RANK TESTS OF PARTIAL CORRELATION

Rank statistics to test the null hypothesis that $ X $ and $ Y $ are conditionally, given $ Z $, independent are given and their asymptotic properties are investigated under the model $ (X, Y, Z) = (U + a_nW, V + b_nW, W) $ where $ (U, V) $ and $ W $ are independent. It is shown that linear rank tests given by $ (X, Y) $ based on the random sample of size $ n $ are asymptotically distribution-free when $ (a_n,b_n) = n^{-1/2}(a,b). It is also shown that Spearman\u27s coefficient of rank correlation and Kendall\u27s coefficient of rank correlation given by $ (X - hat{a}Z, Y-o^^hat{b}Z) are asymptotically distribution-free when $ (a_n,b_n)=(a,b) $ where $ (hat{a},hat{b}) $is some consistent estimator of $ (a,b) $