Answer to Question #131886 in Statistics and Probability for abhi

Question

Let X1, X2, ..., Xn be a random sample from a normal population with mean zero and variance sigma^2. Construct an unbiased estimator of sigma as a function of Summation i=1 to n (Xi)

Answer 1

X¯ is an unbiased estimator. The MSE of X¯ is MSE_X¯ = E(X¯ − µ) ² = V ar(X¯) = σ²/ n

Similarly, as we showed above, E(S² ) = σ² ,

S² is an unbiased estimator for σ²

MSE of S² is given by

MSE_S² = E(S ² − σ ² ) = V ar(S² ) = (2σ⁴)/(n − 1)

Although many unbiased estimators are also reasonable from the standpoint of MSE, be aware that controlling bias does not guarantee that MSE is controlled. In particular, it is sometimes the case that a trade-off occurs between variance and bias in such a way that a small increase in bias can be traded for a larger decrease in variance, resulting in an improvement in MSE.