Negative multinomial distribution

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 ${\displaystyle x_{0}}${\displaystyle x_{0}} 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,...,X_m}, each occurring with non-negative probabilities {p₀,...,p_m} respectively. If sampling proceeded until n observations were made, then {X₀,...,X_m} 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,...,X_m} is negative multinomial. These variables are not multinomially distributed because their sum X₁+...+X_m is not fixed, being a draw from a negative binomial distribution.

If m-dimensional x is partitioned as follows $${\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}$${\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}}} and accordingly ${\displaystyle {\boldsymbol {p}}}${\displaystyle {\boldsymbol {p}}} $${\displaystyle {\boldsymbol {p}}={\begin{bmatrix}{\boldsymbol {p}}^{(1)}\\{\boldsymbol {p}}^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}}$${\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}}} and let $${\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}}$${\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}}

The marginal distribution of ${\displaystyle {\boldsymbol {X}}^{(1)}}${\displaystyle {\boldsymbol {X}}^{(1)}} is ${\displaystyle \mathrm {NM} (x_{0},p_{0}/q,{\boldsymbol {p}}^{(1)}/q)}${\displaystyle \mathrm {NM} (x_{0},p_{0}/q,{\boldsymbol {p}}^{(1)}/q)}. That is the marginal distribution is also negative multinomial with the ${\displaystyle {\boldsymbol {p}}^{(2)}}${\displaystyle {\boldsymbol {p}}^{(2)}} removed and the remaining p's properly scaled so as to add to one.

The univariate marginal ${\displaystyle m=1}${\displaystyle m=1} is said to have a negative binomial distribution.

The conditional distribution of ${\displaystyle \mathbf {X} ^{(1)}}${\displaystyle \mathbf {X} ^{(1)}} given ${\displaystyle \mathbf {X} ^{(2)}=\mathbf {x} ^{(2)}}${\displaystyle \mathbf {X} ^{(2)}=\mathbf {x} ^{(2)}} is ${\textstyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2)}},\mathbf {p} ^{(1)})}${\textstyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2)}},\mathbf {p} ^{(1)})}. That is, $${\displaystyle \Pr(\mathbf {x} ^{(1)}\mid \mathbf {x} ^{(2)},x_{0},\mathbf {p} )=\Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {(1-\sum _{i=1}^{n}{p_{i}^{(1)}})^{x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)}}}{\Gamma (x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)})}}\prod _{i=1}^{n}{\frac {(p_{i}^{(1)})^{x_{i}}}{(x_{i}^{(1)})!}}.}$${\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)})!}}.}

If ${\displaystyle \mathbf {X} _{1}\sim \mathrm {NM} (r_{1},\mathbf {p} )}${\displaystyle \mathbf {X} _{1}\sim \mathrm {NM} (r_{1},\mathbf {p} )} and If ${\displaystyle \mathbf {X} _{2}\sim \mathrm {NM} (r_{2},\mathbf {p} )}${\displaystyle \mathbf {X} _{2}\sim \mathrm {NM} (r_{2},\mathbf {p} )} are independent, then ${\displaystyle \mathbf {X} _{1}+\mathbf {X} _{2}\sim \mathrm {NM} (r_{1}+r_{2},\mathbf {p} )}${\displaystyle \mathbf {X} _{1}+\mathbf {X} _{2}\sim \mathrm {NM} (r_{1}+r_{2},\mathbf {p} )}. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.

If $${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{m}))}$${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{m}))} then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum, $${\displaystyle \mathbf {X} '=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{i}+p_{j},\ldots ,p_{m})).}$${\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})).}

This aggregation property may be used to derive the marginal distribution of ${\displaystyle X_{i}}${\displaystyle X_{i}} mentioned above.

The entries of the correlation matrix are $${\displaystyle \rho (X_{i},X_{i})=1.}$${\displaystyle \rho (X_{i},X_{i})=1.} $${\displaystyle \rho (X_{i},X_{j})={\frac {\operatorname {cov} (X_{i},X_{j})}{\sqrt {\operatorname {var} (X_{i})\operatorname {var} (X_{j})}}}={\sqrt {\frac {p_{i}p_{j}}{(p_{0}+p_{i})(p_{0}+p_{j})}}}.}$${\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})}}}.}

If we let the mean vector of the negative multinomial be $${\displaystyle {\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p} }$${\displaystyle {\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p} } and covariance matrix $${\displaystyle {\boldsymbol {\Sigma }}={\tfrac {x_{0}}{p_{0}^{2}}},円\mathbf {p} \mathbf {p} '+{\tfrac {x_{0}}{p_{0}}},円\operatorname {diag} (\mathbf {p} ),}$${\displaystyle {\boldsymbol {\Sigma }}={\tfrac {x_{0}}{p_{0}^{2}}},円\mathbf {p} \mathbf {p} '+{\tfrac {x_{0}}{p_{0}}},円\operatorname {diag} (\mathbf {p} ),} then it is easy to show through properties of determinants that ${\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}}${\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}}. From this, it can be shown that $${\displaystyle x_{0}={\frac {\sum {\mu _{i}}\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}}}$${\displaystyle x_{0}={\frac {\sum {\mu _{i}}\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}}} and $${\displaystyle \mathbf {p} ={\frac {|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|\sum {\mu _{i}}}}{\boldsymbol {\mu }}.}$${\displaystyle \mathbf {p} ={\frac {|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|\sum {\mu _{i}}}}{\boldsymbol {\mu }}.}

Substituting sample moments yields the method of moments estimates $${\displaystyle {\hat {x}}_{0}={\frac {(\sum _{i=1}^{m}{{\bar {x_{i}}})}\prod _{i=1}^{m}{\bar {x_{i}}}}{|\mathbf {S} |-\prod _{i=1}^{m}{\bar {x_{i}}}}}}$${\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}}}}}} and $${\displaystyle {\hat {\mathbf {p} }}=\left({\frac {|{\boldsymbol {S}}|-\prod _{i=1}^{m}{{\bar {x}}_{i}}}{|{\boldsymbol {S}}|\sum _{i=1}^{m}{{\bar {x}}_{i}}}}\right){\boldsymbol {\bar {x}}}}$${\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}}}}

• Negative binomial distribution
• Multinomial distribution
• Inverted Dirichlet distribution, a conjugate prior for the negative multinomial
• Dirichlet negative multinomial distribution

Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi- nomial distribution. Biometrics 53: 971–82.

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.

• Factorial and binomial topics
• Multivariate discrete distributions

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