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.     

Admissible linear estimation in singular linear models with respect to a restricted parameter set

The admissibility results of Hoffmann (1977), proved in the context of a nonsingular covariance matrix are extended to the situation where the covariance matrix is singular. Admissible linear estimators in the Gauss-Markoff model are characterised and admissibility of the Best Linear Unbiased Estimator is investigated.     

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.     

Admissibilities of matrix linear estimators multivariate linear models

This article respectively provides sufficient conditions and necessary conditions of matrix linear estimators of an estimable parameter matrix linear function in multivariate linear models with and without the assumption that the underlying distribution is a normal one with completely unknown covariance matrix. In the latter model, a necessary and sufficient condition is given for matrix linear estimators to be admissible in the space of all matrix linear estimators under each of three different kinds of quadratic matrix loss functions, respectively. In the former model, a sufficient condition is first provided for matrix linear estimators to be admissible in the space of all matrix estimators having finite risks under each of the same loss functions, respectively. Furthermore in the former model, one of these sufficient conditions, correspondingly under one of the loss functions, is also proved to be necessary, if additional conditions are assumed.     

Identification of the linear factor model

Abstract

This paper provides several new results on identification of the linear factor model. The model allows for correlated latent factors and dependence among the idiosyncratic errors. I also illustrate identification under a dedicated measurement structure and other reduced rank restrictions. I use these results to study identification in a model with both observed covariates and latent factors. The analysis emphasizes the different roles played by restrictions on the error covariance matrix, restrictions on the factor loadings and the factor covariance matrix, and restrictions on the coefficients on covariates. The identification results are simple, intuitive, and directly applicable to many settings.     

On a superiority problem in misspecified restricted linear models

Razzaghi (1987) conjectures that a wrong choice of covariance matrix in a restricted linear model results in loss of efficiency, This conjecture is proved to be correct.     

Computations of predictors/estimators under a linear random-effects model with parameter restrictions

This paper considers the problem of simultaneously predicting/estimating unknown parameter spaces in a linear random-effects model with both parameter restrictions and missing observations. We shall establish explicit formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameters in such a model, and derive a variety of mathematical and statistical properties of the BLUPs under general assumptions. We also discuss some matrix expressions related to the covariance matrix of the BLUP, and present various necessary and sufficient conditions for several equalities and inequalities of the covariance matrix of the BLUP to hold.     

On a problem with singularity in comparison of linear experiments

For consistency, the parameter space in the Gauss-Markov model with singular covariance matrix is usually restricted by observation vector. This restriction arises some difficulties in comparison of linear experiments. To avoid it we reduce the problem of comparison from singular to nonsingular case.     

Predictions and estimations under a group of linear models with random coefficients

This article investigates the problem of establishing best linear unbiased predictors and best linear unbiased estimators of all unknown parameters in a group of linear models with random coefficients and correlated covariance matrix. We shall derive a variety of fundamental statistical properties of the predictors and estimators by using some matrix analysis tools. In particular, we shall establish necessary and sufficient conditions for the predictors and estimators to be equivalent under single and combined equations in the group of models by using the method of matrix equations, matrix rank formulas, and partitioned matrix calculations.     

Estimation of regression vectors in linear mixed models with Dirichlet process random effects

The Dirichlet process has been used extensively in Bayesian non parametric modeling, and has proven to be very useful. In particular, mixed models with Dirichlet process random effects have been used in modeling many types of data and can often outperform their normal random effect counterparts. Here we examine the linear mixed model with Dirichlet process random effects from a classical view, and derive the best linear unbiased estimator (BLUE) of the fixed effects. We are also able to calculate the resulting covariance matrix and find that the covariance is directly related to the precision parameter of the Dirichlet process, giving a new interpretation of this parameter. We also characterize the relationship between the BLUE and the ordinary least-squares (OLS) estimator and show how confidence intervals can be approximated.     

Bartlett-corrected tests for normal linear models when the error covariance matrix is nonscaiar

This paper provides Bartlett corrections to improve likelihood ratio tests for heteroskedastic normal linear models when the error covariance matrix is nonscaiar and depends on a set of unknown parameters. The Bartlett corrections are simple enough to be used algebraically to obtain several closed-form expressions in special cases. The corrections have also advantages for numerical purposes because they involve only simple operations on matrices and vectors.     

Perfect linear models and perfect parametric functions

General linear models with a common design matrix and with various structures of the variance–covariance matrix are considered. We say that a model is perfect for a linearly estimable parametric function, or the function is perfect in the model, if there exists the best linear unbiased estimator. All perfect models for a given function and all perfect functions in a given model are characterized.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.     

A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

Checking identifiability of covariance parameters in linear mixed effects models

To build a linear mixed effects model, one needs to specify the random effects and often the associated parametrized covariance matrix structure. Inappropriate specification of the structures can result in the covariance parameters of the model not identifiable. Non-identifiability can result in extraordinary wide confidence intervals, and unreliable parameter inference. Sometimes software produces implication of model non-identifiability, but not always. In the simulation of fitting non-identifiable models we tried, about half of the times the software output did not look abnormal. We derive necessary and sufficient conditions of covariance parameters identifiability which does not require any prior model fitting. The results are easy to implement and are applicable to commonly used covariance matrix structures.     

Adaptation of the jackknifed ridge methods to the linear mixed models

The purpose of this article is to obtain the jackknifed ridge predictors in the linear mixed models and to examine the superiorities, the linear combinations of the jackknifed ridge predictors over the ridge, principal components regression, r?k class and Henderson's predictors in terms of bias, covariance matrix and mean square error criteria. Numerical analyses are considered to illustrate the findings and a simulation study is conducted to see the performance of the jackknifed ridge predictors.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.     

Asymptotic distribution of least square estimators for linear models with dependent errors

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan (Central limit theorems for time series regression. Probab Theory Relat Fields. 1973;26(2):157–170), who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. Whatever the design satisfying Hannan's conditions, we define an estimator of the covariance matrix and we prove its consistency under very mild conditions. As an application, we show how to modify the usual tests on the linear model in this dependent context, in such a way that the type-I error rate remains asymptotically correct, and we illustrate the performance of this procedure through different sets of simulations.     

On linear lon-odds and estimation of discriminant coefficients

Fisher's Linear Discriminant Function Can be used to classify an individual who has sampled from one of two multivariate normal Populations. In the following, this function is viewed as the other given his data vector it is assumed that the Population means and common covariance matrix are unknown. The vector of discriminant coeffients β(p×1) is the gradient of posterior log-odds and certain of its lineqar functions are directional derivatives which have a practical meaning. Accordingly, we treat the problems of estimating several linear functions of β The usual estimatoes of these functions are scaled versions of the unbiased estmators. In this Paper, these estimators are domainated by explicit alterenatives under a quadratic loss function. we reduce the problem of estimating β to that of estimating the inverse convariance matrix.