The Experts below are selected from a list of 249 Experts worldwide ranked by ideXlab platform

Dominique Fourdrinier - One of the best experts on this subject based on the ideXlab platform.

  • on completeness of the General Linear Model with spherically symmetric errors
    Statistical Methodology, 2014
    Co-Authors: Dominique Fourdrinier, William E Strawderman, Martin T. Wells
    Abstract:

    Abstract We consider the canonical form of the General Linear Model, with spherically symmetric errors, which may be viewed as a random vector in R n partitioned as ( X t U t ) t with a spherically symmetric density σ − n g ( { ‖ x − θ ‖ 2 + ‖ u ‖ 2 } σ − 2 ) around a mean vector, partitioned as ( θ t 0 t ) t , where dim X = dim θ = p and dim U = dim 0 = k with p + k = n . When the location parameter θ and the scale parameter σ are unknown and the generating function g ( ⋅ ) is known, we show that the statistic ( X , ‖ U ‖ 2 ) is minimal sufficient and we investigate whether it is a complete statistic or not. In particular, when g ( t ) has support contained in a compact interval not containing zero, we show non-completeness of the minimal sufficient statistic. Of course if the distribution is normal, well known results for exponential families imply its completeness. We also show that ( X , ‖ U ‖ 2 ) is complete for the Generalized multivariate t distribution.

  • Estimation of a Loss Function for Spherically Symmetric Distributions in the General Linear Model
    Annals of Statistics, 1995
    Co-Authors: Dominique Fourdrinier, Martin T. Wells
    Abstract:

    This paper is concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. We examine the canonical setting of a General Linear Model where the dimension of the parameter space is greater than 4 and less than the dimension of the sampling space. We consider two location estimators ― the least squares estimator and a shrinkage estimator ― and we compare their unbiased loss estimator with an improved loss estimator. The domination results are valid for a large class of spherically symmetric distributions and, in so far as the sampling distribution does not need to be precisely specified, the estimates have desirable robustness properties.

  • shrinkage estimators under spherical symmetry for the General Linear Model
    Journal of Multivariate Analysis, 1995
    Co-Authors: Dominique Cellier, Dominique Fourdrinier
    Abstract:

    This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a General Linear Model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper Linear subspace of the sampling space, we give an unbiased estimator of the difference of the risks between the least squares estimator ?0 and a General shrinkage estimator ? = ?0 ? ?X ? ?0 ?2 · g ? ?0. We obtain a General condition of domination for ? over ?0 which is weaker than that of Brandwein and Strawderman. We do not need any superharmonicity condition on g. Our results are valid for General quadratic loss.

Yongge Tian - One of the best experts on this subject based on the ideXlab platform.

  • A prediction analysis in a constrained multivariate General Linear Model with future observations
    Communications in Statistics - Theory and Methods, 2019
    Co-Authors: Yuqin Sun, Hong Jiang, Yongge Tian
    Abstract:

    AbstractWe give a mathematical analysis to some fundamental prediction problems on a constrained multivariate General Linear Model (CMGLM) with future observations, including the derivation of anal...

  • On decompositions of estimators under a General Linear Model with partial parameter restrictions
    Open Mathematics, 2017
    Co-Authors: Bo Jiang, Yongge Tian, Xuan Zhang
    Abstract:

    Abstract A General Linear Model can be given in certain multiple partitioned forms, and there exist subModels associated with the given full Model. In this situation, we can make statistical inferences from the full Model and subModels, respectively. It has been realized that there do exist links between inference results obtained from the full Model and its subModels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these Models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a General Linear Model with partial parameter restrictions. We will demonstrate how to decompose best Linear unbiased estimators (BLUEs) under the constrained General Linear Model (CGLM) as the sums of estimators under subModels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their Generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of Linear regression Models.

  • On simultaneous prediction in a multivariate General Linear Model with future observations
    Statistics & Probability Letters, 2017
    Co-Authors: Yongge Tian, Cheng Wang
    Abstract:

    We provide a General derivation for the closed-form formula of the best Linear unbiased predictors (BLUPs) of all unknown parameter matrices in a multivariate General Linear Model (MGLM) with future observations by using some new matrix analysis tools, and present a variety of valuable properties and features of the BLUPs under various General assumptions.

  • Some remarks on General Linear Model with new regressors
    Statistics & Probability Letters, 2015
    Co-Authors: Shengjun Gan, Yongge Tian
    Abstract:

    Assume that an original General Linear Model is misspecified by adding some new regressors. We investigate in such a case relationships between the best Linear unbiased estimators under the two Models. In particular, we give necessary and sufficient conditions for the best Linear unbiased estimators to be equal under the two Models.

  • On equalities of estimations of parametric functions under a General Linear Model and its restricted Models
    Metrika, 2009
    Co-Authors: Yongge Tian
    Abstract:

    Estimations of parametric functions under a General Linear Model and its restricted Models involve some complicated operations of matrices and their Generalized inverses. In the past several years, a powerful tool—the matrix rank method was utilized to manipulate various complicated matrix expressions that involve Generalized inverses of matrices. In this paper, we use this method to derive necessary and sufficient conditions for six equalities of the ordinary least-squares estimators and the best Linear unbiased estimators of parametric functions to equal under a General Linear Model and its corresponding restricted Model.

Martin T. Wells - One of the best experts on this subject based on the ideXlab platform.

  • on completeness of the General Linear Model with spherically symmetric errors
    Statistical Methodology, 2014
    Co-Authors: Dominique Fourdrinier, William E Strawderman, Martin T. Wells
    Abstract:

    Abstract We consider the canonical form of the General Linear Model, with spherically symmetric errors, which may be viewed as a random vector in R n partitioned as ( X t U t ) t with a spherically symmetric density σ − n g ( { ‖ x − θ ‖ 2 + ‖ u ‖ 2 } σ − 2 ) around a mean vector, partitioned as ( θ t 0 t ) t , where dim X = dim θ = p and dim U = dim 0 = k with p + k = n . When the location parameter θ and the scale parameter σ are unknown and the generating function g ( ⋅ ) is known, we show that the statistic ( X , ‖ U ‖ 2 ) is minimal sufficient and we investigate whether it is a complete statistic or not. In particular, when g ( t ) has support contained in a compact interval not containing zero, we show non-completeness of the minimal sufficient statistic. Of course if the distribution is normal, well known results for exponential families imply its completeness. We also show that ( X , ‖ U ‖ 2 ) is complete for the Generalized multivariate t distribution.

  • Estimation of a Loss Function for Spherically Symmetric Distributions in the General Linear Model
    Annals of Statistics, 1995
    Co-Authors: Dominique Fourdrinier, Martin T. Wells
    Abstract:

    This paper is concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. We examine the canonical setting of a General Linear Model where the dimension of the parameter space is greater than 4 and less than the dimension of the sampling space. We consider two location estimators ― the least squares estimator and a shrinkage estimator ― and we compare their unbiased loss estimator with an improved loss estimator. The domination results are valid for a large class of spherically symmetric distributions and, in so far as the sampling distribution does not need to be precisely specified, the estimates have desirable robustness properties.

Dennis Leech - One of the best experts on this subject based on the ideXlab platform.

  • testing for spatial heterogeneity in functional mri using the multivariate General Linear Model
    IEEE Transactions on Medical Imaging, 2011
    Co-Authors: Robert Leech, Dennis Leech
    Abstract:

    Much current research in functional magnetic resonance imaging (fMRI) employs multivariate machine learning approaches (e.g., support vector machines) to detect distributed spatial patterns from the temporal fluctuations of the neural signal. The aim of many studies is not classification, however, but investigation of multivariate spatial patterns, which pattern classifiers detect only indirectly. Here we propose a direct statistical measure for the existence of distributed spatial patterns (or spatial heterogeneity) applicable to fMRI datasets. We extend the univariate General Linear Model (GLM), typically used in fMRI analysis, to a multivariate case. We demonstrate that contrasting maximum likelihood estimations of different restrictions on this multivariate Model can be used to estimate the extent of spatial heterogeneity in fMRI data. Under asymptotic assumptions inference can be made with reference to the χ2 distribution. The test statistic is then assessed using simulated timecourses derived from real fMRI data followed by analyzing data from a real fMRI experiment. These analyses demonstrate the utility of the proposed measure of heterogeneity as well as considerations in its application. Measuring spatial heterogeneity in fMRI has important theoretical implications in its own right and may have potential uses for better characterising neurological conditions such as stroke and Alzheimer's disease.

  • testing for spatial heterogeneity in functional mri using the multivariate General Linear Model
    2010
    Co-Authors: Robert Leech, Dennis Leech
    Abstract:

    Much current research in functional MRI employs multivariate machine learning approaches (e.g., support vector machines) to detect fine-scale spatial patterns from the temporal fluctuations of the neural signal. The aim of many studies is not classification, however, but investigation of multivariate spatial patterns, which pattern classifiers detect only indirectly. Here we propose a direct statistical measure for the existence of fine-scale spatial patterns (or spatial heterogeneity) applicable for fMRI datasets. We extend the univariate General Linear Model (typically used in fMRI analysis) to a multivariate case. We demonstrate that contrasting maximum likelihood estimations of different restrictions on this multivariate Model can be used to estimate the extent of spatial heterogeneity in fMRI data. Under asymptotic assumptions inference can be made with reference to the X2 distribution. The test statistic is then assessed using simulated timecourses derived from real fMRI data. This demonstrates the utility of the proposed measure of heterogeneity as well as considerations in its application. Measuring spatial heterogeneity in fMRI has important theoretical implications in its own right and has potential uses for better characterising neurological conditions such as stroke and Alzheimer’s disease.

Robert Leech - One of the best experts on this subject based on the ideXlab platform.

  • testing for spatial heterogeneity in functional mri using the multivariate General Linear Model
    IEEE Transactions on Medical Imaging, 2011
    Co-Authors: Robert Leech, Dennis Leech
    Abstract:

    Much current research in functional magnetic resonance imaging (fMRI) employs multivariate machine learning approaches (e.g., support vector machines) to detect distributed spatial patterns from the temporal fluctuations of the neural signal. The aim of many studies is not classification, however, but investigation of multivariate spatial patterns, which pattern classifiers detect only indirectly. Here we propose a direct statistical measure for the existence of distributed spatial patterns (or spatial heterogeneity) applicable to fMRI datasets. We extend the univariate General Linear Model (GLM), typically used in fMRI analysis, to a multivariate case. We demonstrate that contrasting maximum likelihood estimations of different restrictions on this multivariate Model can be used to estimate the extent of spatial heterogeneity in fMRI data. Under asymptotic assumptions inference can be made with reference to the χ2 distribution. The test statistic is then assessed using simulated timecourses derived from real fMRI data followed by analyzing data from a real fMRI experiment. These analyses demonstrate the utility of the proposed measure of heterogeneity as well as considerations in its application. Measuring spatial heterogeneity in fMRI has important theoretical implications in its own right and may have potential uses for better characterising neurological conditions such as stroke and Alzheimer's disease.

  • testing for spatial heterogeneity in functional mri using the multivariate General Linear Model
    2010
    Co-Authors: Robert Leech, Dennis Leech
    Abstract:

    Much current research in functional MRI employs multivariate machine learning approaches (e.g., support vector machines) to detect fine-scale spatial patterns from the temporal fluctuations of the neural signal. The aim of many studies is not classification, however, but investigation of multivariate spatial patterns, which pattern classifiers detect only indirectly. Here we propose a direct statistical measure for the existence of fine-scale spatial patterns (or spatial heterogeneity) applicable for fMRI datasets. We extend the univariate General Linear Model (typically used in fMRI analysis) to a multivariate case. We demonstrate that contrasting maximum likelihood estimations of different restrictions on this multivariate Model can be used to estimate the extent of spatial heterogeneity in fMRI data. Under asymptotic assumptions inference can be made with reference to the X2 distribution. The test statistic is then assessed using simulated timecourses derived from real fMRI data. This demonstrates the utility of the proposed measure of heterogeneity as well as considerations in its application. Measuring spatial heterogeneity in fMRI has important theoretical implications in its own right and has potential uses for better characterising neurological conditions such as stroke and Alzheimer’s disease.