Question

3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in a rural county follows Beta distribution with probability density function (pdf) fx (20) = 0.00-1 where 0 < x < 1 and parameter 6 > 0. Let X1, ..., X, be a random sample of size n from a population of rural counties. Researchers intend to make statistical inference on the parameter 6 using collected data X1, ..., (a) Let Y; = – log(Xi) for i = 1,..., n. Show that Y; follows exponential distribution with pdf fy(y) = 0 exp(-oy), 0<y< 0,0 >0. (b) Given the fact that Z = - Y; follows Gamma distribution with pdf om fz(2) = T(n) -zn-1 exp(-02), 0<x<00, show that (n - 1)2-1 is an unbiased estimator of e. (c) Derive the Crámer-Rao lower bound (CRLB) for unbiased estimators of 0, and argue whether the variance of (n-1)2-1 attains the lower bound. (d) Find the maximum likelihood estimator of , and derive its expectation Elê) and vari- ance VÔ). Justify whether ê or (n − 1)2-1 has a smaller mean squared error (MSE). (e) Find either exact or asymptotic (1 - a) confidence interval (CI) for 0. Points: (a) 5, (b) 5, (C) 5, (d) 5, (e) 5

Answer 1

Solution: 3 Let xinxar sample of size w ., Xu be a random a distribution from with puf a ? Q-1 Οχ OLX</ Fx 640) where (a 070 is unknocon Let Yi = - Log Xi for 1=1,2,. ., y=- Logh > x= Thena X= 4 da és - dy Thus, the pdf of Yas, Fab) = FgQc) lyn yoty y a @ d yo i o e

И Hencen FGE yzo 070 Thus, the Yi follows exponential distribution with paiame le co We knoco that, za Yi follows Gamma distribution with pay zzo n b rej as ? И | oh OZ f(2)= Z e

Theun E (2_1] = (0-12 Žf (2) dz (n-1) on n-2-on da = Z In-1 a .(n-1)on (M-111n1 oni © Hence, E[(n-1)2-1] = 0 Thus, (n-1)771 is an unbiased estimator of o.

© the likelihood funcha force as ? И L(x) = T1, f(x) i=1 X - T Ox nu 0-1 a 6 T Xi in oro as The loglikelihood famelion for a l(a) = log nlogo +(6-1, logre Elogne CON 드 и И dl(a) = Then, do dlo) O n - de 2 02

Lot I @be Asher uformation, Then - I (C) = EL d²l(0) do2 n 6 - Then Grames Rao love a bound ass CRLB I @) CRLB = a @2 - © И/

we have E ( 2 = -1 E (2/2) = con Sa n- 3.02 e dr in 04 こ [n-2 on-a co2 = (n-1)(na) 02 62 こ Then Var (2) = E() - (E(=)]² (n-1)(n-2) (n-172 (n.) 02 M-152 (1-2) 02. (n-1 (na こ

Then, var a (12) (n-172 var (2) - - a (n-1)(n-1) .: Var ((n-1)==] 02 (1-2) Pom equalian 0 and equation qesh The variance of (n=1121 does not attains the louder bound. We

d we have a И. d d (C) И = de + loga, For the MLE of or we get dl (0) do M + slo logy, ao n И И Elogxi ñ (9 И e in a Z И @ کے h 2 Hencer the MLE of to as, @ = 2

Then a a E (2) = n E (²) no n-1 vas (o a var (2) 2 (th = n2 var n262 (5-12 (5-2) Hence ECâ) = N202 (M-172 (1-2) no n-1 and varē) -

2 var (12) + (bias (131] MSE (721) com + to MSE (131) - Mona -(0) bias (@) = E@)- - nco n-1 no- no to I (n-1) n-1 Msel@): vaale) + bias(8,72 02 2262 t n-1;2(2-2)(n-1, n² -12 02 +} 2-2

@2. (n²+1-2) (n-1)2 (2-2) = (1249-2)602 (ii) MSE (@) = (n-1)=(1-2) from equation (i ) and equation (1) we got

(22+2-2) 02/02 MS ECO MS E{{n-1127] - (0-10-23 102 (5272-2) > 2 (n-1)2 MSE(6) > Ms E( (naz-]-61 Bom equalien (), are good may (1-1721 has a smaller MSE ther cê