# Efficiency of maximum likelihood estimation for a multinomial distribution with known probability sums

## TL;DR

This paper analyzes how knowing the sum of certain categories' probabilities in a multinomial distribution affects the efficiency of maximum likelihood estimation, revealing limited benefits and potential risks with small samples.

## Contribution

It provides an asymptotic expansion of the MLE risk in both full and submodels, highlighting the limited improvement and possible increased risk with small samples.

## Key findings

- Risk reduction is small in some cases when using the submodel.
- Using the submodel can increase risk with small sample sizes.
- Asymptotic expansion of MLE risk for both models is derived.

## Abstract

For a multinomial distribution, suppose that we have prior knowledge of the sum of the probabilities of some categories. This allows us to construct a submodel in a full (i.e., no-restriction) model. Maximum likelihood estimation (MLE) under this submodel is expected to have better estimation efficiency than MLE under the full model. This article presents the asymptotic expansion of the risk of MLE with respect to Kullback--Leibler divergence for both the full model and submodel. The results reveal that, using the submodel, the reduction of the risk is quite small in some cases. Furthermore, when the sample size is small, the use of the subomodel can increase the risk.