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In most places where I've looked, it generally says that the Epanechnikov kernel is optimal for kernel density estimation (KDE), in the sense that it minimizes the mean integrated squared error (MISE). See for instance Wikipedia, which indicates that the uniform kernel has 92.9% the efficiency of the Epanechnikov kernel.

Yet, in the paper "Swanepoel, J. W. H. (1988). Mean intergrated squared error properties and optimal kernels when estimating a diatribution function" it concludes that either a uniform kernel or an exponential kernel (depending on some conditions detailed in the paper) minimize the MISE, and are therefore the optimal ones.

What explains this discrepancy?

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Epanechnikov kernel is quit famous along with Gaussian, etc and one of those of which are easily available in different soft wares. Actually,every kernel performs good in different situations. It just defined the shape. Most important thing in this manner is Bandwidth.

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I know this question is old, but notice that the second paper is considering estimation of the cumulative distribution function, not the density function.

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