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Define a conditional median of Y given X = x to be any median of the conditional distribution of Y given X = x. Suppose that we will get to observe X and then we will need to predict Y. Suppose that we wish to choose our prediction d(X) so as to minimize mean absolute error, E(|Y − d(X)|). Prove that d(x) should be chosen to be a conditional median of Y given X = x. You can modify the proof of Theorem 4.7.3 to handle this case. |
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