Statistical applications of the multivariate skew-normal distribution

Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.Comment: full-length version of the published paper, 32 pages, with 7 figures, uses psfra

Cholesky-ANN models for predicting multivariate realized volatility

Accurately forecasting multivariate volatility plays a crucial role for the financial industry. The Cholesky-Artificial Neural Networks specification here presented provides a twofold advantage for this topic. On the one hand, the use of the Cholesky decomposition ensures positive definite forecasts. On the other hand, the implementation of artificial neural networks allows to specify nonlinear relations without any particular distributional assumption. Out-of-sample comparisons reveal that Artificial neural networks are not able to strongly outperform the competing models. However, long-memory detecting networks, like Nonlinear Autoregressive model process with eXogenous input and long shortterm memory, show improved forecast accuracy respect to existing econometric models

Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise

Contribuições ao estudo de dados longitudinais na teoria de resposta ao item

Orientador: Caio Lucidius Naberezny AzevedoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Na presente tese desenvolvemos classes de modelos longitudinais da Teoria de Resposta o Item (TRI) considerando duas abordagens. A primeira é baseada na decomposição de Cholesky de matrizes de covariância de interesse, relacionadas aos traços latentes. Essa metodologia permite representar um amplo conjunto de estruturas de dependência de maneira relativamente simples, facilita a escolha de distribuições a priori para os parâmetros relacionados à estrutura de dependência, facilita a implementação de algoritmos de estimação (particularmente sob o enfoque Bayesiano), permite considerar diferentes distribuições (multivariadas) para os traços latentes de modo simples, torna bastante fácil a incorporação de estruturas de regressão para os traços latentes, entre outras vantagens. Desenvolvemos, adicionalmente, uma classe de modelos com estruturas de curvas de crescimento para os traços latentes. Na segunda abordagem utilizamos cópulas Gaussianas para representar a estrutura de dependência dos traços latentes. Diferentemente da abordagem anterior, essa metodologia permite o total controle das respectivas distribuições marginais mas, igualmente, permite considerar um grande número de estruturas de dependência. Utilizamos modelos dicotômicos de resposta ao item e exploramos a utilização da distribuição normal e normal assimétrica para os traços latentes. Consideramos indivíduos acompanhados ao longo de várias condições de avaliação, submetidos a instrumentos de medida em cada uma delas, os quais possuem alguma estrutura de itens comuns. Exploramos os casos de um único e de vários grupos como também dados balanceados e desbalanceados, no sentido de considerarmos inclusão e exclusão de indivíduos ao longo do tempo. Algoritmos de estimação, ferramentas para verificação da qualidade de ajuste e comparação de modelos foram desenvolvidos sob o paradigma bayesiano, através de algoritmos MCMC híbridos, nos quais os algoritmos SVE (Single Variable Exchange) e Metropolis-Hastings são considerados quando as distribuições condicionais completas não são conhecidas. Estudos de simulação foram conduzidos, os quais indicaram que os parâmetros foram bem recuperados. Além disso, dois conjuntos de dados longitudinais psicométricos foram analisados para ilustrar as metodologias desenvolvidas. O primeiro é parte de um estudo de avaliação educacional em larga escala promovido pelo governo federal brasileiro. O segundo foi extraído do Amsterdam Growth and Health Longitudinal Study (AGHLS) que monitora a saúde e o estilo de vida de adolescentes holandesesAbstract: In this thesis we developed families of longitudinal Item Response Theory (IRT) models considering two approaches. The first one is based on the Cholesky decomposition of the covariance matrices of interest, related to the latent traits. This modeling can accommodate several dependence structures in a easy way, it facilitates the choice of prior distributions for the parameters of the dependence matrix, it facilitates the implementation of estimation algorithms (particularly under the Bayesian paradigm), it allows to consider different (multivariate) distributions for the latent traits, it makes easier the inclusion of regression and multilevel structures for the latent traits, among other advantages. Additionally, we developed growth curve models for the latent traits. The second one uses a Gaussian copula function to describes the latent trait structure. Differently from the first one, the copula approach allows the entire control of the respective marginal latent trait distributions, but as the first one, it accommodates several dependence structures. We focus on dichotomous responses and explore the use of the normal and skew-normal distributions for the latent traits. We consider subjects followed over several evaluation conditions (time-points) submitted to measurement instruments which have some structure of common items. Such subjects can belong to a single or multiple independent groups and also we considered both balanced and unbalanced data, in the sense that inclusion or dropouts of subjects are allowed. Estimation algorithms, model fit assessment and model comparison tools were developed under the Bayesian paradigm through hybrid MCMC algorithms, such that when the full conditionals are not known, the SVE (Single Variable Exchange) and Metropolis-Hastings algorithms are used. Simulation studies indicate that the parameters are well recovered. Furthermore, two longitudinal psychometrical data sets were analyzed to illustrate our methodologies. The first one is a large-scale longitudinal educational study conducted by the Brazilian federal government. The second was extracted from the Amsterdam Growth and Health Longitudinal Study (AGHLS), which monitors the health and life-style of Dutch teenagersDoutoradoEstatisticaDoutor em Estatística162562/2014-4,142486/2015-9CNPQCAPE

Efficient inference about the tail weight in multivariate Student tt distributions

We propose a new testing procedure about the tail weight parameter of multivariate Student $t$ distributions by having recourse to the Le Cam methodology. Our test is asymptotically as efficient as the classical likelihood ratio test, but outperforms the latter by its flexibility and simplicity: indeed, our approach allows to estimate the location and scatter nuisance parameters by any root-$n$ consistent estimators, hereby avoiding numerically complex maximum likelihood estimation. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method on a financial data set. We conclude the paper by indicating how to use this framework for efficient point estimation.Comment: 23 page

Statistical analysis of factor models of high dimension

This paper considers the maximum likelihood estimation of factor models of high dimension, where the number of variables (N) is comparable with or even greater than the number of observations (T). An inferential theory is developed. We establish not only consistency but also the rate of convergence and the limiting distributions. Five different sets of identification conditions are considered. We show that the distributions of the MLE estimators depend on the identification restrictions. Unlike the principal components approach, the maximum likelihood estimator explicitly allows heteroskedasticities, which are jointly estimated with other parameters. Efficiency of MLE relative to the principal components method is also considered.Comment: Published in at http://dx.doi.org/10.1214/11-AOS966 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The SENSE-Isomorphism Theoretical Image Voxel Estimation (SENSE-ITIVE) Model for Reconstruction and Observing Statistical Properties of Reconstruction Operators

The acquisition of sub-sampled data from an array of receiver coils has become a common means of reducing data acquisition time in MRI. Of the various techniques used in parallel MRI, SENSitivity Encoding (SENSE) is one of the most common, making use of a complex-valued weighted least squares estimation to unfold the aliased images. It was recently shown in Bruce et al. [Magn. Reson. Imag. 29(2011):1267-1287] that when the SENSE model is represented in terms of a real-valued isomorphism,it assumes a skew-symmetric covariance between receiver coils, as well as an identity covariance structure between voxels. In this manuscript, we show that not only is the skew-symmetric coil covariance unlike that of real data, but the estimated covariance structure between voxels over a time series of experimental data is not an identity matrix. As such, a new model, entitled SENSE-ITIVE, is described with both revised coil and voxel covariance structures. Both the SENSE and SENSE-ITIVE models are represented in terms of real-valued isomorphisms, allowing for a statistical analysis of reconstructed voxel means, variances, and correlations resulting from the use of different coil and voxel covariance structures used in the reconstruction processes to be conducted. It is shown through both theoretical and experimental illustrations that the miss-specification of the coil and voxel covariance structures in the SENSE model results in a lower standard deviation in each voxel of the reconstructed images, and thus an artificial increase in SNR, compared to the standard deviation and SNR of the SENSE-ITIVE model where both the coil and voxel covariances are appropriately accounted for. It is also shown that there are differences in the correlations induced by the reconstruction operations of both models, and consequently there are differences in the correlations estimated throughout the course of reconstructed time series. These differences in correlations could result in meaningful differences in interpretation of results