Large Sample Theory

For a random i.i.d. sample X1 , . . . , Xm assuming nite fourth moments1 1. show that the joint asymptotic distribution of the sample mean Xn : % 2:121 X1- and the sample variance 3% 1 Z?=1(Xi X\")2 is Xn p 1: 02 #3 m((3)_(2))w(o,(#3 , where ,u = EXl, a"? : Var(X1],,u3 = E(X1 1M3, and #4 = E(X1 pf. 2. Use the above results to show that X.\" and 33: are asymptotically independent if the random sample is from a Normal distribution. 3. Find the asymptotic distribution of 5;; [Km the sample coefcient of variation. 4. Find the asymptotic distribution of log 3}