Notation NM
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{displaystyle { extrm {NM}}(k_{0},,p)} Parameters k0 ∈ N0 — the number of failures before the experiment is stopped,
p ∈ R — m-vector of “success” probabilities,
p0 = 1 − (p1+…+pm) — the probability of a “failure”. Support k
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{displaystyle k_{i}in {0,1,2,ldots },1leq ileq m} PDF Γ
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{displaystyle Gamma !left(sum _{i=0}^{m}{k_{i}}
ight){ rac {p_{0}^{k_{0}}}{Gamma (k_{0})}}prod _{i=1}^{m}{ rac {p_{i}^{k_{i}}}{k_{i}!}},}
where Γ(x) is the Gamma function. Mean k
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{displaystyle { frac {k_{0}}{p_{0}}},p} Variance k
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{displaystyle { frac {k_{0}}{p_{0}^{2}}},pp'+{ frac {k_{0}}{p_{0}}},operatorname {diag} (p)} | ||
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.
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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 k0, 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.
Negative multinomial distribution example
The table below shows an example of 400 melanoma (skin cancer) patients where the Type and Site of the cancer are recorded for each subject.
The sites (locations) of the cancer may be independent, but there may be positive dependencies of the type of cancer for a given location (site). For example, localized exposure to radiation implies that elevated level of one type of cancer (at a given location) may indicate higher level of another cancer type at the same location. The Negative Multinomial distribution may be used to model the cancer rates at a given site and help measure some of the cancer type dependencies within each location.
If
Different columns in the table (sites) are considered to be different instances of the random multinomially distributed vector, X. Then we have the following estimates of expected counts (frequencies of cancer):
For the first site (Head and Neck, j=0), suppose that
Notice that the pair-wise NM correlations are always positive, whereas the correlations between multinomial counts are always negative. As the parameter
The marginal distribution of each of the