Question

Let f(X⃗ ) be some estimator, and let y be the “true” value that f(X⃗ ) is estimating. For example, X⃗ might be a vector of n iid random numbers with mean µ, while f(X⃗ ) is the sample mean. In this case, y = µ.

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Problem 1 In lecture, we saw that there is a trade-off between the bias and variance of a model. This problem will make that trade-off precise. Let X - (Xi,.....Xn) be a vector of random numbers. Let f(X) be some estimator, and let y be the "true" value that f(X) is estimating. For example, X might be a vector of n iid random numbers with mean , while f(X) is the sample mean. In this case, y p The squared error (f(X) )s often used to quantify how good the estimator is. Show that the expected squared error can be written as: variance bias2 Hints: 1) X is a random vector, but in this problem you can essentially treat it as a single random number 2) Let a be some number. The quantity (f(X) - a+a u)2 is the same as (f(X) - y)2; we've just added zero. But by picking a carefully, you can expand the square to get something more desirable.

Answer 1