7 Limit properties of kernel estimators of con- ditional mean function. Suppose that with d$_{y}$ = 1 the conditional mean function m ( x ) = E$_{y}$$_{|}$$_{x}$y exists; by (21) it then can be represented as Consider the usual kernel estimator that can also be represented as Then for any continuously differentiable ˜ ψ ( x ) 29 ( ˆ m, ˜ ψ ) = ∫ Σ$_{v}$ ∫ y ˆ f$_{x,y}$ ( x, y ) ψ$_{v}$ ( y ) dy ˆ f$_{x}$ ( x ) ˜ ψ ( x ) dx = − Σ$_{v}$ ∫ ∫ ∂ d$_{x}$ ˆ F$_{x,y}$ ( x, y ) d $_{dy}$[ yψ$_{v}$ ( y )] dy ˆ f$_{x}$ ( x ) ˜ ψ ( x ) dx = − Σ$_{v}$( ˆ m, ˜ ψψ$_{v}$ ) . ∫ y ˆ f$_{x,y}$ ( x, y ) dy ˆ f$_{x}$ ( x ) = Σ$_{v}$ ∫ y ˆ f$_{x,y}$ ( x, y ) ψ$_{v}$ ( y ) dy ˆ f$_{x}$ ( x ) . ˆ m ( x ) = Σ y$_{i}$K ( x$_{i}$ − x h ) Σ K x$_{j}$ − x h , ( m, ψ ) (22) = Σ$_{v}$( − 1) d$_{x}$ +1 ∫ ... ∫ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x ))[ yψ ′ $_{v}$( y ) + ψ$_{v}$ ( y )] dF$_{x}$ ( x ) dy$_{1}$...dy$_{d}$$_{y}$.