INFERENCE STATISTICS

\fQUESTION 2 Refer to QUESTION 1. Suppose that o' = 1 and that it is desired to test: Ho : ( = 0 versus H1 : # # 0 at the o E (0, 1) level of significance. (a) What are the maximum likelihood estimates of u : (i) under Ho; and (i) under Hi? (3) (b) Evaluate the likelihood functions under Ho and H1, and hence show that the likelihood ratio test statistic for the hypotheses is: A(x) = exp ixi 121 = exp HE 1= 1 i=1 i=1 where a is the maximum likelihood estimate found in part a(i) under H1. (8) (c) Show that the likelihood ratio test rejects Ho if p? ) 12 > ca where ca isatisfies a = P ( "? [ ? > call = 0 ). i=1 (3) (d) Prove or disprove that the distribution of the test statistic in part (c) is x under Ho. (4)