15:31 zhitlukhin.pythonanywhere.com Suppose the population distribution is unifo

Question

15:31 zhitlukhin.pythonanywhere.com Suppose the population distribution is uniform on interval [0, a] where a 0 is ar unknown parameter. 1. A sample of a single element X1 is going to be used to estimate a and the estimator = cx, is proposed. What value of c makes this estimator unbiased? 2 (1 point) Correct 2. Suppose a sample of two elements X1, X2 is going to be used to estimate a and the estimator Y2 cXit c2Y2 is pro What values of c, c2 make this estimator unbiased and most efficient? Write them separated by a comma posed 0.5, 0.5 (1 point) Incorrect 3. (*) Again, a sample of two elements X1, X2 is going to be used, but the estimator is Y3 = d . max (Xi, X2) is proposed. What value of d makes this estimator unbiased? 1.5 Correct (1 point) Comment: it can be proved that Ys is more efficient than Y2 (but not required in this problem)

Explanation / Answer

Y = c1 X1 + c2 X2

E(Y) = c1 a/2 + c2 a/2 = a

hence

c1 + c2 = 2

var(Y) =( c1^2 + c2^2) a^2/12

hence

we have to minimize

c1^2 + c2^2 given c1 + c2 = 2

hence

c1 = c2 = 1

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