Inference for the Multinomial Distribution

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


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