ABSTRACT
Proceeding exactly in the same way as in Theorem 8.7.3 ( being the complement of C) we can get
where A is a constant. To prove part (b), consider a family of regions given by
These regions are either intervals or complements of intervals. When 0<a< b, R(a, b) b) is a finite interval not including zero (excluding the trivial extreme case). Consider a random variable Y such that Y/s (s>0) is distributed as the ratio of
independent random variables. Let It can be shown by differentiation that ß(d)>ß(1) if d lies in the open interval with endpoints d, 1. Define a random variable Z by
where k=1, 2, and suppose that ?2=···=?p=1. Then the distribution of Z is independent of ?1 and is independent of the first factor in the right-hand side of (8.209). From Exercise 5b the power of the rejection regions C(a, b) is less than its size if ?1 lies strictly between d and 1. Q.E.D.
