and by the Cramer-Wold theorem convergence to a limit Gaussian process for ˆ S − 1 / 2 n − → η nh and thus for S − 1 / 2 − → η hn follows. ■ Proof of Theorem 3. Since for a smooth kernel ˆ F ( x, y ) ∈ Φ$_{c}$ by the Lemma the value of the functional for ψ ∈ D (0 , 1) d$_{x}$ , ( ˆ F$_{y}$$_{|}$$_{x}$, ψ ) is the same as ( ˆ F$_{y}$$_{|}$$_{x}$, ˜ ψ ) , with the latter defined by (13) where ˜ ψ = ˆ f$_{x}$ψ ( ˆ F$_{x}$ ) . Thus for any ψ ∈ D (0 , 1) : More concisely it is ( ˆ F$_{y}$$_{|}$$_{x}$, ψ ) = ( ˆ F$_{y}$$_{|}$$_{x}$, ψ ) (26) = ( − 1) d$_{x}$ ∫ 1 n Σ ¯ G ( y − y$_{i}$ h$_{y}$ ) ¯ K ( x$_{i}$ − x h ) ∂ $^{d$_{x}$}$ψ ( Σ ¯ K ( x$_{i}$ − x h )) d ( Σ ¯ K ( x$_{i}$ − x h )) . (27) ( − 1) d$_{x}$ ∫ ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˆ F$_{x}$ ( x ) ) d ( ˆ F$_{x}$ ( x ) ) + ( − 1) d$_{x}$ [ ∫ ˜ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˜ F$_{x}$ ( x ) ) d ( ˜ F$_{x}$ ( x ) ) − ∫ ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˆ F$_{x}$ ( x ) ) d ˆ F$_{x}$ ( x )] . ( − 1) d$_{x}$ [ ∫ ˜ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˜ F$_{x}$ ( x ) ) d ( ˜ F$_{x}$ ( x ) ) − ∫ ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˆ F$_{x}$ ( x ) ) d ˆ F$_{x}$ ( x )] = O$_{p}$ ( h 2 ) . Here ”hat” indicates empirical distribution function and ”tilde” the kernel estimated distribution function. By standard arguments the smooth kernel introduces a bias; by the usual expansions using differentiability of ψ we get that for the second order kernel 37