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\title{The Bias and Variance of Finite Element Discretizations \\
for FFT-Based Kernel Density Estimation}
\author{Ramesh Natarajan\\
IBM T. J. Watson Research Center \\
                P. O. Box 218 \\
          Yorktown Heights, N.Y. 10598. \\
          e-mail: ramesh@watson.ibm.com}
\date{}
\maketitle
A computationally-efficient procedure for kernel density
estimation has been given by Silverman (1982), with
extensions by Jones and Lotwick (1984).  The use of the FFT
algorithm in this procedure requires the  input data to be
discretized on a regular mesh, which leads to high-frequency
errors in the evaluation of the density estimate and its derivatives.
The smoothing effect of the kernel multiplier typically damps
these high-frequency errors.  Nevertheless,  these errors can
still have significant implications for many statistical
procedures that compute the kernel density estimates as an
intermediate step.  In this paper, we describe finite element
discretization procedures that can be systematically used to
yield higher-order accuracy while retaining the implementation
efficiency of the FFT-based algorithms.  The bias and variance
of the resulting kernel density estimates are derived, which
suggest how these fast computational algorithms can be modified
to yield bias-free estimates. Simulation studies that verify the
results of the theoretical analysis are also presented.

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