Suppose that instead of estimators of the form đđ+đ , we consider estimators of the form ÎË=đđ and ask for the value of đ that minimizes the mean squared error. Mimic the derivation you have just seen and find the optimal value of đ . Your answer should be an algebraic expression involving some of the constants đ , đ , đ , where đ=đ[Î2] , đ=đ[Îđ] , đ=đ[đ2] .
c/d
To find the optimal value of đ that minimizes the mean squared error for the estimator ÎĚ = đđ, we need to minimize the mean squared error function.
The mean squared error (MSE) is defined as the average squared difference between the estimator and the true value. In this case, the MSE is given by:
MSE = E[(ÎĚ - Î)^2]
Let's substitute ÎĚ = đđ into the MSE formula:
MSE = E[(đđ - Î)^2]
Expanding the square:
MSE = E[(đ^2đ^2 - 2đđÎ + Î^2)]
Now, let's simplify this expression. We can split it into three terms:
MSE = đ^2E[đ^2] - 2đE[đÎ] + E[Î^2]
We know that đ = Î[Î^2], đ = Î[Îđ], and đ = Î[đ^2]. Substituting these values into the MSE equation:
MSE = đ^2đ - 2đđ + đ
Now, we want to find the value of đ that minimizes the MSE. To do that, we take the derivative of the MSE with respect to đ and set it equal to zero:
d(MSE)/dđ = 2đđ - 2đ = 0
Solving this equation for đ:
2đđ - 2đ = 0
2đđ = 2đ
đ = đ/đ
Therefore, the optimal value of đ that minimizes the mean squared error is given by đ = đ/đ, where đ = Î[Î^2], đ = Î[Îđ], and đ = Î[đ^2].