10. Testing a correlation coefficient. Let (Xl' yl)•... '(Xn , y") be a sample from a bivariate normal distribution. (i) For testing P S Po against P > Po there exists a UMP invariant test with respect to the group of all transformations X[ = aX; +

b, Y;' = CY; + d for which

a. C > O. This test rejects when the sample correlation coefficient R is too large. (ii) The problem of testing P = 0 against p=;'O remains invariant in addition under the transformation Y;' = - Y;, X[ = X;. With respect to the group generated by this transformation and those of (i) there exists a UMP invariant test. with rejection region IRI

c. [(i) : To show that the probability density pp(r) of R has monotone likelihood ratio, apply the condition of Chapter 3. Problem 8(i), to the expression (88) given for this density in Chapter 5. Putting t = pr + 1, the second derivative a2 10g pp(r)/ap Br up to a positive factor is 00L c;cjt;+j- 2[U - i)2(t - 1) + (i + j)] i.j-O 2[ .f c;t;]2 ,-0 To see that the numerator is positive for all t > 0, note that it is greater than 00 00 2Lc/-2 L cjt j[U - i)2( t - 1) + ( i + j )]. ; - 0 j-;+l Holding i fixed and using the inequality cj + 1 < tCj' the coefficient of t! in the interior sum is 0.]