To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # Negative multinomial distribution

Notation ${\textrm {NM}}(x_{0},\,p)$ x0 ∈ N0 — the number of failures before the experiment is stopped,p ∈ Rm — m-vector of "success" probabilities,p0 = 1 − (p1+…+pm) — the probability of a "failure". $x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m$ $\Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {p_{0}^{x_{0}}}{\Gamma (x_{0})}}\prod _{i=1}^{m}{\frac {p_{i}^{x_{i}}}{x_{i}!}},$ where Γ(x) is the Gamma function. ${\tfrac {x_{0}}{p_{0}}}\,p$ ${\tfrac {x_{0}}{p_{0}^{2}}}\,pp'+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (p)$ ${\bigg (}{\frac {p_{0}}{1-p'e^{it}}}{\bigg )}^{\!x_{0}}$ 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.

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, 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

$\mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}$ 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 the negative binomial distribution.

### 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

If

$\mathbf {X} =(X_{1},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{m}))$ 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

${\boldsymbol {\mu }}={\frac {x_{0}}{p_{0}}}\mathbf {p}$ ${\boldsymbol {\Sigma }}={\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {p} \mathbf {p} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )$ ,

then it is easy to show through properties of determinants that $|{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}$ . 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

Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.