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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hermite distribution ： ウィキペディア英語版 Hermite distribution | kurtosis = $\backslash frac$ | pgf = $\backslash exp(a\_1(s-1)+a\_2(s^2-1))\backslash ,$ | mgf = $\backslash exp(a\_1(e^t-1)+a\_2(e^-1))\backslash ,$ | char = $\backslash exp(a\_1(e^-1)+a\_2(e^-1))\backslash ,$ }} In probability theory and statistics, the Hermite distribution, named after Charles Hermite, is a discrete probability distribution used to model ''count data'' with more than one parameter. This distribution is flexible in terms of its ability to allow a moderate over-dispersion in the data. The Hermite distribution is a special case of the Poisson binomial distribution, when ''n'' = 2. The authors Kemp and Kemp have called it "Hermite distribution" from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of (modified) Hermite polynomials. == History == The distribution first appeared in the paper ''Applications of Mathematics to Medical Problems'', by Anderson Gray McKendrick in 1926. In this work the author explains several mathematical methods that can be applied to medical research. In one of this methods he considered the bivariate Poisson distribution and showed that the distribution of the sum of two correlated Poisson variables follow a distribution that later would be known as Hermite distribution. As a practical application, McKendrick considered the distribution of counts of bacteria in leucocytes. Using the method of moments he fitted the data with the Hermite distribution and found the model more satisfactory than fitting it with a Poisson distribution. The distribution was formally introduced and published by C. D. Kemp and Adrienne W.Kemp in 1965 in their work ''Some Properties of ‘Hermite’ Distribution''. The work is focused on the properties of this distribution for instance a necessary condition on the parameters and their Maximum Likelihood (MLE), the analysis of the probability generating function (PGF) and how it can be expressed in terms of the coefficients of (modified) Hermite polynomials. An example they have used in this publication is the distribution of counts of bacteria in leucocytes that used McKendrick but Kemp and Kemp estimate the model using the maximum likelihood method. Hermite distribution is a special case of discrete compound Poisson distribution with only 2 parameters. 〔Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, ISBN 978-0-471-27246-5.〕 The same authors published in 1966 the paper ''An alternative Derivation of the Hermite Distribution''. In this work established that the Hermite distribution can be obtained formally by combining a Poisson distribution with a Normal distribution. In 1971, Y. C. Patel did a comparative study of various estimation procedures for the Hermite distribution in his doctoral thesis. It included maximum likelihood, moment estimators, mean and zero frequency estimators and the method of even points. In 1974, Gupta and Jain did a research on a generalized form of Hermite distribution. In the probabilistic number theory, due to Bekelis's work, when a strongly additive function $(m),(x\; \backslash ge\; 1)$ only takes value on prime number ''p'', under some conditions, then the frequent number of $(m)$ convergent to a Hermite distribution for $x\; \backslash to\; \backslash infty$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hermite distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.