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.     

Comparing Stochastically Restricted Linear Estimators in a Regression Model

Conditions for superiority of the minimum dispersion estimator over another with respect to the covariance matrix are derived when the vector parameter of a regression model is subject to competing stochastic restrictions. The restrictions may also consist both of a deterministic part and a stochastic part.     

Improved covariance matrix estimators for weighted analysis of microarray data.

Empirical Bayes models have been shown to be powerful tools for identifying differentially expressed genes from gene expression microarray data. An example is the WAME model, where a global covariance matrix accounts for array-to-array correlations as well as differing variances between arrays. However, the existing method for estimating the covariance matrix is very computationally intensive and the estimator is biased when data contains many regulated genes. In this paper, two new methods for estimating the covariance matrix are proposed. The first method is a direct application of the EM algorithm for fitting the multivariate t-distribution of the WAME model. In the second method, a prior distribution for the log fold-change is added to the WAME model, and a discrete approximation is used for this prior. Both methods are evaluated using simulated and real data. The first method shows equal performance compared to the existing method in terms of bias and variability, but is superior in terms of computer time. For large data sets (>15 arrays), the second method also shows superior computer run time. Moreover, for simulated data with regulated genes the second method greatly reduces the bias. With the proposed methods it is possible to apply the WAME model to large data sets with reasonable computer run times. The second method shows a small bias for simulated data, but appears to have a larger bias for real data with many regulated genes.     

Estimation of a covariance matrix with zeros

We consider estimation of the covariance matrix of a multivariaterandom vector under the constraint that certain covariancesare zero. We first present an algorithm, which we call iterativeconditional fitting, for computing the maximum likelihood estimateof the constrained covariance matrix, under the assumption ofmultivariate normality. In contrast to previous approaches,this algorithm has guaranteed convergence properties. Droppingthe assumption of multivariate normality, we show how to estimatethe covariance matrix in an empirical likelihood approach. Theseapproaches are then compared via simulation and on an exampleof gene expression.