The Double Kernel Method in Density Estimation
نویسنده
چکیده
Let fnh be the Parzen-Rosenblatt kernel estimate of a density f on the real line, based upon a sample of n i.i.d. random variables drawn from f , and with smoothing factor h. Let gnh be another kernel estimate based upon the same data, but with a different kernel. We choose the smoothing factor H so as to minimize ∫ |fnh− gnh|, and study the properties of fnH and gnH . It is shown that the estimates are consistent for all densities provided that the characteristic functions of the two kernels do not coincide in an open neighborhood of the origin. Also, for some pairs of kernels, and all densities in the saturation class of the first kernel, we show that lim sup n→∞ E {∫ |fnH − f |) } E { infh ∫ |fnh − f | } ≤ C, where C is a constant depending upon the pair of kernels only. This constant can be arbitrarily close to one.
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