Question

22. Suppose that X1, X2,...,x, Fp, where F, is a discrete distribution with probability mass function p(x) = p(1– p)* for x=0,1,2,.... (See Example 7.3.9.) The MLE Ộ=1/(1+X) has the asymptotic distribution below: VnCô–p) DY~N(0,(1 – p)p?). (8.54) Use the normal distribution in (8.54) to obtain, via a variance stabilizing transformation, an approximate 100(1 – a)% confidence interval for p.

Answer 1

as a Solution: 22 Suppose that X,Xar.... In be a random sample from a distribuliten with puut P(x) = pl-pl ; x = 0,1,2,- Tee likelihood function of pas L(PG) - t P(x4) as ? ic, И. Xi = TP (1-P) il 볼 de ph (1-P) Fi The loglikelihood funelion of pas Logl (P(x) = n logp + Mi log(1-P)

u xe Leifferentiate wit pruegen. a logu (PA) = 1 For MLE of Pr. We des dloge 6(x) 1-Р =0 de. n P S4 =) - 6 1 — Р И. di = sle te n -> I-P X I-P dh = X = 1

1-P = px 7. Itx Itx Exi 3 p+pX=1 8 = 1th Hence the MLE of pas, alog L(P(x) = de (1-P) q²log L(P(x) = -2 dez (1-p72 Them Fisher information on pas? dtog L(P(x) we have a И Eli И P p2 [ ICP) = El- of rog let

E(4) > I (P2= nt p2 (1-p)? n t & I(P) = N (1-P) p(1-P)2 p2 n > I(P) - P(1-P) Pz + 3) p = n(1-P+P) p2(1-P) n I (P) = p2 (1-P) Then the asymptolae Distribulia) of ß = its as how N(P. Je ên N(P, P2C1-P) In ( P-p) uN (0, (1-0)p?) J

We have a p þ v N(P, PZ (1-1) Thein, an approximale 100(1-29% confidence inlisval for pass ệ t za PVI-B where he Thy and ZwN(0,1) vn