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1
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Least squares estimator consistency: a geometric approach

100%
Mexia J., da Silva J.
Discussiones Mathematicae Probability and Statistics
|
2006
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tom 26
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nr 1
19-45
EN
Consistency of LSE estimator in linear models is studied assuming that the error vector has radial symmetry. Generalized polar coordinates and algebraic assumptions on the design matrix are considered in the results that are established.
2
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An alternative approach to bonus malus

100%
Guerreiro G., Mexia J.
Discussiones Mathematicae Probability and Statistics
|
2004
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tom 24
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nr 2
197-213
EN
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, a few optimal bonus scales were calculated, such as Norberg's (1979), Borgan, Hoem's and Norberg's (1981), Gilde and Sundt's (1989) and Andrade e Silva's (1991). To compare our results, since this was the first application of the model, we used the Classic Model for Bonus Malus and the Open Model developed by Centeno and Andrade e Silva (2001). The results of the Stochastic Vortices and the Open Modelare highly similar and quite different from those of the Classic Model. Besides this the distribution of policyholders in the various bonus classes was derived assuming that the entrances followed adequatestochastic models.
3
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Compact hypothesis and extremal set estimators

100%
Mexia J., Corte Real P.
Discussiones Mathematicae Probability and Statistics
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2003
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tom 23
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nr 2
103-121
EN
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 parameter value $\vec{β₀}^{k}$ is the sole point in ∇, strongly consistent pointwise estimators, ${ \^{\vec{βₙ}}^{k}: n ∈ ℕ }$ for $\vec{β₀}^{k}$ are derived and confidence ellipsoids for $\vec{β₀}^{k}$ centered at $\^{\vec{βₙ}}^{k}$ are obtained, as well as, strongly consistent tests. Lastly an application to binary data is presented.
4
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Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

100%
Mexia J., da Silva J.
Discussiones Mathematicae Probability and Statistics
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2007
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tom 27
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nr 1-2
27-45
EN
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.
5
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Stochastic vortices in periodically reclassified populations

100%
Guerreiro G., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2008
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tom 28
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nr 2
209-227
EN
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 of independent Poisson distributed random variables; we use Markov chains to obtain limit results for the relative sizes of the sub-populations corresponding to the states of the chain. Namely we will obtain conditions for stability of the relative sizes for transient and recurrent states as well as for all states. The existence of such stability corresponds to the existence of a stochastic structure based either on the transient or on the recurrent states or even on all states. We call these structures stochastic vortices because the structure is maintained despite entrances, departures and reallocations.
6
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ANOVA using commutative Jordan algebras, an application

100%
Rodrigues P., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2006
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tom 26
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nr 2
179-191
EN
Binary operations on commutative Jordan algebras are used to carry out the ANOVA of a two layer model. The treatments in the first layer nests those in the second layer, that being a sub-model for each treatment in the first layer. We present an application with data retried from agricultural experiments.
7
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Selective F tests for sub-normal models

100%
Nunes C., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2003
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tom 23
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nr 2
167-174
EN
F tests that are specially powerful for selected alternatives are built for sub-normal models. In these models the observation vector is the sum of a vector that stands for what is measured with a normal error vector, both vectors being independent. The results now presented generalize the treatment given by Dias (1994) for normal fixed-effects models, and consider the testing of hypothesis on the ordering of mean values and components.
8
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Non-central generalized F distributions

100%
Nunes C., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2006
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tom 26
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nr 1
47-61
EN
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.
9
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Strong law of large numbers for additive extremum estimators

100%
Mexia J., Real P.
Discussiones Mathematicae Probability and Statistics
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2001
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tom 21
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nr 2
81-88
EN
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 hold. A strong law of large numbers is derived for additive extremum estimators. This law requires only the existence of first order moments and may be of interest in connection with maximum likelihood estimators, since the usual assumption that the observations are identically distributed is discarded.
10
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Canonic inference and commutative orthogonal block structure

81%
Carvalho F., Mexia J., Oliveira M.
Discussiones Mathematicae Probability and Statistics
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2008
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tom 28
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nr 2
171-181
EN
It is shown how to define the canonic formulation for orthogonal models associated to commutative Jordan algebras. This canonic formulation is then used to carry out inference. The case of models with commutative orthogonal block structures is stressed out.
11
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Algebraic structure for the crossing of balanced and stair nested designs

81%
Fernandes C., Ramos P., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2014
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tom 34
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nr 1-2
71-88
EN
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 observations than the usual cross balanced nested designs and it is easy to carry out inference.
12
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F and selective F tests with balanced cross-nesting and associated models

81%
Nunes C., Pinto I., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2006
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tom 26
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nr 2
193-205
EN
F tests and selective F tests for fixed effects part of balanced models with cross-nesting are derived. The effects of perturbations in the numerator and denominator of the F statistics are considered.
13
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Cross additivity - an application

81%
Ferreira S., Ferreira D., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2006
|
tom 26
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nr 2
207-219
EN
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 arises from the introduction of the concept of regions of controled posterior risk.
14
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Variance components estimation in generalized orthogonal models

81%
Fernandes C., Ramos P., Ferreira S., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2007
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tom 27
|
nr 1-2
99-115
EN
The model $y = ∑_{j=1}^w X_j β̲_j + e̲$ is generalized orthogonal if the orthogonal projection matrices on the range spaces of matrices $X_j$, j = 1, ..., w, commute. Unbiased estimators are obtained for the variance components of such models with cross-nesting.
15
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Inference for random effects in prime basis factorials using commutative Jordan algebras

81%
Jesus V., Rodrigues P., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2007
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tom 27
|
nr 1-2
15-25
EN
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.
16
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Confidence intervals for large non-centrality parameters

81%
Inacio S., Oliveira M., Mexia J.
Discussiones Mathematicae Probability and Statistics
|
2015
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tom 35
|
nr 1-2
45-56
EN
We use asymptotic linearity to derive confidence intervals for large non-centrality parameters. These results enable us to measure relevance of effects and interactions in multifactors models when we get highly statistically significant the values of F tests statistics. We show how to use our approach by considering two sets of data as application examples.
17
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Stable hypothesis for mixed models with balanced cross-nesting

81%
Ferreira D., Mexia J., Ferreira S.
Discussiones Mathematicae Probability and Statistics
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2005
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tom 25
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nr 2
241-249
EN
Stable hypothesis are hypothesis that may refer either for the fixed part or for the random part of the model. We will consider such hypothesis for models with balanced cross-nesting. Generalized F tests will be derived. It will be shown how to use Monte-Carlo methods to evaluate p-values for those tests.
18
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Estimators and tests for variance components in cross nested orthogonal designs

81%
Fonseca M., Mexia J., Zmyślony R.
Discussiones Mathematicae Probability and Statistics
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2003
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tom 23
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nr 2
175-201
EN
Explicit expressions of UMVUE for variance components are obtained for a class of models that include balanced cross nested random models. These estimators are used to derive tests for the nullity of variance components. Besides the usual F tests, generalized F tests will be introduced. The separation between both types of tests will be based on a general theorem that holds even for mixed models. It is shown how to estimate the p-value of generalized F tests.
19
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Small perturbations with large effects on value-at-risk

81%
Esquível M., Dimas L., Mexia J., Didier P.
Discussiones Mathematicae Probability and Statistics
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2013
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tom 33
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nr 1-2
151-169
EN
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.
20
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Orthogonal models: Algebraic structure and explicit estimators for estimable vectors

81%
Pereira A., Fonseca M., Mexia J.
Discussiones Mathematicae Probability and Statistics
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2015
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tom 35
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nr 1-2
29-44
EN
We study the algebraic structure of orthogonal models thus of mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known pairwise orthogonal projection matrices, POOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expression for the LSE of these models.
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