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Optimal Choice of the Kernel Function for the Parzen Kernel-type Density Estimators

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Title: Optimal Choice of the Kernel Function for the Parzen Kernel-type Density Estimators
Author: Kazakos, D.
Type: Tech Report
Keywords: kernel; parzen
Citation: D. Kazakos, "Optimal Choice of the Kernel Function for the Parzen Kernel-type Density Estimators," Rice University ECE Technical Report, no. TR7501, 1975.
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>(- &#8734; ,+ &#8734;), for p &#8712; [1, + &#8734;]. 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 Kernel-type 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> [1-T<sup>-a</sup> |y|<sup>a</sup>], for |y|<T where [-T,T] is the support interval, and a=2-p<sup>-1</sup>.
Date Published: 1975-04-20
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  • ECE Publications [1028 items]
    Publications by Rice University Electrical and Computer Engineering faculty and graduate students