Statistics?

A statistic S is said to be an unbiased estimator of the unknown parameter 胃 if E[S]= 胃, otherwise S is said to be biased and the bias (B)= E[S]-胃. Suppose that two independent measurements, m(1) and m(2), are made of the length of a side of a square piece of sheet metal of unknown length, L. The measurements are subject to a measurement error, e(i), that has a mean = 0 and a standard deviation of 蟽. That is m(i) = L + e(i). Two suggestions are made for estimators to use as an estimate the unknown area L^2. The two suggested estimators are: S(1) = [(m(1) + m(2))/2]^2 and S(2) = [(m(1))^2 +(m(2))^2]/2 That is S(1) averages the two measurements and then squares them, whereas S(2) squares the two measurements and then averages them. Which one of the following statements is true? Hint: Recall that Var(x) = E[X^2]-[E[X]]^2 and use the fact that if X and Y are independent E[XY]=E[X]*E[Y]....





1.) Both estimators S(1) and S(2) are unbiased.


2.) Both estimators S(1) and S(2) are biased and|||S(1) = [(m(1) + m(2))/2]^2


S(2) = [(m(1))^2 +(m(2))^2]/2





m(i) = L + e(i)


So E[m(i)] = L + 0 = L





E[m(i)^2] = E[m(i)] *E[m(i)] = L^2





So E[S(1)] = 1/4*E[(m(1) + m(2)]*E[(m(1) + m(2)]


= 1/4 * (2L)*(2L)


= L^2


So S(1) is unbiased





E[S(2)] ={ E[m(1)^2] + E[m(2)^2]} / 2


Now E[m(1)^2] = Var[m(1)] + E[m(1)]^2


= 蟽 + L^2


Similarly E[m(2)^2] = 蟽 + L^2


So E[S(2)] = L^2 + 蟽


This has a bias of 蟽