Notation NM ( x 0 , p ) {\displaystyle {\textrm {NM}}(x_{0},\,\mathbf {p} )} Parameters x 0 > 0 {\displaystyle x_{0}>0} — the number of failures before the experiment is stopped, p {\displaystyle \mathbf {p} } ∈ R m — m -vector of "success" probabilities,p 0 = 1 − (p 1 +…+p m ) — the probability of a "failure". Support x i ∈ { 0 , 1 , 2 , … } , 1 ≤ i ≤ m {\displaystyle x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m} PMF Γ ( ∑ i = 0 m x i ) p 0 x 0 Γ ( x 0 ) ∏ i = 1 m p i x i x i ! , {\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}!}},} where Γ(x ) is the Gamma function . Mean x 0 p 0 p {\displaystyle {\tfrac {x_{0}}{p_{0}}}\,\mathbf {p} } Variance x 0 p 0 2 p p ′ + x 0 p 0 diag ( p ) {\displaystyle {\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {pp} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )} MGF ( p 0 1 − ∑ j = 1 m p j e t j ) x 0 {\displaystyle {\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{t_{j}}}}{\bigg )}^{\!x_{0}}} CF ( p 0 1 − ∑ j = 1 m p j e i t j ) x 0 {\displaystyle {\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{it_{j}}}}{\bigg )}^{\!x_{0}}}
In probability theory and statistics , the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x 0 , p )) to more than two outcomes.[1]
As with the univariate negative binomial distribution, if the parameter 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, {X 0 ,...,X m }, each occurring with non-negative probabilities {p 0 ,...,p m } respectively. If sampling proceeded until n observations were made, then {X 0 ,...,X m } would have been multinomially distributed . However, if the experiment is stopped once X 0 reaches the predetermined value x 0 (assuming x 0 is a positive integer), then the distribution of the m -tuple {X 1 ,...,X m } 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 .
Properties [ edit ] Marginal distributions [ edit ] If m -dimensional x is partitioned as follows
X = [ X ( 1 ) X ( 2 ) ] with sizes [ n × 1 ( m − n ) × 1 ] {\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
p {\displaystyle {\boldsymbol {p}}} p = [ p ( 1 ) p ( 2 ) ] with sizes [ n × 1 ( m − n ) × 1 ] {\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
q = 1 − ∑ i p i ( 2 ) = p 0 + ∑ i p i ( 1 ) {\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}} The marginal distribution of X ( 1 ) {\displaystyle {\boldsymbol {X}}^{(1)}} is N M ( x 0 , p 0 / q , 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 p ( 2 ) {\displaystyle {\boldsymbol {p}}^{(2)}} removed and the remaining p' s properly scaled so as to add to one.
The univariate marginal m = 1 {\displaystyle m=1} is said to have a negative binomial distribution.
Conditional distributions [ edit ] The conditional distribution of X ( 1 ) {\displaystyle \mathbf {X} ^{(1)}} given X ( 2 ) = x ( 2 ) {\displaystyle \mathbf {X} ^{(2)}=\mathbf {x} ^{(2)}} is N M ( x 0 + ∑ x i ( 2 ) , p ( 1 ) ) {\textstyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2)}},\mathbf {p} ^{(1)})} . That is,
Pr ( x ( 1 ) ∣ x ( 2 ) , x 0 , p ) = Γ ( ∑ i = 0 m x i ) ( 1 − ∑ i = 1 n p i ( 1 ) ) x 0 + ∑ i = 1 m − n x i ( 2 ) Γ ( x 0 + ∑ i = 1 m − n x i ( 2 ) ) ∏ i = 1 n ( 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)})!}}.} Independent sums [ edit ] If X 1 ∼ N M ( r 1 , p ) {\displaystyle \mathbf {X} _{1}\sim \mathrm {NM} (r_{1},\mathbf {p} )} and If X 2 ∼ N M ( r 2 , p ) {\displaystyle \mathbf {X} _{2}\sim \mathrm {NM} (r_{2},\mathbf {p} )} are independent , then X 1 + X 2 ∼ N M ( r 1 + r 2 , 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 .
Aggregation [ edit ] If
X = ( X 1 , … , X m ) ∼ NM ( x 0 , ( p 1 , … , 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,
X ′ = ( X 1 , … , X i + X j , … , X m ) ∼ NM ( x 0 , ( p 1 , … , p i + p j , … , 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 X i {\displaystyle X_{i}} mentioned above.
Correlation matrix [ edit ] The entries of the correlation matrix are
ρ ( X i , X i ) = 1. {\displaystyle \rho (X_{i},X_{i})=1.} ρ ( X i , X j ) = cov ( X i , X j ) var ( X i ) var ( X j ) = 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})}}}.} Parameter estimation [ edit ] Method of Moments [ edit ] If we let the mean vector of the negative multinomial be
μ = x 0 p 0 p {\displaystyle {\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p} } and
covariance matrix Σ = x 0 p 0 2 p p ′ + x 0 p 0 diag ( p ) , {\displaystyle {\boldsymbol {\Sigma }}={\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {p} \mathbf {p} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} ),} then it is easy to show through properties of
determinants that
| Σ | = 1 p 0 ∏ i = 1 m μ i {\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}} . From this, it can be shown that
x 0 = ∑ μ i ∏ μ i | Σ | − ∏ μ i {\displaystyle x_{0}={\frac {\sum {\mu _{i}}\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}}} and
p = | Σ | − ∏ μ i | Σ | ∑ μ i μ . {\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
x ^ 0 = ( ∑ i = 1 m x i ¯ ) ∏ i = 1 m x i ¯ | S | − ∏ i = 1 m 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
p ^ = ( | S | − ∏ i = 1 m x ¯ i | S | ∑ i = 1 m x ¯ i ) 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}}}} Related distributions [ edit ] References [ edit ] ^ 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 [ edit ] 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 with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families