INFERENCE STATISTICS

QUESTION 1 [32] 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, B > 0 and x > 0; 0 otherwise.QUESTION 2 Refer to QUESTION 1 above. Suppose that a : 1 and that it is desired to test Hg : ,8 : 1 versus H1 : 1'3 i 1 at the {105 level of significance. {a} What are the maximum likelihood estimates of it : (i) under Hg] and {ii} under H1? [4} {b} Evaluate the likelihood functions under H0 and H1, and hence show that the likelihood ratio test statistic for the hypotheses is: iixi = 3\" exp {m1 3131} where 3 is the maximum likelihood estimate of ,8 found in part a[ii) under H1. [6} {c} Completely specify the decision rule of the likelihood ratio test in terms of iii} in part {b}. [3} {d} Sketcha graphwhich shows that the decision rule in part [c] is equivalent to rejecting Hg if either}? if a or}? b where a <.: b are the solutions for in inequality prove or disprove that distribution of is fin under hr hint: what x hi>