Question

7.41. Let X. X. ..., X, denote a random sample from a distribution that is N(0, 0). Then Y- X is a complete sufficient statistic for 0. Find the unbiased minimum variance estimator of .

Answer 1

soln 7.41:- Let xx,x₂,.... Xn denote the random sample. from a distribution that is more). Then Y= 5x2 is a complete sufficient statistic for o minimum variance for @² Xn Noo?) nzo,... N-1 2 N-L ô 2 ets xn² hac bras (62) = Elo 2) - 0² EL 0 2] El t xu] var (xn) = 02 E (xn²) - (E (xw))² Elxn²) = 02 E [?] - EL Xn?) to E[xx] et 7 h 02 - 0² bias [ ô2] =0 var cô2] = EL C82) 2] - TE 'Cô 2)] ² ET COP)]= en]?} no - N-1 N-1 ELEEF {en.X} N-1 Mul nzo mzo 1 since an is Gaussian with zero mean and var 02 Elxn.xn xm xm] = E(Xn ] E [xmJ + E [xen am 17 + € [xhamil ) E[ancd] = Elab) El Cd)+ Elac) Elbo) + Elad). Elbe) This relationship basically derived from

Gaussian product process. = L page @. M { [E (x)] + 2 ( E(Xn Xm)] }. who + tak tx} + 2E) TCH This term will be zero mo E [6 27 2] = 02 + + 204 var [62] = 04 + 204 (82) ² = 204 This is the estimator of the variance. so the limit of the var Ilim var t ô 2] = lim 20t - o rience for oz. unit of the variance Proved. # Hence the sample variance estimator is the minimum variance unbiased estimator in the asymptotic sense. Thank you!