Negative multinomial distribution

Notation	${\textrm {NM}}(x_{0},\,\mathbf {p} )$
Parameters	$x_{0}>0$ — the number of failures before the experiment is stopped, $\mathbf {p}$ ∈ R^m — m-vector of "success" probabilities, p₀ = 1 − (p₁+…+p_m) — the probability of a "failure".
Support	$x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m$
PMF	$\Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {p_{0}^{x_{0}}}{\Gamma (x_{0})}}\prod _{i=1}^{m}{\frac {p_{i}^{x_{i}}}{x_{i}!}},$ where Γ(x) is the Gamma function.
Mean	${\tfrac {x_{0}}{p_{0}}}\,\mathbf {p}$
Variance	${\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {pp} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )$
MGF	${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{t_{j}}}}{\bigg )}^{\!x_{0}}$
CF	${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{it_{j}}}}{\bigg )}^{\!x_{0}}$

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x₀, p)) to more than two outcomes.^[1]

As with the univariate negative binomial distribution, if the parameter $x_{0}$ 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.

Properties

Marginal distributions

If m-dimensional x is partitioned as follows

\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

{\boldsymbol {p}}

{\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-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)}

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

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

Conditional distributions

The conditional distribution of $\mathbf {X} ^{(1)}$ given $\mathbf {X} ^{(2)}=\mathbf {x} ^{(2)}$ is ${\textstyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2)}},\mathbf {p} ^{(1)})}$ . That is,

\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

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

Aggregation

\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,

\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 $X_{i}$ mentioned above.

Correlation matrix

The entries of the correlation matrix are

\rho (X_{i},X_{i})=1.

\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

Method of Moments

If we let the mean vector of the negative multinomial be

{\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p}

and covariance matrix

{\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

{\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}}

. From this, it can be shown that

x_{0}={\frac {\sum {\mu _{i}}\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}}

and

\mathbf {p} ={\frac {|{\boldsymbol {\Sigma }}|-\prod {\mu _{i}}}{|{\boldsymbol {\Sigma }}|\sum {\mu _{i}}}}{\boldsymbol {\mu }}.

Substituting sample moments yields the method of moments estimates

{\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

{\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

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

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.

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda