Suppose (X1, Y1),... (Xn, Yn) are i.i.d. bivariate observations in the plane, and let denote the

Suppose (X1, Y1),... (Xn, Yn) are i.i.d. bivariate observations in the plane, and let ρ denote the correlation between X1 and Y1. Let ˆρn be the sample correlation

ρˆn =

(Xi − X¯n)(Yi − Y¯n)

[



i(Xi − X¯n)2 

j (Xj − Y¯n)2]

2 .

(i) For testing independence of Xi and Yi, construct a randomization test based on the test statistic Tn = n1/2|ρˆn| .

(ii) For testing ρ = 0 versus ρ > 0 based on the test statistic ˆρn, determine the limit behavior of the randomization distribution when the underlying population is bivariate Gaussian with correlation ρ = 0. Determine the limiting power of the randomization test under local alternatives ρ = hn−1/2. Argue that the randomization test and the optimal UMPU test (5.75) are asymptotically equivalent in the sense of Problem 13.24.

(iii) Investigate what happens if the underlying distribution has correlation 0, but Xi and Yi are dependent.

Section 15.3