Question

Assignment 3 Deadline: Wednesday : Nov. 13 in class. Total mark=30 Note : Please answer the following questions. You must give complete details in your solutions and clearly justify your final answer. and 1. Consider the random sample X1, X2, X3, X, from a population with mean standard deviation o. Consider the following two estimators for p A - 2X,-XA = (5/8) X, + (1/8)X2 + (1/8)X, + (1/8)X, (a) (2) Are they unbiased estimators of ? (Why?) (b) (2) Which of the two estimators is best for estimating u? (Why?)

Answer 1

(a) E(1) = E(2X1 - X4) = 2E(X1) - E(X4) = 24 - H = 1 (since E(X= , i = 1,2,3,4) E (2) = F (Ex1 + x2 + 3x3 + 5x1) = E(X1) + E(X3) + {E(X3) + F(X) (since E(X) = H, i = 1, 2, 3, 4) Hence h1 and 12 are unbiased estimators of H.

(6) Varî1) = E(2X1-X4) = 4Var (X1)+Var (X4) (since X1 and X4 are independent) = 40² + o² = 50° (since Var(Xi) = o?, i = 1, 2, 3, 4) Var() = Var (Ex1 +3x2 + 3x3 +şxa) - Var(xv) toy Var(83) + V ar (Xa) + Var(xs) (Since XẮs are independent) 25 +1+1+1 –02 (since Var (Xi) = o?, i=1,2,3,4) 64 _702 - 16 Since Var(hz) < Var(1) and both are unbiased estimators of u so ü2 is best estimator for estimating u.uz are unbiased estimators of u.