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Many distributions are known to be log-concave: Poisson, Negative Binomial,...

However, we are often interested the likelihood function (the density as a function of the parameters, not as a function of the data). Hence my question:

Does (or in what cases) log-concavity of the distribution implies log-concavity of the likelihood in the paramater space?

I'm especially interested in Negative Multinomials, parametrized as:

$$ \frac{\Gamma(\sum_k k_i)}{\Gamma(k_0)\prod_ik_i!} (1-\sum_i p_i)^{k_0}p_i^{k_{i}} $$

where $k_0$ is known and fixed. My parameters are the probabilities $p_1,...,p_m$

Any reference would be also appreciated.

alberto
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    What would you mean by "parameter space"? A distribution is simply a distribution, whereas "parameter space" seems to refer to parameterized sets of distributions. The answer to your question depends not on the distribution but on that entire set in which it is embedded. – whuber Mar 31 '17 at 15:48
  • Yes, I'm interested on parameterized distributions. I mean, for instance, the $\lambda$ parameter of a Poisson. I'm studying the MLE of the sum of a product of Negative Multinomials. – alberto Mar 31 '17 at 15:59
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    My point is that the answer depends on the set of distributions you are considering. Even worse, it depends on exactly how you parameterize them. – whuber Mar 31 '17 at 16:01
  • @whuber oh, thanks, I edited the question with my parameterization. – alberto Mar 31 '17 at 16:19

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