Discretized and Interpolated Kernel Density Estimates

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Abstract
The kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator. Nevertheless, for computational (and especially graphical) reasons, exact kernel density estimates are often presented in (piecewise constant) discretized or (piecewise linear) interpolated form, using the exact estimate only at some grid of points. The asymptotic integrated mean squared error properties of these modifications are studied. It is seen that discretization may adversely affect the order of magnitude of this risk criterion, so care needs to be taken in practice to limit the degree of discretization employed. This roughness effect essentially disappears when interpolation is used instead; then, it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate's performance. Data discretization prior to applying the kernel density estimation prescription is also addressed. Such prebinning does not affect the smoothness of the resulting estimate and has a reassuringly minor effect on overall performance, both in conjunction with postbinning schemes and without.

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