Consider a linear regression model with p parameters, fit by least squares to a set of training data (x1,y1),...,(xN,yN) drawn at random from a population. Let β^ be the least squares estimate. Suppose we have some test data (x~1,y~1),...,(x~M,y~M) drawn at random from the same population as the training data. If Rtr(β) = 1/NΣ1,N(yi-β^Txi)^2 and Rte(β) = 1/MΣ1,M(y~i-β^Tx~i)^2, prove that E[Rtr(β^)] <= E[Rte(β^)], where the expectations are over all that is random in each expression.
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