The Annals of Mathematical Statistics

On the Normal Approximation to the Hypergeometric Distribution

W. L. Nicholson

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In this paper a new normal approximation to a sum of hypergeometric terms is derived, which is a direct generalization of Feller's normal approximation to the binomial distribution [2]. For intervals that are asymmetric with respect to the mean, or when the distribution is skewed, the new approximation is a marked improvement over the classical procedure. The hypergeometric distribution is discussed in Section 2, along with the classical norming and the resulting approximation. Feller's remarkable normal approximation for the related binomial distribution is given in Section 3 with an indication of how it can be extended to cover the hypergeometric case. The result of such an extension is presented in Theorem 2 of Section 4. This theorem gives upper and lower bounds on the hypergeometric sum and hence provides a useful estimate of the relative error. Preliminary results to proving Theorem 2 are exhibited in Section 5. The proof follows in Section 6.

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Ann. Math. Statist., Volume 27, Number 2 (1956), 471-483.

First available in Project Euclid: 28 April 2007

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Nicholson, W. L. On the Normal Approximation to the Hypergeometric Distribution. Ann. Math. Statist. 27 (1956), no. 2, 471--483. doi:10.1214/aoms/1177728270.

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