The multivariate regression of a p × 1 vector Y of random variables on a q × 1 vector X of explanatory variables is considered. It is assumed that linear transformations of the components of Y can be the bas...
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The multivariate regression of a p × 1 vector Y of random variables on a q × 1 vector X of explanatory variables is considered. It is assumed that linear transformations of the components of Y can be the basis for useful interpretation whereas the components of X have strong individual identity. When p ≥ q a transformation is found to a new q × 1 vector of responses Y ∗ such that in the multiple regression of, say, Y 1 ∗ on X , only the coefficient of X 1 is nonzero, i.e. such that Y 1 ∗ is conditionally independent of X 2 , …, X q , given X 1 . Some associated inferential procedures are sketched. An illustrative example is described in which the resulting transformation has aided interpretation.
Bayes estimates are derived in multivariate linear models with unknown distribution. The prior distribution is defined using a Dirichlet prior for the unknown error distribution and a normal-Wishart distribution for t...
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Bayes estimates are derived in multivariate linear models with unknown distribution. The prior distribution is defined using a Dirichlet prior for the unknown error distribution and a normal-Wishart distribution for the parameters. The posterior distribution is determined and explicit expressions are given in the special cases of location-scale and two-sample models. The calculation of self-in formative limits of Bayes estimates yields standard estimates.
Given a random singular matrix X, in the present article we find the Jacobian of the transformation Y = X+, where X+ is the Moore-Penrose inverse of X, both in the general case and when X is a non-negative definite ma...
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Given a random singular matrix X, in the present article we find the Jacobian of the transformation Y = X+, where X+ is the Moore-Penrose inverse of X, both in the general case and when X is a non-negative definite matrix. Expressions for the densities of the Moore-Penrose inverse of the singular Wishart and Pseudo-Wishart matrices are obtained. Similarly, an expression for the density of the matrix-variate singular T-distribution is proposed. Finally, these results are applied to the Bayesian inference of the multivariate linear model. (c) 2004 Elsevier B.V. All fights reserved.
Envelopes have been proposed in recent years as a nascent methodology for sufficient dimension reduction and efficient parameter estimation in multivariate linear models. We extend the classical definition of envelope...
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Envelopes have been proposed in recent years as a nascent methodology for sufficient dimension reduction and efficient parameter estimation in multivariate linear models. We extend the classical definition of envelopes in to incorporate a nonlinear conditional mean function and a heteroscedastic error. Given any two random vectors and , we propose two new model-free envelopes, called the martingale difference divergence envelope and the central mean envelope, and study their relationships to the standard envelope in the context of response reduction in multivariate linear models. The martingale difference divergence envelope effectively captures the nonlinearity in the conditional mean without imposing any parametric structure or requiring any tuning in estimation. Heteroscedasticity, or nonconstant conditional covariance of , is further detected by the central mean envelope based on a slicing scheme for the data. We reveal the nested structure of different envelopes: (i) the central mean envelope contains the martingale difference divergence envelope, with equality when has a constant conditional covariance;and (ii) the martingale difference divergence envelope contains the standard envelope, with equality when has a linear conditional mean. We develop an estimation procedure that first obtains the martingale difference divergence envelope and then estimates the additional envelope components in the central mean envelope. We establish consistency in envelope estimation of the martingale difference divergence envelope and central mean envelope without stringent model assumptions. Simulations and real-data analysis demonstrate the advantages of the martingale difference divergence envelope and the central mean envelope over the standard envelope in dimension reduction.
The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate reg...
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The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariatemodels with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.
The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linearmodels with ARMA error term. In contrast with earlier uni...
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The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linearmodels with ARMA error term. In contrast with earlier univariate results, the central sequence here is correlogram-based, i.e. expressed in terms of a generalized concept of residual cross-covariance function.
Restricted parameter spaces for covariance matrices, such as Sigma = sigma(2)I or Sigma = alpha I + beta J, are often used to simplify estimation. In addition, fixed upper and/or lower bounds may be needed to ensure t...
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Restricted parameter spaces for covariance matrices, such as Sigma = sigma(2)I or Sigma = alpha I + beta J, are often used to simplify estimation. In addition, fixed upper and/or lower bounds may be needed to ensure that estimates satisfy a priori hypotheses. With multivariate variance components models, several covariance matrices need to be simultaneously estimated and, even with a reduced parameter space, estimation can be difficult. In earlier work we have discussed estimation for a widely-used class of models where the variance components matrices need only be nonnegative definite. In this article we extend these results to handle a wide class of restricted parameter spaces. We state the conditions required for a parameterization to be a member of the class, discuss the implementation of the results for several different members of the class, and discuss estimation with both balanced and unbalanced data. We give several examples to demonstrate the results.
The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained a...
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The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained as matrix-t which are identical to those obtained under matrix normal and matrix-t distributions. This gives inference robustness with respect to departures from the reference case of independent sampling from the matrix normal or dependent but uncorrelated sampling from matrix-t distributions. Some successful applications of matrix-t distribution in the field of spatial prediction have been addressed. (C) 2005 Elsevier Inc. All rights reserved.
For the multivariate linear model, the joint asymptotic distribution of scale and coordinatewise M-estimators are obtained under rather general conditions. The main objective of this paper is to obtain these results w...
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For the multivariate linear model, the joint asymptotic distribution of scale and coordinatewise M-estimators are obtained under rather general conditions. The main objective of this paper is to obtain these results without assuming any symmetry-like conditions on the error distribution. By centering the design matrix, a simple closed form expression is obtained for the covariance matrix of the estimators of regression and scale parameters. This simple form is useful for understanding the behaviour of these estimators. Test of linear hypotheses about slope parameters is also discussed.
A survey is given of papers which have influenced or have been influenced by the Growth Curve model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical ve...
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A survey is given of papers which have influenced or have been influenced by the Growth Curve model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical version of the model, tests, extensions, incomplete data, Bayesian approaches and covariance structures.
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