Asymptotic Results for Multinomial Models

Akoto, Isaac Osei; Mexia, João T.; Marques, Filipe J.

doi:10.3390/sym13112173

Cited by 2 publications

(3 citation statements)

References 26 publications

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“…In this section, we obtain and consider estimators for the multinomial model. By limit distributions, see [8], we show that the estimators are consistent.…”

Section: Multinomial Models and Estimatorsmentioning

confidence: 64%

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Consistency of Decision in Finite and Numerable Multinomial Models

Akoto

Mexia

2023

Mathematics

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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 cθ,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=nin,i=1,⋯, where ni 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.

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“…In this section, we obtain and consider estimators for the multinomial model. By limit distributions, see [8], we show that the estimators are consistent.…”

Section: Multinomial Models and Estimatorsmentioning

confidence: 64%

“…vector of the indexes of the h largest components of v v v n ∈ X , we have see[10] dd d h ( p p p n ) s − −− → n→∞ d d d h (p p p)(31)if the h + 1 largest components of p p p are distinct.Proof.…”

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confidence: 99%

Consistency of Decision in Finite and Numerable Multinomial Models

Akoto

Mexia

2023

Mathematics

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“…This paper examined the issues of bias, variance, and sampling distribution properties and was concerned with two standard approaches for non-iteratively estimating the linearby-linear parameter of an ordinal log-linear model. We have done so assuming a Poisson sampling scheme, but it can also be adapted for a multinomial sampling scheme; see, for example, [62] for a study of asymptotic results for multinomial models. These approaches to estimation are easy to determine mathematically and so are easier to compute than their iterative counterparts (including Newton's method and iterative proportional fitting).…”

Section: Discussionmentioning

confidence: 99%

Asymptotic Characteristics of the Non-Iterative Estimates of the Linear-by-Linear Association Parameter for Ordinal Log-Linear Models

et al. 2022

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Over the past decade, a series of procedures has been introduced to estimate, using a non-iterative method, the linear-by-linear association parameter of an ordinal log-linear model. This paper will examine the two key non-iteratively determined estimates of the parameter for the analysis of the association between the two categorical variables that form a contingency table; these are the log and the Beh-Davy non-iterative estimates, referred to simply as the LogNI and the BDNI estimates, respectively. Such an examination will focus on determining their asymptotic characteristics. To do so, a computational study was undertaken for tables of varying sizes to show that these two estimates are asymptotically unbiased. It is also shown that both estimates are asymptotically normally distributed. On the basis of the standard errors, their relative efficiency was established for the 13 commonly analysed contingency tables that appear throughout the literature.

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Asymptotic Results for Multinomial Models

Cited by 2 publications

References 26 publications

Consistency of Decision in Finite and Numerable Multinomial Models

Consistency of Decision in Finite and Numerable Multinomial Models

Asymptotic Characteristics of the Non-Iterative Estimates of the Linear-by-Linear Association Parameter for Ordinal Log-Linear Models

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