Least Squares and Minimum MSE Estimators of Variance Components in Mixed Linear Models

The paper deals with the quadratic invariant estimators of the linear functions of variance components in mixed linear model. The estimator with locally minimal mean square error with respect to a parameter ? is derived. Under the condition of normality of the vector Y the theoretical values of MSE of several types of estimators are compared in two different mixed models; under a different types of distributions a simulation study is carried out for the behaviour of derived estimators.     

Variance Component Estimation for Mixed Model Analysis of cDNA Microarray Data

Microarrays provide a valuable tool for the quantification of gene expression. Usually, however, there is a limited number of replicates leading to unsatisfying variance estimates in a gene‐wise mixed model analysis. As thousands of genes are available, it is desirable to combine information across genes. When more than two tissue types or treatments are to be compared it might be advisable to consider the array effect as random. Then information between arrays may be recovered, which can increase accuracy in estimation. We propose a method of variance component estimation across genes for a linear mixed model with two random effects. The method may be extended to models with more than two random effects. We assume that the variance components follow a log‐normal distribution. Assuming that the sums of squares from the gene‐wise analysis, given the true variance components, follow a scaled χ²‐distribution, we adopt an empirical Bayes approach. The variance components are estimated by the expectation of their posterior distribution. The new method is evaluated in a simulation study. Differentially expressed genes are more likely to be detected by tests based on these variance estimates than by tests based on gene‐wise variance estimates. This effect is most visible in studies with small array numbers. Analyzing a real data set on maize endosperm the method is shown to work well. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unbiasedness of Iterated Aitken Estimators

The iterated Aitken estimator of the parameter vector of the general linear model is shown to be unbiased on the assumption that the errors underlying the model follow a symmetric probability law with mean zero.     

On the Optimality of F-test on Variance Components in Mixed Linear Models

The analysis-of-variance tests for hypotheses on random effects in regular linear models are considered. Conditions are given for these tests to be uniformly most powerful unbiased or uniformly most powerful invariant unbiased. An example shows that the difference between these conditions can be serious.     

The Estimation of Skewness and Kurtosis of Random Effects in the Linear Model

Generalising the ANOVA method of estimating variance components in mixed linear models a simple procedure is presented to estimate skewness and kurtosis of the distributions of the random effects of the model. For the model II of a one-way classification this procedure is demonstrated explicitly.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

On the Probability of a Negative Estimator of Heteroscedasticity in the Random Coefficient Model

In the estimation of a linear regression model with random coefficients, sometimes negative estimates of variances of random coefficients are obtained–an undesirable feature. In this paper we have obtained the asymptotic bounds of the probability of obtaining negative estimators. A simple illustration is also provided for the purpose.     

Modification of the Minimum Logit Chi-Squared Estimator in Simple Linear Logistic Regression

A new modification of Berkson's minimum logit chi-squared estimator in simple linear logistic regression is suggested in order to achieve reduction of first order bias of the estimator as well as in the model. Furthermore, unlike estimators currently available, our procedure is quite simple to apply in practice and is valid even in the presence of zero frequencies in the table.     

Comparing Stochastically Restricted Estimators in a Linear Regression Model

The minimum dispersion linear unbiased estimators of the vector of parameters in a linear regression model are compared when the parameters of the model are subject to stochastic linear restrictions with different dispersion matrices of the disturbances involved in them.     

On the Estimation of Intra-Sire Regression Heritability Subject to Linear Constraint

The derivation of the restricted intra-sire regression heritability estimator is provided. Procedures for obtaining a stable estimate of residual error variance σ² are outlined. A small illustration based on live data is given.     

Weighted Multivariate Regression Estimates Solved by Random Effects Approach

The paper deals with the random effects model, where the expectation vector and the covariance matrix of the effect influencing the population are to be estimated. The iterated estimator of expectation vector is derived, based on the invariant estimator of the combined covariance matrix, and some of its statistical properties are shown.     

A Nonparametric Estimator of Heterogeneity Variance with Applications to SMR‐ and Proportion‐Data

In this paper the situation of extra population heterogeneity is discussed from a analysis of variance point of view. We first provide a non‐iterative way of estimating the variance of the heterogeneity distribution without estimating the heterogeneity distribution itself for Poisson and binomial counts. The consequences of the presence of heterogeneity in the estimation of the mean are discussed. We show that if the homogeneity assumption holds, the pooled mean is optimal while in the presence of strong heterogeneity, the simple (arithmetic) mean is an optimal estimator of the mean SMR or mean proportion. These results lead to the problem of finding an optimal estimator for situations not represented by these two extreme cases. We propose an iterative solution to this problem. Illustrations for the application of these findings are provided with examples from various areas.     

A Note on Comparing Stochastically Restricted Linear Estimators in a Regression Model

In his recent paper Liski (1989) derived conditions for superiority of the minimum dispersion estimator over another with respect to the covariance matrix when the parameter vector of a regression model is subject to competing stochastic restrictions. The aim of this note is to provide another necessary and sufficient condition which admits an easier interpretation of superiority related to the covariance matrix criterion.     

方向数据密度函数的最近邻估计

设Ｘ为取值于ｄ＋１维空间中单位球面上的单位随机向量,具有概率密度函数ｆ（ｘ）本文讨论密度函数ｆ（ｘ）的估计问题,给出了基于方向数据的最近邻估计,并建立这种最近邻估计的逐点强,弱相合性,一致强相合性及渐近正态性,得到了与欧氏情形（Ｒ＾ｄ）基本一致的结果。     

On Admissible Estimators in a Linear Model

Let Y be observable random vector such that EY=Xβ and D(Y)=ρ²V. Linear estimation of a parameter p′β under the squared loss is considered. RAO, 1976 and 1979, obtained a necessary and sufficient condition for admissibility of an estimator t′Y in the case X=I. This result will be extended for arbitrary X. AMS 1970 subject classifications. Primary 62J05; secondary 62C15.     

A Mixed Model for Binary Data: The Modified X2-test

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions P_g in treatment group g(g = 1, 2, …, G). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H₀:E(P₁) = E(P₂) = … = E(P_G). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects) follows the X²-distribution.     

A Bayesian Estimator of the Optimum for a Single Factor Quadratic Response Model

Estimation of the location and magnitude of the optimum has long been considered an important problem in response surface methodology. In the industrial context, prior information accumulated by the subject matter specialist bears special significance. In this paper we use the Bayesian approach to estimating the optimum in a single factor quadratic regression model. Following the Bayesian general linear model development by Broemeling the normal/gamma conjugate prior is used. Explicit formulas for the generalized maximum likehood estimates of the characteristic parameters are obtained from the joint posterior distribution.