Question

3. It is known that the test MSE has the U-shape curve, then given a value xo, please show the equation of the test MSE, and prove how you derive this equation

Answer 1

Test Mse ->

Sometimes, a statistical model or estimator must be “tweaked” to get the best possible model or estimator. The MSE criterion is a tradeoff between (squared) bias and variance and is defined as:

“T is a minimum [MSE] estimator of θ if MSE(T, θ) ≤ MSE(T’ θ), where T’ is any alternative estimator of θ.

he equality E(XY)= E(X)E(Y)  holds if XX and YY are independent and their expected values exist, but "only if" is wrong: generally this equality holds if XX and YY are uncorrelated even if they are not independent. For example, let XX be −1−1, 00, or 11 each with equal probability and let Y=X^{2 ,} Y=X² then that equality holds although XX and YY are far from independent.

However, there is no need for that here. What is claimed is this

a constant the same constant

As for E(θ^−E(θ^)) = E(θ^)−E(θ^) , that is like E(θ^−5) = E(θ^)−E(5) = E(θ^)−5 What is done with 55 in this last sequence of equalities is done with another constant in the sequence of equalities that you ask about.

Finally, (E(θ^)−θ)² is the square of the bias, and E((E(θ^)−θ)²) being the expected value of a constant, is the same thing.