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I am trying to compute the variance of the multiplicative multinomial distribution. I know that it can be computed using the MOMENT DEFINITION but I don't understand which kind of technique this is. I have never heard about it.

In particular the multiplicative multinomial has a density function like $$P(\textbf{Y} = \textbf{y}) = \binom{m}{y_1 ... y_k}p_1^{y_1} \dots p_k^{y_k} \prod_{l=1}^{k}\prod_{j=1}^{k}\nu_{l_j}^{y_l y_j}$$ where in particular $m=\sum_{i_1}^{k} y_i$, while $p_1...p_k$ is the vector of probabilities associated to each one of the k categories and $\nu_{l_j}$ for $j,l \in {1,...,k}$ and $j<l$.

Can somebody help me? Also just defining what moment definition means. thanks in advance!

Bibi
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  • At https://stats.stackexchange.com/a/155019/919 I define the multivariate moment. The rest is just algebra. – whuber Apr 19 '21 at 12:02
  • Sorry but it is not very clear what you wrote at that link. Can you just establish which is the starting point in my example? Then I can keep going with all the algebra... – Bibi Apr 23 '21 at 08:49

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