Abstract:

Let W<sub>p</sub><sup>(2)</sup> be the Sobolev space of probability density functions f(X) whose first derivative is absolutely continuous and whose second derivative is in L<sub>p</sub>( ∞ ,+ ∞), for p ∈ [1, + ∞]. Using an upper bound to the mean square error for a fixed X E [f(X)  f<sub>n</sub>(X)0]<sup>2</sup>, found by G. Wahba, where f<sub>n</sub>(X) is the Parzen Kerneltype estimate of f(X), we find the finite support Kernel function K(X) that minimizes the said upper bound. The optimal Kernel funciton is: K(y) = (1+a<sup>1</sup>) (2T)<sup>1</sup> [1T<sup>a</sup> y<sup>a</sup>], for y<T where [T,T] is the support interval, and a=2p<sup>1</sup>. 