Question

6. Let ?1, ?2, ..., ?? be a random sample from a Normal population (?, ?2).

Hint: For exponential families we have that E [∂ ln f (x | θ)] 2 = −E [∂2 ln f (x | θ)] θ ∂θ θ ∂θ2

c) Does s2 reach the C-R level? Justify

n-1 6. Sea X1, X2, ..., Xn una muestra aleatoria de una población Normal(u, o?). a) Demostrar que s? ET-;(x1-1) Fes un estimador insesgado de oz. b) Encontrar la cota de Crammer-Rao (C-R) para cualquier estimador insesgado de oz. Hint: Para familias exponenciales se tiene que E Co in f(x|o)]* = -Eo Cook In f(x)0)] c) s alcanza la cota de C-R? Justifique.

Answer 1

2 Å Elx, M) I! is, We Solution are given a Sandom sample xn from N(H, 02) 2 E (X; - E(X)) n X., X2, the pelo il Then vexi) % x; i's Exi ca) mai (23-1)? n 2 e o au 1-0<X;< @ 2 unbiased “Д < A< o 70 s is not an 2 =) estimator of 2 because E(s) to E (X:) H and v(xi) = 02 . 2 as Let 3(x; -m)? En S $ ] nai nal 2 2 is n an unbiased estimator S ¿Q :-) E(5²) = 60 all 2 - of

(b) the likelihood funebion i's She Fisher Infoemation is given La f (a! HO?) by (X:-1)? I (02)=-=[ 02 Log L zo Flatou e 2o4 G {1; 23 2 log. L 2 in logo " - n logan rog 25 = 2(2-4) 6 264 2 202 ini 2. dloga A(x:-H) ? t. 204 - an tise(2; & E(2-4) 204 ob ao? 2 202 n 11 2 n 2²log L 2 {(xi-H² tn V(x;) n 4 ob 204 6 a dot n 2 = + Cart - 8 (x:xH) 204 204 ial 06

an + 264 04 n 11 - n 264 Let 7(x) 그 n 2 5 is an nal Unbiased estimator of 2 Then the Cramera Rao lower bound for o 2 is 2 Var (769) > Choir U)] 2

Vair(n-1) 52 in n/04 a 204 (0) Since X; noNlH,a2) then Xi H N(o,) 2 2 (xi -1) - 7 (1) n (8-H) -24 G 2 ie (ni)s? 2 ~) 7 2 n o ) = 21 2 (n-1)" vairs 2 ) an 2 4 0-4 Vou (s) and --- U) 2

Since Val (8²) & Co Isusel brand So the estimatol ? does not reach c R level. *