Suppose that instead of estimators of the form 𝑎𝑋+𝑒 , we consider estimators of the form Θˆ=𝑎𝑋 and ask for the value of 𝑎 that minimizes the mean squared error. Mimic the derivation you have just seen and find the optimal value of 𝑎 . Your answer should be an algebraic expression involving some of the constants 𝑏 , 𝑐 , 𝑑 , where 𝑏=𝐄[Θ2] , 𝑐=𝐄[Θ𝑋] , 𝑑=𝐄[𝑋2] .

Question

Suppose that instead of estimators of the form  𝑎𝑋+𝑒 , we consider estimators of the form  Θˆ=𝑎𝑋  and ask for the value of  𝑎  that minimizes the mean squared error. Mimic the derivation you have just seen and find the optimal value of  𝑎 . Your answer should be an algebraic expression involving some of the constants  𝑏 ,  𝑐 ,  𝑑 , where  𝑏=𝐄[Θ2] ,  𝑐=𝐄[Θ𝑋] ,  𝑑=𝐄[𝑋2] .

PS-VP · Accepted Answer

c/d

Explain Bot · Answer

To find the optimal value of 𝑎 that minimizes the mean squared error for the estimator Θ̂ = 𝑎𝑋, 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[(Θ̂ - Θ)^2] Let's substitute Θ̂ = 𝑎𝑋 into the MSE formula: MSE = E[(𝑎𝑋 - Θ)^2] Expanding the square: MSE = E[(𝑎^2𝑋^2 - 2𝑎𝑋Θ + Θ^2)] Now, let's simplify this expression. We can split it into three terms: MSE = 𝑎^2E[𝑋^2] - 2𝑎E[𝑋Θ] + E[Θ^2] We know that 𝑏 = Ε[Θ^2], 𝑐 = Ε[Θ𝑋], and 𝑑 = Ε[𝑋^2]. Substituting these values into the MSE equation: MSE = 𝑎^2𝑑 - 2𝑎𝑐 + 𝑏 Now, we want to find the value of 𝑎 that minimizes the MSE. To do that, we take the derivative of the MSE with respect to 𝑎 and set it equal to zero: d(MSE)/d𝑎 = 2𝑎𝑑 - 2𝑐 = 0 Solving this equation for 𝑎: 2𝑎𝑑 - 2𝑐 = 0 2𝑎𝑑 = 2𝑐 𝑎 = 𝑐/𝑑 Therefore, the optimal value of 𝑎 that minimizes the mean squared error is given by 𝑎 = 𝑐/𝑑, where 𝑏 = Ε[Θ^2], 𝑐 = Ε[Θ𝑋], and 𝑑 = Ε[𝑋^2].