This paper deals with the estimation of the parameters (cell probabilities) of a multinomial distribution. For the binomial distribution it is known that the maximum likelihood estimator (MLE) is admissible with respect to the squared error loss. It is shown that the MLE is admissible also in the case of the multinomial distribution.
"Estimation of Multinomial Probabilities." Ann. Statist. 7 (2) 282 - 283, March, 1979. https://doi.org/10.1214/aos/1176344612