Econometrics Seminar - Shin Kanaya (Essex)
| Dates: | 30 November 2023 |
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| Times: | 12:00 - 13:00 |
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| What is it: | Seminar |
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| Organiser: | School of Social Sciences |
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Tittle: Optimal kernels for density estimation revisited
Shin Kanaya (University of Essex) and Yuta Okamoto (Kyoto University)
Abstract: In this paper, we revisit the optimality of kernel functions in probability density function estimation. Gasser, Muller, and Mammitzsch (1985, the Journal of the Royal Statistical Society: Series B) present the so-called optimal kernels that minimize the asymptotic (integrated) mean-squared error (AMSE) of the kernel density estimator (KDE): i.e., the Epanechnikov kernel for the second order case and polynomial-function based ones for the higher order cases. These classical optimal kernels are derived using 1) a restriction on the number of sign changes (of an optimal solution/kernel function) and 2) the Taylor-expansion based (integrated) AMSE. We discuss that there are not necessarily good theoretical grounds for the former restriction and the use of the Taylor-expansion approximation may not necessarily capture bias-variance properties of the KDE in finite samples, resulting in inadmissibility of the classical kernels (with respect to the exact integrated MSE). We derive and present a class of kernel functions that are optimal with respect to the Fourier-transformation (FT) based MSE expression (which can better capture finite-sample and higher-order properties of the KDE), and then show that these optimal kernels also satisfy the admissibility (with respect to the exact integrated MSE) and mini-max optimality (with respect to the FT based IMSE). As an interesting finding, our optimal kernel function derived under the continuity of the underlying density function (possibly possessing discontinuous derivatives) coincides with Silverman's (1984, the Annals of Statistics) kernel function, based on which one can construct a kernel regression estimator that is asymptotically equivalent to a spline smoothing regression estimator.
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