On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the "unbiased" estimator of the discriminant coefficients.