Consistency of Decision in Finite and Numerable Multinomial Models

Publisher Copyright: © 2023 by the authors. This research received no external funding.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 (Formula presented.) whose decision is indexed by a parameter (Formula presented.) and having a cost (Formula presented.) depending on (Formula presented.) and on (Formula presented.), 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 (Formula presented.) with components (Formula presented.), where (Formula presented.) is the number of times we obtain the ith result when we have a sample of size n, is a consistent estimator of (Formula presented.). 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.publishersversionpublishe

Inference for types and structured families of commutative orthogonal block structures

Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.info:eu-repo/semantics/publishedVersio

Estimation in mixed models through three step minimization

The aim of this article is to present an estimation procedure for both fixed effects and variance components in linear mixed models. This procedure consists of a maximum likelihood method which we call Three Step Minimization, TSM. The major contribution of this method is that when variances tend to be null standard algorithms behave badly, unlike the TSM method, which uses a grid search algorithm in a compact set. A numerical application with real and simulated data is provided.info:eu-repo/semantics/publishedVersio

Preliminary results on confidence intervals for open bonus malus

This work was partially supported by Financiamento Base 2009 ISFL-1-297 from FCT/MCTES/PT.Considering open portfolios, we analyze bonus–malus systems (BMS) under a realistic approach, as we already did in Guerreiro and Mexia (Discuss. Math. Probab. Stat. 24(2):197–213, 2004). Using stochastic vortices model we are now able to predict long-run distribution through confidence intervals.authorsversionpublishe

Limit distributions for asymptotically linear statistics with spherical error

The aim of this work is to obtain general results for the limit distributions of asymptotically linear statistics when the error is spherical, increasing non-centrality. These results apply directly to homoscedastic normal error thus to high precision measurements. We present a numerical example on cylinder volume to illustrate the usefulness of our approach.info:eu-repo/semantics/publishedVersio

Randomized sample size F tests for the one-way layout

Distributions and densities for F test statistics are obtained assuming random sample sizes, thus getting random degrees of freedom and non-centrality parameters. Classical optimum properties are extended to this new setup as well as Scheffé Theorem for simultaneous confidence intervals.publishersversionpublishe

Tests and relevancies for the hypotheses of an orthogonal family in a model with orthogonal block structure

A model has an orthogonal block structure if it has, as covariance matrix, a linear combination of pairwise orthogonal projection matrices, that add up to the iden- tity matrix. The range space of these matrices are associated to hypotheses of an orthogonal family. In this paper we show how to obtain tests for these hypotheses when normality is assumed and how to consider their relevance when normality is discarded. Besides the notion of relevance, we formulate hypotheses in a general way that may be applied to models with orthogonal block structure, whose factors may have xed and/or random e ects. The results are applied to prime basis factorial models and an example is presented.info:eu-repo/semantics/publishedVersio

One-way random effects ANOVA with random sample sizes: An application to a Brazilian database on cancer registries

ANOVA is routinely used in many situations, namely in medical research, where the sample sizes may not be previously known. This leads us to consider the samples sizes as realizations of random variables. The aim of this paper is to extend one-way random effects ANOVA to those situations and apply our results to a Brazilian database on cancer registries.info:eu-repo/semantics/publishedVersio

Estimation in additive models and ANOVA-like applications

A well-known property of cumulant generating function is used to estimate the first four order cumulants, using least-squares estimators. In the case of additive models, empirical best linear unbiased predictors are also obtained. Pairs of independent and identically distributed models associated with the treatments of a base design are used to obtain unbiased estimators for the fourth-order cumulants. An application to real data is presented, showing the good behaviour of the least-squares estimators and the great flexibility of our approach.info:eu-repo/semantics/publishedVersio

One-way fixed effects ANOVA with missing observations

The aim of this paper is to extend the theory of F-tests with random sample sizes to situations when missing observations may occur. We consider the one-way ANOVA with fixed effects. This approach is illustrated through an application to patients affected by melanoma skin cancer, from three different states of Brazil.info:eu-repo/semantics/publishedVersio