This morning, in our mathematical statistical class, we’ve seen briefly the multinomial distribution, and statistical inference. has a
distribution if its probability function is
with and
.
The maximum likelihood estimator is then the optimum of
We use Lagrange multiplier to solve this constrained optimization problem,
First order conditions are here
and
Thus,
From
we can easily get that Lagrande multiplier is . And then
One can easily get that this maximum likelihood estimator is unbiased, since . Actually, we can easily prove that
and that , while
. The trick to get the later is simple,
and . Thus, we can easily get the covariance. From that term, we can write that
with
while
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (November 3, 2015). Inference for the Multinomial Distribution. Freakonometrics. Retrieved April 30, 2025 from https://doi.org/10.58079/ov11