Improved Predictions in Linear Regression Models with Stochastic Linear Constraints

In this article we have considered two families of predictors for the simultaneous prediction of actual and average values of study variable in a linear regression model when a set of stochastic linear constraints binding the regression coe cients is available These families arise from the method of mixed regression estimation Performance properties of these families are analyzed when the objective is to predict values outside the sample and within the sample Introduction Quite often we come across situations demanding simultaneous prediction of the actual and average values of study variable As an illustration consider a new drug for increasing the duration of sleep in patients su ering from high blood pressure When a medical practitioner is told to prescribe some new drug he she would like to enquire What will be the increase on the average in the duration of sleep when a speci c dose of this drug is administered On the other strand a patient would like to know What will be the actual increase in the duration of my sleep if I take the prescribed dose of this drug Thus the medical practitioner is more interested in the prediction of average value in comparison to the prediction of actual value The opposite is true in case of the patient who is more concerned with the prediction of actual value It is therefore imperative to consider the simultaneous prediction of actual and average values in such a manner that actual and average values are assigned possibly unequal weightage Such situations occur not only in medical sciences but in other disciplines too like Economics see Shalabh Zellner Toutenburg and Shalabh have considered the problem of simultaneous prediction of actual and average values of study variable in a linear regression model assuming the availability of some prior information in the form of few linear constraints binding the regression coe cients When these constraints are stochastic in nature they have studied the performance properties of pre dictors arising from pure and mixed regression methods of estimation for the coe cients For forecasting values actual or average or a combination of both Institute of Statistics LMU M unchen Ludwigstr M unchen Germany toutenb stat uni muenchen de Department of Statistics University of Jammu Jammu India of study variable outside the sample such as future values their investigations have revealed that the predictions based on mixed regression estimation are superior at least asymptotically with respect to the criterion of predictive dis persion matrix to the predictions based on pure regression estimation provided that the degrees of freedom i e excess of the numbers of observations over the numbers of unknown coe cients are three or more This is however not true when the aim is to predict values within the sample Here mixed regression based predictions are superior only when the predictions of average values is as signed a higher weight in comparison to the prediction of actual values provided that the degrees of freedom are at least three In view of the above ndings reported by Toutenburg and Shalabh a natural question is as follows Can we further improve the performance of mixed regression based predictions In other words is it possible to construct predictors having superior performance properties at least in these situations where mixed regression based predictions are known to be better than the pure regression based predictions This article presents a simple e ort in this direc tion Our presentation is as follows In Section we describe the model and present estimators of regression coe cients Section considers the prediction of values outside the sample and analyzes the properties of some predictors Similarly Section deals with the prediction of values within the sample Some remarks are then placed in Section Finally the Appendix presents derivation of results Model Speci cation and the Estimators Consider the following linear regression model y X u where y is a n vector of observations on the study variable X is n k full column rank matrix of n observations on k explanatory variables is the vector of regression coe cient is a scalar and u is a vector of disturbances following a multivariate normal distribution with mean vector and variance covariance matrix I In addition to let us be given some prior information in the form of a set of g stochastic linear constraints binding the regression coe cients as follows r R v where r is g vector with known elements R is a g k matrix with known elements and v is a g random vector with mean vector and variance covariance matrix assumed to be known and positive de nite It is assumed that u and v are stochastically independent and n k exceeds The pure regression estimator of is the least squares estimator of in