Question

Let Yı, Y2, Ys, and Y4 be independent, identically distributed random variables from a mean u and a variance 02. Consider a different estimator of u: W=Y+Y2+2Y3+ Y 00 This is an example of a weighted average of the Y a) Show that W is a linear estimator. b) Is W an unbiased estimator of u? Show that it is - or it isn't (E(W) = Find the variance of W and compare it to the variance of the sample mean d) Is W as good an estimator as Y? Explain your answer.

Answer 1

Let Y₁,Y₂,Y₃ and Y₄ be independent, identically distributed random variables from a population

Let Y,, 42, 43 and Y4 be independent, identically distributed random Variables from a population with a mean n and a variance a2 consider a different estimator of me W = 1/3 4 + 1/3 3 Y ₂ + 1 / 2 Y 3 + 3 3 3 74 This is an example of a weighted average of the Y: (a) show that w is a linear estimator (b) Is W an unbiased estimator of el? show that it is aor it isn't (E(W) = ?) (C) Find the variance of w and compare it to the variance of the sample mean ý. (d) is wo as good an estimator as ý? Explain your answer. Answers (a) Since w is a linear combination of YI, Y2, 43, 44 so w is alinear estimator of o

b) E(W)= (4.+E (42) + € (Y3) + F(44) (since Ely:)-0, 1:1,2,3,4) . Hence w is unbiased estimator of o variw) - by War (+ var (Y2) + uz vas (43) I 64 + 82 Va I 1 +I 36 +9 64 288 = 850th (since var (Y; ) = 02, i=1,2,3,4 and y's are independent) var (y)van ( . * ,+x+YY) - - Ž van (lt) (since his are independent) 288 Hence varlý)< van (w). (d, since van (y)<var (w) So w is not better estimaton than 7.