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.     

Optimal design for linear regression models in the presence of heteroscedasticity caused by random coefficients

Random coefficients may result in heteroscedasticity of observations. For particular situations, where only one observation is available per individual, we derive optimal designs based on the geometry of the design locus.     

Testing for the parametric component of partially linear EV models under random censorship

This paper investigates the hypothesis test of the parametric component in partially linear errors-in-variables (EV) model with random censorship. We construct two test statistics based on the difference of the corrected residual sum of squares and empirical likelihood ratio under the null and alternative hypotheses. It is shown that the limiting distributions of the proposed test statistics are both weighted sum of independent standard chi-squared distribution with one degree of freedom under the null hypothesis. Based on the adjusted test statistics, we further develop two new types of test procedures. Finite sample performance of the proposed test procedures is evaluated by extensive simulation studies.     

A prediction analysis in a constrained multivariate general linear model with future observations

Abstract

We give a mathematical analysis to some fundamental prediction problems on a constrained multivariate general linear model (CMGLM) with future observations, including the derivation of analytical formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameter matrices, and the presentation of many novel and valuable properties of the BLUPs.     

Optimal design for quasi-likelihood estimation in Poisson regression with random coefficients

Count data may be described by a Poisson regression model. If random coefficients are involved, maximum likelihood is not feasible and alternative estimation methods have to be employed. For the approach based on quasi-likelihood estimation a characterization of design optimality is derived and optimal designs are determined numerically for an example with random slope parameters.     

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.     

On weak consistency in linear models with equi-correlated random errors

A new estimator in linear models with equi-correlated random errors is postulated. Consistency properties of the proposed estimator and the ordinary least squares estimator are studied. It is shown that the new estimator has smaller variance than the usual least squares estimator under some mild conditions. In addition, it is observed that the new estimator tends to be weakly consistent in many cases where the usual least squares estimator is not.     

Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

A new test for random effects in linear mixed models with longitudinal data

As it is known, testing the existence of random effects is often transferred to testing their zero variances/covariance matrices. It is a nonstandard testing problem because the hypothetical values are on the boundary of the whole space. In the literature, a difference-based test was proposed, which has asymptotically tractable null distribution and is then easy to implement. However, the projection method on which the difference-based test relies may affect and deteriorate its performance when covariates associated with fixed effects and covariates associated with random effects are highly correlated. In the paper, for linear mixed models (LMM) with longitudinal data, a new test is proposed to avoid this problem. The new test is also asymptotically distribution-free and more powerful than the difference-based test, particularly when the above correlation is high. The new test is consistent against all global alternatives and can detect local alternatives converging to the null at a rate as close as to m^−1/2$m^{-1/2}$ with m being the number of subjects. Simulations are carried out to examine the performance and a real data analysis is performed for illustration.     

Modeling of covariance structures of random effects and random errors in linear mixed models

Abstract

In this paper, we discuss how to model the mean and covariancestructures in linear mixed models (LMMs) simultaneously. We propose a data-driven method to modelcovariance structures of the random effects and random errors in the LMMs. Parameter estimation in the mean and covariances is considered by using EM algorithm, and standard errors of the parameter estimates are calculated through Louis’ (1982 Louis, T.A. (1982). Finding observed information using the EM algorithm. J. Royal Stat. Soc. B 44:98–130. [Google Scholar]) information principle. Kenward’s (1987 Kenward, M.G. (1987). A method for comparing profiles of repeated measurements. Appl. Stat. 36:296–308.[Crossref], [Web of Science ®] , [Google Scholar]) cattle data sets are analyzed for illustration,and comparison to the literature work is made through simulation studies. Our numerical analysis confirms the superiority of the proposed method to existing approaches in terms of Akaike information criterion.     

Three-step estimation in linear mixed models with skew-t distributions

Linear mixed models based on the normality assumption are widely used in health related studies. Although the normality assumption leads to simple, mathematically tractable, and powerful tests, violation of the assumption may easily invalidate the statistical inference. Transformation of variables is sometimes used to make normality approximately true. In this paper we consider another approach by replacing the normal distributions in linear mixed models by skew-t distributions, which account for skewness and heavy tails for both the random effects and the errors. The full likelihood-based estimator is often difficult to use, but a 3-step estimation procedure is proposed, followed by an application to the analysis of deglutition apnea duration in normal swallows. The example shows that skew-t models often entail more reliable inference than Gaussian models for the skewed data.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.     

Convergence rates in the law of large numbers for END linear processes with random coefficients

Abstract

In this paper, we consider convergence rates in the Marcinkiewicz–Zygmund law of the large numbers for the END linear processes with random coefficients. We extend some results of Baum and Katz (1965 Baum, L. E., and M. Katz. 1965. Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 120 (1):108–23. doi: 10.2307/1994170.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of dependent linear processes with the random coefficients.     

Prediction in multilevel generalized linear models

Summary.  We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.     

Estimation of linear models with anonymised panel data

The paper analyses the biasing effect of anonymising micro data by multiplicative stochastic noise on the within estimation of a linear panel model. In short panels, additional bias results from serially correlated regressors. Results in this paper are related to the project “Firms’ Panel Data and Factual Anonymisation,” which is financed by Federal Ministry of Education and Research. We would like to thank the anonymous referees for helpful comments.