Unbiasedness of sample mean and variance

Unbiasedness of Sample Mean and Variance
Download Document
Showing pages : 1 of 1
This preview has blurred sections. Sign up to view the full version! View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Suppose S = {X1 , . . . , Xn } is a simple random sample from a population with and nite variance 2 < . Show that the sample mean X is an unbiased estimator for , so is sample variance s2 for 2 . 1. Before moving to formal proofs, there are several properties regarding expectation and variance as we metioned in early chapters: E (aX + b) = aE (X ) + b. (1) This property is called linearity of expectation, which naturally implies n E (X1 + . . . + Xn ) = E (X1 ) + E (X2 + . . . + Xn ) = E (X1 ) + E (X2 ) + E (X3 + . . . + Xn ) = E (Xi ). i=1 The other properties are regarding variance, V ar(aX + b) = a2 V ar(X ), V ar(X1 + X2 ) = V ar(X1 ) + V ar(X2 ) + 2Cov (X1 , X2 ). (2) The property given in (2) implies: (a) If X1 , . . . , Xn are independent, then they are uncorrelated, i.e. Cov (Xi , Xj ), i = j . Therefore, n V ar{ n Xi } = i=1 n V ar(Xi ) + i=1 Cov (Xi , Xj ) = V ar(Xi ). i=1 i=j (b) If X1 , . . . , Xn are independent and identically distributed, n n V ar(X ) = V ar{n1 Xi } = n2 nV ar(X1 ) = Xi } = n2 V ar{ i=1 i=1 V ar(X1 ) . n 2. The following equality is helpful in many derivations, n n (Xi a)2 n{X a}2 , a. (Xi X )2 = (3) i=1 i=1 This is because n (Xi X )2 = i=1 Note that n {(Xi a) (X a)}2 = i=1 n i=1 (Xi [(Xi a)2 2(Xi a)(X a) + (X a)2 ]. i=1 a)(X a) = n(X a)(X a), the result in equation (3) follows. 3. There are two equivalent ways of expressiong variance, they are V ar(X1 ) = E {X E (X )}2 = E (X 2 ) {E (X )}2 . Now, n E ( X ) = n 1 E (Xi ) = n1 n = . i=1 E (s2 ) = E n i=1 (Xi X )2 n1 n = 1 E { (Xi X )2 } = n 1 i=1 n i=1 E (Xi )2 nE (X )2 . n1 Note that E (Xi )2 = V ar(Xi ) = 2 and E (X )2 = V ar(X ) = n1 V ar(Xi ) = n1 2 , we then have E (s2 ) = n 2 n n1 2 = 2 . n1 1 ...
View Full Document