Consistency of Decision in Finite and Numerable Multinomial Models

Isaac Akoto () and João T. Mexia
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Isaac Akoto: Center of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal
João T. Mexia: Center of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal

Mathematics, 2023, vol. 11, issue 11, 1-8

Abstract: 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 | n , p whose decision is indexed by a parameter θ and having a cost c θ , p depending on θ and on p , we show that, under general conditions, the probability of taking the least cost decision tends to 1 when n tends to ∞ , 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 ˜ n with components p ˜ n , i = n i n , i = 1 , ⋯ , where n i is the number of times we obtain the i th 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.

Keywords: stochastic convergence; decision theory; estimators (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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