Consistency of Decision in Finite and Numerable Multinomial Models

The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model M (vertical bar n, p) whose decision is indexed by a parameter theta and having a cost c (theta, p) depending on q and on p, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when n tends to infinity, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator (p) over tilde (n) with components (p) over tilden, i = n(i)/n, i = 1, ... , where n(i) is the number of times we obtain the ith result when we have a sample of size n, is a consistent estimator of p. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model.

Automated summary

The multinomial distribution is commonly used for modeling categorical data. A study shows that, under certain conditions, the probability of choosing the least costly decision tends to 1 as the sample size approaches infinity, indicating consistency in decision making. The result also demonstrates that the estimator (p) over tilde (n), which is based on the observed frequencies, is a consistent estimator of the parameter p for both finite and countable models. This finding allows for a more general form of consistency for the cost function of a multinomial model.