ABSTRACT
From (6.32), since ? is a diagonal matrix, Sii/?i, i=1,…, p are independently
distributed random variables and for any and the Sii (or any function thereof) are mutually independent. Furthermore, the distribution of
is independent of ?1,…, ?p. From Exercise 5 it follows that there
exists a constant such that and
irrespective of the value of C chosen. Hence if we evaluate the probability with
Thus the acceptance region ? of the likelihood ratio test satisfies P(? |H1)- P(? |H0) >0. Hence the likelihood ratio test is biased. From Exercise 5 it follows that if 2r=m, then ß(?) increases as |?- 1| increases. Hence from the fact noted in connection with the proof of (i) we get the proof of (ii). Q.E.D.
