SOME PROPERTIES OF THE LEAST SQUARES ESTIMATOR IN REGRESSION ANALYSIS WHEN THE INDEPENDENT VARIABLES ARE STOCHASTIC

For the linear regression of y on x observations the loss in estimating the true regression function by another function is considered as a loss function. For the loss function, it is shown under certain conditions that if the class of estimates which are linear in ys and have bounded risk is non-empty, then the estimate obtained by the method of least squares belongs to this class and has uniformly minimum risk in this class. A necessary and sufficient condition on the distribution function of x observations is obtained for this class to be non-empty, which unfortunately is not easy to verify in particular cases and is violated in a ver simple situation. owever, by a sequential modification of the sampling scheme, this condition may always be satisfied at the cost of an arbitrarily small increase in the expected sa ple size. I T IS ALSO SHOWN UNDER CERTAIN FURTHER C NDITIONS ON THE FAMILY OF ADMISSIBLE DISTRIB TIONS THAT THE LEAST SQUARES ESTIMATOR IS MINIMAX IN THE CLASS OF ALL ESTIMATORS. Author