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Stochastic vortices in periodically reclassified populations

Gracinda Rita GuerreiroJoão Tiago Mexia — 2008

Discussiones Mathematicae Probability and Statistics

Our paper considers open populations with arrivals and departures whose elements are subject to periodic reclassifications. These populations will be divided into a finite number of sub-populations. Assuming that: a) entries, reclassifications and departures occur at the beginning of the time units; b) elements are reallocated at equally spaced times; c) numbers of new elements entering at the beginning of the time units are realizations...

Non-central generalized F distributions

Célia NunesJoão Tiago Mexia — 2006

Discussiones Mathematicae Probability and Statistics

The quotient of two linear combinations of independent chi-squares will have a generalized F distribution. Exact expressions for these distributions when the chi-square are central and those in the numerator or in the denominator have even degrees of freedom were given in Fonseca et al. (2002). These expressions are now extended for non-central chi-squares. The case of random non-centrality parameters is also considered.

Compact hypothesis and extremal set estimators

João Tiago MexiaPedro Corte Real — 2003

Discussiones Mathematicae Probability and Statistics

In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when the true...

An alternative approach to bonus malus

Gracinda Rita GuerreiroJoão Tiago Mexia — 2004

Discussiones Mathematicae Probability and Statistics

Under the assumptions of an open portfolio, i.e., considering that a policyholder can transfer his policy to another insurance company and the continuous arrival of new policyholders into a portfolio which can be placed into any of the bonus classes and not only in the "starting class", we developed a model (Stochastic Vortices Model) to estimate the Long Run Distribution for a Bonus Malus System. These hypothesis render the model quite representative of the reality. With the obtained Long Run Distribution,...

F-tests for generalized linear hypotheses in subnormal models

Joao Tiago MexiaGerberto Carvalho Dias — 2001

Discussiones Mathematicae Probability and Statistics

When the measurement errors may be assumed to be normal and independent from what is measured a subnormal model may be used. We define a linear and generalized linear hypotheses for these models, and derive F-tests for them. These tests are shown to be UMP for linear hypotheses as well as strictly unbiased and strongly consistent for these hypotheses. It is also shown that the F-tests are invariant for regular transformations, possess structural stability and are almost strongly consistent for generalized...

Strong law of large numbers for additive extremum estimators

João Tiago MexiaPedro Corte Real — 2001

Discussiones Mathematicae Probability and Statistics

Extremum estimators are obtained by maximizing or minimizing a function of the sample and of the parameters relatively to the parameters. When the function to maximize or minimize is the sum of subfunctions each depending on one observation, the extremum estimators are additive. Maximum likelihood estimators are extremum additive whenever the observations are independent. Another instance of additive extremum estimators are the least squares estimators for multiple regressions when the usual assumptions...

Likelihood and parametric heteroscedasticity in normal connected linear models

Joao Tiago MexiaPedro Corte Real — 2000

Discussiones Mathematicae Probability and Statistics

A linear model in which the mean vector and covariance matrix depend on the same parameters is connected. Limit results for these models are presented. The characteristic function of the gradient of the score is obtained for normal connected models, thus, enabling the study of maximum likelihood estimators. A special case with diagonal covariance matrix is studied.

Joint estimation for normal orthogonal mixed models

Vera de JesusSandra Saraiva FerreiraJoão Tiago Mexia — 2007

Discussiones Mathematicae Probability and Statistics

Commutative Jordan algebras are used to express the structure of mixed orthogonal models and to derive complete sufficient statistics. From these statistics, UMVUE, (Uniformly Minimum Variance Unbiased Estimators), are derived for the relevant parameters, first of single models then of several such models. These models may correspond to experiments designed separately so our results may be seen as a contribution to this meta-analysis.

Algebraic structure for the crossing of balanced and stair nested designs

Célia FernandesPaulo RamosJoão Tiago Mexia — 2014

Discussiones Mathematicae Probability and Statistics

Stair nesting allows us to work with fewer observations than the most usual form of nesting, the balanced nesting. In the case of stair nesting the amount of information for the different factors is more evenly distributed. This new design leads to greater economy, because we can work with fewer observations. In this work we present the algebraic structure of the cross of balanced nested and stair nested designs, using binary operations on commutative Jordan algebras. This new cross requires fewer...

Algebraic structureof step nesting designs

Célia FernandesPaulo RamosJoão Tiago Mexia — 2010

Discussiones Mathematicae Probability and Statistics

Step nesting designs may be very useful since they require fewer observations than the usual balanced nesting models. The number of treatments in balanced nesting design is the product of the number of levels in each factor. This number may be too large. As an alternative, in step nesting designs the number of treatments is the sum of the factor levels. Thus these models lead to a great economy and it is easy to carry out inference. To study the algebraic structure of step nesting designs we introduce...

Cross additivity - an application

Sandra Saraiva FerreiraDário FerreiraJoão Tiago Mexia — 2006

Discussiones Mathematicae Probability and Statistics

We try to show that Discriminant Analysis can be considered as a branch of Statistical Decision Theory when viewed from a Bayesian approach. First we present the necessary measure theory results, next we briefly outline the foundations of Bayesian Inference before developing Discriminant Analysis as an application of Bayesian Estimation. Our approach renders Discriminant Analysis more flexible since it gives the possibility of classing an element as belonging to a group of populations. This possibility...

Generalized F tests and selective generalized F tests for orthogonal and associated mixed models

Célia NunesIola PintoJoão Tiago Mexia — 2008

Discussiones Mathematicae Probability and Statistics

The statistics of generalized F tests are quotients of linear combinations of independent chi-squares. Given a parameter, θ, for which we have a quadratic unbiased estimator, θ̃, the test statistic, for the hypothesis of nullity of that parameter, is the quotient of the positive part by the negative part of such estimator. Using generalized polar coordinates it is possible to obtain selective generalized F tests which are especially powerful for selected families of alternatives. We build both classes...

Small perturbations with large effects on value-at-risk

Manuel L. EsquívelLuís DimasJoão Tiago MexiaPhilippe Didier — 2013

Discussiones Mathematicae Probability and Statistics

We show that in the delta-normal model there exist perturbations of the Gaussian multivariate distribution of the returns of a portfolio such that the initial marginal distributions of the returns are statistically undistinguishable from the perturbed ones and such that the perturbed V@R is close to the worst possible V@R which, under some reasonable assumptions, is the sum of the V@Rs of each of the portfolio assets.

Inference for random effects in prime basis factorials using commutative Jordan algebras

Vera M. JesusPaulo Canas RodriguesJoão Tiago Mexia — 2007

Discussiones Mathematicae Probability and Statistics

Commutative Jordan algebras are used to drive an highly tractable framework for balanced factorial designs with a prime number p of levels for their factors. Both fixed effects and random effects models are treated. Sufficient complete statistics are obtained and used to derive UMVUE for the relevant parameters. Confidence regions are obtained and it is shown how to use duality for hypothesis testing.

Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

João Tiago MexiaJoão Lita da Silva — 2007

Discussiones Mathematicae Probability and Statistics

Let Y i = x i T β + e i , 1 ≤ i ≤ n, n ≥ 1 be a linear regression model and suppose that the random errors e₁, e₂, ... are independent and α-stable. In this paper, we obtain sufficient conditions for the strong consistency of the least squares estimator β̃ of β under additional assumptions on the non-random sequence x₁, x₂,... of real vectors.

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