A linear regression with unobserved dependent variables
Let (⊖,X) be a random vector such that E(X|⊖) = ⊖ and Var(x|⊖) a + b⊖ + c⊖2 for some known constants a, b and c. Assume X1,…,Xn are independent observations which have the same distribution as X. Let t(X) be the linear regression of ⊖ on X. The linear empirical Bayes estimator is used to approximate the linear regression function. It is shown that under appropriate conditions, the linear empirical Bayes estimator approximates the linear regression well in the sense of mean squared error.