MULTIVARIATE MULTIPLE REGRESSION WITH STOCHASTIC PREDICTOR VARIABLES.
STANFORD UNIV CALIF DEPT OF STATISTICS
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The paper treats the regression of one or more dependent variables predictands upon one or more independent variables predictors the variables are assumed to be jointly normally distributed. Given a random sample of observations from the joint normal distribution of both sets of variables and an additional independent observation on the predictors, the problem is to predict the corresponding value of the predictands, when the loss function is the conditional mean square of the distance between the predicted and the actual values in the metric of the residual covariance matrix, given the sample of observations from the joint distribution. Some relations between this prediction problem and the problem of estimating the regression function and regression coefficients are given. Steins proof for the case of one predictand showed that the maximum likelihood estimator MLE is minimax in the class of nonrandomized estimators which depend upon the sufficient statistics is generalized to the case of more than one dependent variable this result is extended to show that the MLE is minimax in the class of all estimators. It is also shown that the predicting function based on the MLE is minimax in certain classes of predicting functions. Author
- Statistics and Probability