On least squares estimation for categorical data

We consider a multinomial distribution in which the cell probabilities are known arbitrary functions of a vector parameter θ. It is desired to estimate θ by least squares. Three variations of the least squares approach are investigated, and each is found to be equivalent, in the very strong sense of being algebraically identical, to one of the following estimation procedures: maximum likelihood, minimum χ² and minimum modified χ². Two of these results also apply to the multiple hypergeometric distribution.