Tests for independence in nonparametric regression ( supplement )

Proof of (2.17) From (2.10) we have with high probability for large n and uniformly in x and y √ n(F n (x, y) − ˆ F X (x) ˆ G(y)) ≤ α n x, y + log 2 n n − G(y)α n (x, ∞) − ˆ F X (x) α n ∞, y − log 2 n n + 2C log 2 n √ n , √ n(F n (x, y) − ˆ F X (x) ˆ G(y)) ≥ α n x, y − log 2 n n − G(y)α n (x, ∞) − ˆ F X (x) α n ∞, y + log 2 n n − 2C log 2 n √ n. Set V n,0 = √ n(F n − ˆ F X ˆ G). From (2.12) and Proposition 2.1, we have, using the Skorohod construction for (2.12) (but keeping the same notation), sup x∈D X y∈IR |α n (x, y) − V (x, y)| → 0 a.s. (0.2) and sup x∈D X y∈IR |V n,0 (x, y) − V 0 (x, y)| → 0 a.s. (0.3) Set M (x, y) = F X (x)G(y)(1 − F X (x))(1 − G(y)) andˆM (x, y) = ˆ F X (x) ˆ G(y)(1 − ˆ F X− (x))(1 − ˆ G − (y)). 1 4 be arbitrary and let δ(ε) > 0 be a function of ε to be chosen later on, such that lim ε↓0 δ(ε) = 0. Denote with q 1ε and q 1ε the δ(ε)-th and (1 − δ(ε))-th quantiles of F X , respectively, and with q 2ε , q 2ε the same quantiles of G. Write S ε = (q 1ε , q 1ε) × (q 2ε , q 2ε). We have S ε V 2 n,0 (x, y) ˆ M (x, y) d ˆ F X (x)d ˆ G(y) − S ε V 2 0 (x, y) M (x, y) dF X (x)dG(y)