy and Lv (2020) constructed prediction intervals for hypergeometric distributions. Themangani et al (2020) introduced a generalized hypergeometric distribution and discussed its applications on univalent functions. Li and Tian (2010) provided a simple algorithm for the high-order origin moments of the hypergeometric distribution. The analytical calculations of the high-order origin moments of the hypergeometric distribution are quite challenging. We provide a novel method to analytically calculate the high-order origin moments of the hypergeometric distribution, that is, the expectation identity method. . .) origin moment of the hypergeometric distribution by the expectation identity method. We will use the expectation identity of the hypergeometric distribution to calculate its first four origin moments. We will use the expectation identity of the hypergeometric distribution to calculate its general kth
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