Question

Question.1 (11 Marks] Let X be a random variable with the following pdf: f(x; 8) 1>0, >0. 1 (1) (a) Show that I f(x;0)d.r = 1 (b) is the pdf f(c;) member of the Exponential family distributions? Justify in details your answer (c) Find a sufficient statistic for the unknown parameter 8. (d) Find a maximum likelihood estimator for 6.

Answer 1

oth Answer - у к». ) с hp hah oh (ве 0), ә (40) 10 dx=0 dy КО-Х. - ҒУ txe z de f(0,0) dal = 1 (1,0) = ғ) »!) os xeze (0) овок 7 T Question

(6) The pdf. of one - Parameter exponential family is written as, 010).T(x) $(1,0) = c(0). hſx). e Given that, *1940) 1 x (%) » ; a>0 . 1 Here clo) C(O)= z ihra)= x; 010) = ; 7(4) = x Therefore it is a member of Exponential family o di kelihood function, L= {(u', 0) = ial n n L= Com и 여 o ie 1-(4)? 6-*** ()

nu 2 zn M a(t,0)= (6) SWI и Iя, я, - Comparing it with Neyman factorization Theorem . L=GCT, O). Hall 2n comparing 0 & ②, we have (where there -- Mo) = π ou Xi is sufficient for /Answer For sample sized re for n=1, T = X is sufficient for a (d) dikelihood frenetien, L= (0)** log L=-2n dogo the log Xi - het te a logL +6 +5 x Thus T= n ДХ 2n "Xe + slo 2 a o

a lo8L and & DogL 2 loge eo ao For M. L. E. of o, we must have a lose =0 gives - 20 / + 2 h et le Xi 2 n ie Exe yo ie Xi an M. L.E of a Exi an izl Ausna