Simultaneous Prediction Based on Shrinkage Estimator

In the literature of multiple regression models, the customary analysis is the estimation and hypothesis testing about the parameters of the model. However, in various applications, it is utmost important for a practitioner to predict the future values of the response variable. The most common way to tackle with such a problem is the use of Best Linear Unbiased Predictors (BLUP) discussed by Theil (1971), Hendersion (1972) and Judge, Griffiths, Hill, Lütkepohl and Lee (1985). for further details of the BLUP one can see Toyooka (1982) and Kariya and Toyooka (1985). The Stein-rule predictors and the shrinkage rules based on Stein-rule technique to forecast have also got considerable attention of the researchers in recent past. Copas (1983) considered the prediction in regression using a Stein-rule predictor. Copas and Jones (1987) applied the regression shrinkage technique for the prediction in an autoregressive model. Zellner and Hong (1989) used Bayesian shrinkage rules to forecast international growth rate. Hill and Fomby (1992) analyzed the performance of various improved estimators under an out-of-sample prediction mean square error criterion. Gotway and Cressie (1993) considered a class of linear and non-linear predictors in the context of a general linear model with known disturbances covariance matrix and observed that, under the quadratic loss function, the proposed class of predictors has uniformly smaller risk than the BLUP. Khan and Bhatti (1998) obtained the prediction distribution for a set of future responses from a multiple linear regression model following an equi-correlation structure. Tuchscheres, Herrendorfer and Tuchscheres (1998) proposed Estimated Best Linear Unbiased Predictor (EBLUP) with the help of a designated simulation experiment using MSE and GSD technique for evaluation.

The present paper deals with the problem of prediction based on shrinkage estimator in a general linear model with nonspherical disturbances. A general family of predictors for the composite target function, considered by Shalabh (1995), has been proposed and its asymptotic distribution has been derived employing large sample asymptotic theory. The risk based on quadratic loss structure of the proposed family of predictors has been obtained. The performance of proposed family of predictors is compared with the feasible Best Linear Unbiased Predictor (FBLUP) under the MSE matrix criterion and Quadratic loss function criterion. Further, we obtain the expression for an estimator for the MSE matrix of the proposed predictor. The results of a numerical simulation have been presented and discussed.