Question

Let X1, X2, ..., X, be a r.s. from P(X), f(x) = (a) Show that X1 is the unbiased estimator for 1. (b) Find îmle for X. (c) Derive the Fisher information I(). (d) Show that Amle is the unbiased estimator (UE) for 1 and Var(AMLE) attains CRLB(= mas). i.e., İmle is minimum variance unbiased estimator (MVUE).

Answer 1

let X., X2, 43, ., tan linda PCX) a) here x~PCA) >> fex) = MAL → E(X) = Znese i § e-dq²-1 (x,-1 ! unbiased X is the estinator for a b) ek lekial) - † fcx, 4) e-nd q{X TX;! log Lef) = ind+ Ex, loga - log 17 TK: → - alo.««,^) co => -- +*=0 22 25 SX is the MLE for a

Now fisher information 1(1) is defined as I (1) = -El 2² log L) 22² l X L = -nats : log L= und + Ex, tog x - log i ti aber keine -+ de tine -** IW= ( ) ► 16 = E(**) - manj let us consider Ünie = {xt = X Cossider Elîmuo) = E(222). 1 (d) Consider MLG să imce is the unbiased estimator 1. ford. To find vC Kime) = V 3*) - na ver.) .. ^= macam ve Times ) = m = L ERLE

sue causa since - CRLB = g(x) de I(1) tan τλ) 9. is MVUE. MLE