Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Variable selection in elliptical linear mixed model

Variable selection in elliptical Linear Mixed Models (LMMs) with a shrinkage penalty function (SPF) is the main scope of this study. SPFs are applied for parameter estimation and variable selection simultaneously. The smoothly clipped absolute deviation penalty (SCAD) is one of the SPFs and it is adapted into the elliptical LMM in this study. The proposed idea is highly applicable to a variety of models which are set up with different distributions such as normal, student-t, Pearson VII, power exponential and so on. Simulation studies and real data example with one of the elliptical distributions show that if the variable selection is also a concern, it is worthwhile to carry on the variable selection and the parameter estimation simultaneously in the elliptical LMM.     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

On f-tests and linear hypothesis in a general linear model

A simple method of setting linear hypotheses testable by F-tests in a general linear model when the covariance matrix has a general form and is completely unknown, is provided. With some additional conditions imposed on the covariance matrix, there exist the UMP invariant tests of certain linear hypotheses. We derive them to compare the powers with those of F-tests obtained under no restrictions on the covariance matrix. The results are illustrated in a multiple regression model with some examples.     

Ratios of linear combinations in the general linear model

We consider the problem of estimating. the ratio of two linear combinations of thevector of parameters in the general linear model. The nonexistence of an unbiased estimator under normal errorsisdiscussed. Properties of an often used estimator, the maximum likelihood estimator under normal errors, are presented, This is done both for fixed sample size and asymptotically, in the

presence of normal and non-normal errors.     

The polynomial trend model with autocorrelated residuals

The paper considers the fitting of polynomial trends to data when the residuals are autocorrelated. Although OLS is asymptoti­cally efficient it can be quite inefficient in small samples. Hence it is suggested that a test for autocorrelation be carried out and to this end we present a table of exact critical values of the Durbin-Watson test for this model.     

Prediction distribution for a linear regression model with multivariate student-t error distribution

The distribution(s) of future response(s) given a set of data from an informative experiment is known as prediction distribution. The paper derives the prediction distribution(s) from a linear regression model with a multivari-ate Student-t error distribution using the structural relations of the model. We observe that the prediction distribution(s) are multivariate t-variate(s) with degrees of freedom which do not depend on the degrees of freedom of the error distribution.     

Comparisons of parameter and hypothesis definitions in a general linear model

In the context of the general linear model E(Y)=Xβ possibly subject to restrictions Rβ=r two secondary parameters may be well defined by Θ_i=G_iE(Y)-Θ_oi=C_i β-Θ_oi,i=1,2, and corresponding nonconstant hypotheses, H_i:Θ_i=0. The following possible relations are defined: Θ₁: is dependent upon /equivalent to/identical to Θ₂:H₁is a subhypothesis of/is identical to H₂. Necessary and sufficient conditions, involving straightforward matrix computations, are presented for each relation. Comparisons of secondary parameters and hypotheses are illustrated with an incomplete, unbalanced 3 × 4 factorial design from Searle in which, using a constrained version of Searle's model, parameters and hypotheses in the incomplete, unbalanced design are shown to be indentical to parameters one would define if complete balanced data were available. Techniques for computing simplified definitions are illustrated.     

Corrected f-tests in the general linear model

The theory of corrected F-tests in the general linear modeL with correlated errors is studied in this paper. Strictly unbiased F-tests are constructed in the presence of a large class of known non-identity correlation structures. These are generalized likelihood ratio tests which are easily computed. Critical points from the standard F-tables can be used to provide an exact test of desired size.     

The posterior distribution of the fixed and random effects in a mixed-effects linear model

The subject of this paper is Bayesian inference about the fixed and random effects of a mixed-effects linear statistical model with two variance components. It is assumed that a priori the fixed effects have a noninformative distribution and that the reciprocals of the variance components are distributed independently (of each other and of the fixed effects) as gamma random variables. It is shown that techniques similar to those employed in a ridge analysis of a response surface can be used to construct a one-dimensional curve that contains all of the stationary points of the posterior density of the random effects. The “ridge analysis” (of the posterior density) can be useful (from a computational standpoint) in finding the number and the locations of the stationary points and can be very informative about various features of the posterior density. Depending on what is revealed by the ridge analysis, a multivariate normal or multivariate-t distribution that is centered at a posterior mode may provide a satisfactory approximation to the posterior distribution of the random effects (which is of the poly-t form).     

On the best linear unbiased estimator and the linear sufficiency of a general growth curve model

Growth curve models are used to analyze repeated measures data (longitudinal data), which are functions of time. In this paper, some necessary and sufficient conditions for linear function B₁YB₂ to be the best linear unbiased estimator (BLUE) of estimable functions X₁ΘX₂ (or K₁ΘK₂) under the general growth curve model were established. In addition, the representations of BLUE(K₁ΘK₂) (or BLUE(X₁ΘX₂)) were derived when the conditions are satisfied. Two special notions of linear sufficiency with respect to the general growth curve model are given in the end. The findings of this paper enrich some known results in the literature.     

Quasi-minimax estimation in the general linear regression model

This paper is mainly concerned with minimax estimation in the general linear regression model y=Xβ+ε$y=X\beta +\epsilon$ under ellipsoidal restrictions on the parameter space and quadratic loss function. We confine ourselves to estimators that are linear in the response vector y  . The minimax estimators of the regression coefficient β$\beta$ are derived under homogeneous condition and heterogeneous condition, respectively. Furthermore, these obtained estimators are the ridge-type estimators and mean dispersion error (MDE) superior to the best linear unbiased estimator b=(X^′W^-1X)^-1X^′W^-1y$b=(X^{\prime }{W}^{-1}X{)}^{-1}{X}^{\prime }{W}^{-1}y$ under some conditions.     

Generalized f variates in the general linear model

In this paper the theory of the generalized F variate is examined. A large class of statistics is constructed for linear for linear models when the errors are distributed N(0, ∑), where ∑ is positive definite. These statistics are shown to have a generalized F distribution. The role of the generalized F variate is then studied in detail for a linear model of constant intraclass correlation.     

Admissible linear estimators in the general Gauss-Markov model

This paper derives a complete characterization of estimators that are admissible for a given identifiable vector of parametric functions among the set of linear estimators under the general Gauss-Markov model with a dispersion matrix possibly singular. The characterization obtained implies some corollaries, which are then compared with the results known in the literature.     

Two-stage tests of hypotheses in the general linear model

Two-stage (double sample) tests of hypotheses are presented for testing linear hypotheses in the general linear model. General and one-sided alternatives are considered. Computational techniques for computing critical points are discussed. Tables of critical points are presented. An example suggests that two-stage tests can achieve the same power as a fixed sample size test while reducing considerably the expected number of observations required for the test     

Exact distribution of a linear combination of a variable and order statistics from the other two variables of a trivariate elliptical random vector as a mixture of skew-elliptical distributions

We consider here a univariate skew-elliptical distribution, which is a special case of the unified multivariate skew-elliptical distribution studied recently by Arellano-Valle and Azzalini (2006) [1]. We then derive the exact distribution of a linear combination of a variable and order statistics from the other two variables in the case of a trivariate elliptical distribution. We show that the cumulative distribution function (cdf) of this linear combination is a mixture of the univariate skew-elliptical distribution functions.     

Some equalities for estimations of partial coefficients under a general linear regression model

Estimations of partial coefficients in a general regression models involve some complicated operations of matrices and their generalized inverses. In this note, we use the matrix rank method to derive necessary and sufficient conditions for the ordinary least-squares estimator and the best linear unbiased estimator of partial coefficients in a general linear regression model to equal.