Stat 5101 Lecture Notes - School of Statistics

194 Stat 5101 (Geyer) Course Notes7.3.3 MomentsIn this section we calculate moments of sample moments. At first this soundsconfusing, even bizarre, but sample moments are random variables and like anyrandom variables they have moments.Theorem 7.12. If X 1 , ..., X n are identically distributed random variables withmean µ and variance σ 2 , thenE(X n )=µ.(7.21a)If in addition, they are uncorrelated, thenvar(X n )= σ2n .(7.21b)If instead they are samples without replacement from a population of size N,thenvar(X n )= σ2·N−nn N−1 .(7.21c)Note in particular, that because independence implies lack of correlation,(7.21a) and (7.21b) hold in the i. i. d. case.Proof. By the usual rules for linear transformations, E(a + bX) =a+bE(X)and var(a + bX) =b 2 var(X)and(E(X n )= 1 n)n E ∑X ii=1(var(X n )= 1 n)n 2 var ∑X iNow apply Corollary 1 of Theorem 9 of Chapter 4 in Lindgren and (7.11) and(7.13).Theorem 7.13. If X 1 , ..., X n are uncorrelated, identically distributed randomvariables with variance σ 2 , thenandi=1E(V n )= n−1n σ2 , (7.22a)E(S 2 n)=σ 2 .(7.22b)