Question

Suppose you have a random sample yi, i = 1, ..., n, from a distribution such that E[yi) = 0 and Var(yi) = 02. - Yi is the sample (i.) Find the asymptotic distribution of ny, where y = average. (ii.) Find the asymptotic distribution of C(GP) = ?. Is C(02) asymptotically pivotal for o2? Explain. (iii.) Using the result in (ii) provide an asymptotic 95% confidence interval for o(if you did not find the asymptotic distribution at point (ii) write the confidence interval in terms of a generic random variable Z, such that Co2

Answer 1

Given ElY;) = 0 and val();) = 0, i=1,2, ..., (in Elinj) = { (in & 9) = 1 / 2 & E (Y) = 0 and ver con y) = var t & 9) = 1 { varly;) = 1 & =(0-2 zot Hence, arzriftotically in g w0(0,02) (ii) : dej nN(0,02) in sný ~ No. 1) o nege w x { chi-square with a deguerof freedom? e clon) ~ X r's p.def is given by flasy = 2 x 1,600 & Clo2) is argeptotically fivstal for 2 as p. d. f. down't defend on 2 (iii) p (Xi, parse(02) < X 10.02s) = 0.95 & P(x1,0.975 < neye < x,0.025) = 0.95 P ( nye kona 172 ) = 0.95 .: as', confidence intowal for resim Galerie s ) 52 2 X 1,0 1025 0 D.975