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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 Negative multinomial distribution ： ウィキペディア英語版 Negative multinomial distribution \prod_^m}, where Γ(''x'') is the Gamma function. | cdf = | mean = $\backslash tfrac\backslash ,p$ | variance = $\backslash tfrac\backslash ,pp\text{'}\; +\; \backslash tfrac\backslash ,\backslash operatorname(p)$ | mode = | entropy = | mgf = | cf = $\backslash bigg(\backslash frac$ }} In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(''r'', ''p'')) to more than two outcomes.〔Le Gall, F. The modes of a negative multinomial distribution, Statistics & Probability Letters, Volume 76, Issue 6, 15 March 2006, Pages 619-624, ISSN 0167-7152, (10.1016/j.spl.2005.09.009 ).〕 Suppose we have an experiment that generates ''m''+1≥2 possible outcomes, , each occurring with non-negative probabilities respectively. If sampling proceeded until ''n'' observations were made, then would have been multinomially distributed. However, if the experiment is stopped once ''X''0 reaches the predetermined value ''k''0, then the distribution of the ''m''-tuple is ''negative multinomial''. These variables are not multinomially distributed because their sum ''X''1+…+''X''''m'' is not fixed, being a draw from a negative binomial distribution. ==Negative multinomial distribution example== The table below shows the an example of 400 Melanoma (skin cancer) Patients where the Type and Site of the cancer are recorded for each subject. The sites (locations) of the cancer may be independent, but there may be positive dependencies of the type of cancer for a given location (site). For example, localized exposure to radiation implies that elevated level of one type of cancer (at a given location) may indicate higher level of another cancer type at the same location. The Negative Multinomial distribution may be used to model the sites cancer rates and help measure some of the cancer type dependencies within each location. If $x\_$ denote the cancer rates for each site ($0\backslash leq\; i\; \backslash leq\; 2$) and each type of cancer ($0\backslash leq\; j\; \backslash leq\; 3$), for a fixed site ($i\_0$) the cancer rates are independent Negative Multinomial distributed random variables. That is, for each column index (site) the column-vector X has the following distribution: : $X=\backslash \; \backslash sim\; NM(k\_0,\backslash )$. Different columns in the table (sites) are considered to be different instances of the random multinomially distributed vector, X. Then we have the following estimates of expected counts (frequencies of cancer): : $\backslash hat\_\; =\; \backslash frac\}\; =\; \backslash sum\_^\; =\; \backslash sum\_^\; =\; \backslash sum\_^\backslash sum\_^\}$ : Example: $\backslash hat\_\; =\; \backslash frac\}=5.78$ For the first site (Head and Neck, j=0), suppose that $X=\backslash left\; \backslash$ and $X\; \backslash sim\; NM(k\_0=10,\; \backslash )$. Then: : $p\_0\; =\; 1\; -\; \backslash sum\_^3=0.5$ : $NM(X|k\_0,\backslash )=\; 0.00465585119998784$ : $cov()\; =\; \backslash frac=1.6$ : $\backslash mu\_2=\backslash frac\; =\; \backslash frac=2.0$ : $\backslash mu\_3=\backslash frac\; =\; \backslash frac=4.0$ : $corr()\; =\; \backslash left\; (\backslash frac\; \backslash right\; )^\}$ and therefore, $corr()\; =\; \backslash left\; (\backslash frac\; \backslash right\; )^\}\; =\; 0.21821789023599242.$ Notice that the pair-wise NM correlations are always positive, whereas the correlations between multinomial counts are always negative. As the parameter $k\_0$ increases, the paired correlations tend to zero! Thus, for large $k\_0$, the Negative Multinomial counts $X\_i$ behave as ''independent'' Poisson random variables with respect to their means $\backslash left\; (\; \backslash mu\_i=\; k\_0\backslash frac\backslash right\; )$. The marginal distribution of each of the $X\_i$ variables is negative binomial, as the $X\_i$ count (considered as success) is measured against all the other outcomes (failure). But jointly, the distribution of $X=\backslash$ is negative multinomial, i.e., $X\; \backslash sim\; NM(k\_0,\backslash )$ . 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「Negative multinomial 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.