(c). K ( w ) is an l − th order kernel: for w = ( w$_{1}$, ...w$_{k}$ ) the integral 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 6 ̂ F ( x ) = 1 n n ∑ i =1 ¯ K ( x$_{i}$ − x h ) (4) ∫ w j 1 1 ...w j$_{k}$ k K ( w ) dw$_{1}$$_{...dw$_{k}$}$    = 0 if j$_{1}$ + ... + j$_{k}$ < l ; < ∞ if j$_{1}$ + ... + j$_{k}$ = l .