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 = aXi +

b, Y

i = cY1 + d for which

a, c > 0. This test rejects when the sample correlation coefficient R is too large.

(ii) 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 3.27(i), to the expression 5.87 given for this density. Putting t = ρr + 1, the second derivative ∂2 log pρ(r)/∂ρ∂r up to a positive factor is



∞

i,j=0 cicj t i+j−2 

(j − i)

2(t − 1) + (i + j)



2



∞

i=0 citi

2 .

To see that the numerator is positive for all t > 0, note that it is greater than 2

∞

i=0 cit i−2 ∞

j=i+1 cj t j 

(j − i)

2

(t − 1) + (i + j)



.

Holding i fixed and using the inequality cj+1 < 1 2 cj , the coefficient of t j in the interior sum is ≥ 0.]