STATISTICS

Qustrom 2 Refer to oustion , above, let a=1 and that it is destined to test Ho : P = 1 Versus HI: RF I at 2 = 0. 05 ( a) Maximum Likelihood gotimates of B : ( i) under Ho and (is ) under Hi. Mote that the likelihood and the lay likelihood functions of B are? L ( 8(x ) = -on = i = 1 1= 1 Bm e e 1 = 1 and I (8 ) = - meB - + ( i) under Ho the malimum likelihood slimator of B is B= 01 ( is ) under HI C (B) = _ m + +, 2 xi = 0 B = 1 . 2 X x = X 2 XN ( b ) under Ho: B= 1, L (ilx ) = e 1 = 1 under " HI : B + 1 , L celx ) = Anee A - M Xe B e m Now the likelihood natwo is 2 (x) - LCI(X ) e Z L ( 8 / X ) - on e You) = on exp [ - Exit Exit 8 - 1 ) Note that p = X = + Z Xi = 8 m Exi = me, then M AIL x ( x ) = Bm up [me ( # -1)] 7 (xx) = on exp [m ( 1 - B)] -: Proved.QUESTION 1 Suppose that X1, X2, ..., Xn is a random sample from a distribution with probability density function: 1 f(xla, B) = I(a) Ba x 3 if a > 0, 8 > 0 and x > 0; 0 otherwise.QUESTION 2 [211 Refer to QUESTION 1 above. Suppose that a = l and that it is desired to test Hg : ,6 = 1 versus H. : 3:5 l at the 0.05 level of signicance. (a) What are the maximum likelihood estimates of}? : (i) under H\"; and (ii) under HI? {4) to) Evaluate the likelihood functions under H0 and H1, and hence show that the likelihood ratio test statistic for the hypotheses is: Mir.) : Fin cxp{n(1 ~ 3)} where I? is the maximum likelihood estimate of [i found in part a(ii) under H]. {E} (c) Completely specify the decision rule of the likelihood ratio test in terms of 3(1) in part (b). {3) (d) Sketch a graph which shows that the decision rule in part (c) is equivalent to rejecting Hg if either _ b where a