content_image/317371.jpg content_image/317370.jpg
(c). K ( w ) is an l − th order kernel: for w = ( w$_{1}$, ...w$_{k}$ ) the integral 

<img src='content_image/317371.jpg'>

The finite support and boundedness assumptions can be relaxed and are introduced to simplify assumptions and derivations; K is not restricted to be symmetric or non-negative. 

Denote by ¯ K the integral of the kernel function, then 

is an estimator of the distribution function, F ( x ) . The properties of these estimators depend on K and h and are well established (Azzalini, 1981). Generally for h → 0 as n → ∞ with nh → ∞ , ̂ F ( x ) is a root-n consistent and asymptotically Gaussian estimator of F ( x ) at any point of continuity; the uniform norm of the difference, sup ∣ ̂ F ( x ) − F ( x ) ∣ , converges to zero. 

Known convergence properties of ̂ f ( x ) are more complicated; they rely on assumptions about the existence and smoothness of the density, f ( x ); the convergence rate is slower than root-n and depends on the order of the kernel and the rate of the bandwidth h → 0 (Pagan and Ullah, 1999). As shown in Examples 3-5 in Zinde-Walsh (2008), the estimator ̂ f ( x ) fails to converge pointwise if the distribution is not absolutely continuous (e.g. at a mass point or for a fractal measure); of course, in those cases density itself 

<img src='content_image/317370.jpg'>

