Question

Question 4 15 marks] The random variables X1, ... , Xn random variables with common pdf independent and identically distributed are 0 E fx (x;01) 0 independent of the random variables Y^,..., Y, which and are indepen are dent and identically distributed random variables with common pdf 0 fy (y; 02) 0 (a) Show that the MLE8 of 01 and 02 are 1 = X i=1 Y (b) Show that the MLE of 0 when 01 = 0, = 0 is ? = (X+Y)/2 (c) Let g (x)loga2 log (x 1)- 2log 2 (i) Show that (ax) 0 if and only if = 1, and g (x) 2 0; = Show that 11 g (1/a) g (x); (iii) Hence show that ax; g (ax)cx;x > c2}U{x;x < 1/c2}, for suitable c and c2 (d) Let T X/Y. Show that the likelihood ratio test of Ho : 0, = 02 vs HA 0102 rejects Ho when T> k or T<1/k, for suitable k (e) What is the exact distribution of T under Ho?

Answer 1

be d rvs wth cmm m pdf #() e We, 10.W LIkeliheod iS l0412) f(i, 0) 0 NPI When 0 Yiein 0.W (0112) mantimum .rt iff nle,12 is matimum N.T.t. inle1x)=mne d Ln le))- if to if Hence Anle1 ) when 8h iS manimum re. mle of 6, 15 n i Smilaly, mle of 92 b) When &-02-0, Ne can trat X,x as a rs. of Se 2n from the dist Hith Pef -8 a (8) 0. W C Xi t27; 1 Similarly, mle g 0 22 2n

be d rvs wth cmm m pdf #() e We, 10.W LIkeliheod iS l0412) f(i, 0) 0 NPI When 0 Yiein 0.W (0112) mantimum .rt iff nle,12 is matimum N.T.t. inle1x)=mne d Ln le))- if to if Hence Anle1 ) when 8h iS manimum re. mle of 6, 15 n i Smilaly, mle of 92 b) When &-02-0, Ne can trat X,x as a rs. of Se 2n from the dist Hith Pef -8 a (8) 0. W C Xi t27; 1 Similarly, mle g 0 22 2n

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As, a)-0 1<X A0(), ( X A Z2 (<1, H 741 2 1 OY tis olear thet ()-c)x- or f7 있이>아 =자 자약면지하 r- and (e70, e27) for Suitab le ei & ca d) Te Likelihrod ratio test statistic for H: 8 02 Y H:to 오 (이즈,2) Sup lomlel,) Imle mle mie Oomle imle (Bmle Cpente YBe-63) Rejeat Ho if a(,2) <e 1-R

.x. 2 1e. 4 2x < c' 2ll)-l2 7 e le ie glt)c"+ ¥5 ,6he f g a as defined before Reje ct Ho nhen TYK ar TS fr Suitable K c) under Ho : G82-8, te 2 2/ T- AS 2xiEnp(man- ) imdependently 2 Exp mean=2) X2 Y; id 2n 2n T under Ho ty/2n 2n