Testing a correlation coefficient. Let (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate normal distribution.

(i) For testing ρ ≤ ρ0 against ρ > ρ0 there exists a UMP invariant test with respect to the group of all transformations X i = a Xi +

b, Y i = cY1 + d for which

a, c >

0. This test rejects when the sample correlation coefficient R is too large.

(i) The problem of testing ρ = 0 against ρ = 0 remains invariant in addition under the transformation Y i = −Yi, X i = Xi . With respect to the group generated by this transformation and those of (i) there exists a UMP invariant test, with rejection region |R| ≥ C.

[(i): To show that the probability density pρ(r) of R has monotone likelihood ratio, apply the condition of Problem 3.28(i), to the expression given for this density in Problem 5.67. Putting t = ρr + 1, the second derivative ∂2 log pρ(r)/∂ρ∂r up to a positive factor is

∞

i,j=0 ci c j ti+ j−2 $

(j − i)2(t − 1) + (i + j)

%

2

$∞

i=0 ci ti

%2 To see that the numerator is positive for all t > 0, note that it is greater than 2 ∞
i=0 ci t i−2 ∞
j=i+1 c j t j $
(j − i)
2 (t − 1) + (i + j)
%
.
Holding i fixed and using the inequality c j+1 < 1 2 c j , the coefficient of t j in the interior sum is ≥ 0.]