Notation | ![{\displaystyle {\textrm {NM}}(x_{0},\,\mathbf {p} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac31c96882a88810db843edc37eec6a70b1bf69) |
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Parameters | — the number of failures before the experiment is stopped, ∈ Rm — m-vector of "success" probabilities,
p0 = 1 − (p1+…+pm) — the probability of a "failure". |
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Support | ![{\displaystyle x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/372a5e2da37049818ebe0181fcf08292be96a95d) |
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PMF | ![{\displaystyle \Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {p_{0}^{x_{0}}}{\Gamma (x_{0})}}\prod _{i=1}^{m}{\frac {p_{i}^{x_{i}}}{x_{i}!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248440158da2dcb451f517c5f1ded0b93372a8f9) where Γ(x) is the Gamma function. |
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Mean | ![{\displaystyle {\tfrac {x_{0}}{p_{0}}}\,\mathbf {p} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7036c2a202661f1d0b62f38e481e2bdd048a459a) |
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Variance | ![{\displaystyle {\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {pp} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1a763d990e069c4cfe88ca429418256478718a) |
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MGF | ![{\displaystyle {\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{t_{j}}}}{\bigg )}^{\!x_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a593dc1c0a4beb428ef39846138fc9c8212633cd) |
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CF | ![{\displaystyle {\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{it_{j}}}}{\bigg )}^{\!x_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bef192269adba47f541605b304032801cb3b332) |
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In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes.[1]
As with the univariate negative binomial distribution, if the parameter
is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value x0 (assuming x0 is a positive integer), then the distribution of the m-tuple {X1,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.
Properties
Marginal distributions
If m-dimensional x is partitioned as follows
![{\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d5750a5699ba6b7ae1a7f7bc46dba1480f3817)
and accordingly
![{\displaystyle {\boldsymbol {p}}={\begin{bmatrix}{\boldsymbol {p}}^{(1)}\\{\boldsymbol {p}}^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fd0745fb9ee7e258210eb0cefb3c897c16c9ac)
and let
![{\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf26813f72ba80b993bd83c5d9074a3f463b6938)
The marginal distribution of
is
. That is the marginal distribution is also negative multinomial with the
removed and the remaining p's properly scaled so as to add to one.
The univariate marginal
is said to have a negative binomial distribution.
Conditional distributions
The conditional distribution of
given
is
. That is,
![{\displaystyle \Pr(\mathbf {x} ^{(1)}\mid \mathbf {x} ^{(2)},x_{0},\mathbf {p} )=\Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {(1-\sum _{i=1}^{n}{p_{i}^{(1)}})^{x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)}}}{\Gamma (x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)})}}\prod _{i=1}^{n}{\frac {(p_{i}^{(1)})^{x_{i}}}{(x_{i}^{(1)})!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3527ac3c4a7646f363c165061da071b40810792)
Independent sums
If
and If
are independent, then
. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.
Aggregation
If
![{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{m}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19b9027dcd4ff1c7810d646e5f589800e0c77794)
then, if the random variables with subscripts
i and
j are dropped from the vector and replaced by their sum,
![{\displaystyle \mathbf {X} '=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{i}+p_{j},\ldots ,p_{m})).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c42f5b4d48a1f2270e601c2283f94115797f54)
This aggregation property may be used to derive the marginal distribution of
mentioned above.
Correlation matrix
The entries of the correlation matrix are
![{\displaystyle \rho (X_{i},X_{i})=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/effc4f57fb2573ab387032eee185a53fa089c2be)
![{\displaystyle \rho (X_{i},X_{j})={\frac {\operatorname {cov} (X_{i},X_{j})}{\sqrt {\operatorname {var} (X_{i})\operatorname {var} (X_{j})}}}={\sqrt {\frac {p_{i}p_{j}}{(p_{0}+p_{i})(p_{0}+p_{j})}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93269c0ab3dd332a3eab866d244e436c5833bfa8)
Parameter estimation
Method of Moments
If we let the mean vector of the negative multinomial be
![{\displaystyle {\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/277f1cadc8e9d6529c5a19062ef5e6a76214c81e)
and covariance matrix
![{\displaystyle {\boldsymbol {\Sigma }}={\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {p} \mathbf {p} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/052262a92e4e159efbd7ac03b9d0a391c58cfd68)
then it is easy to show through properties of determinants that
![{\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43380f721a48fcb649f4cc21ab32ab01c9a4f081)
. From this, it can be shown that
![{\displaystyle x_{0}={\frac {\sum {\mu _{i}}\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95541a5485d44714f4f9aef718afa093c79037b5)
and
![{\displaystyle \mathbf {p} ={\frac {|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|\sum {\mu _{i}}}}{\boldsymbol {\mu }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4539a234f6042ca6ac52c819a85650d7c1506e43)
Substituting sample moments yields the method of moments estimates
![{\displaystyle {\hat {x}}_{0}={\frac {(\sum _{i=1}^{m}{{\bar {x_{i}}})}\prod _{i=1}^{m}{\bar {x_{i}}}}{|\mathbf {S} |-\prod _{i=1}^{m}{\bar {x_{i}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bab4ee2ea85e0df4ac81bbcbc066bbc9db3d221)
and
![{\displaystyle {\hat {\mathbf {p} }}=\left({\frac {|{\boldsymbol {S}}|-\prod _{i=1}^{m}{{\bar {x}}_{i}}}{|{\boldsymbol {S}}|\sum _{i=1}^{m}{{\bar {x}}_{i}}}}\right){\boldsymbol {\bar {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/446c7f692458dc8fe6e88e22228df9a216e96389)
Related distributions
References
- ^ 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.
Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi- nomial distribution. Biometrics 53: 971–82.
Further reading
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1997). "Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions". Discrete Multivariate Distributions. Wiley. ISBN 978-0-471-12844-1.
Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and singular | |
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Families | |
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Category Commons |