Universal smoothing factor selection in density estimation: theory and practice. (With discussion). (English) Zbl 0949.62026
Summary: In earlier work with Gabor Lugosi [Ann. Stat. 24, No. 6, 2499-2512 (1996; Zbl 0867.62024); ibid. 25, No. 6, 2626-2637 (1997; Zbl 0897.62035)], we introduced a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the \(L_1\) error of the corresponding kernel estimate is not larger than \(3+ \varepsilon\) times the error of the estimate with the optimal smoothing factor plus a constant times \(\sqrt {\log n/n}\), where \(n\) is the sample size, and the constant only depends on the complexity of the kernel used in the estimate. The result is nonasymptotic, that is, the bound is valid for each \(n\). The estimate uses ideas from the minimum distance estimation work of Y.G. Yatracos [ibid. 13, 768-774 (1985; Zbl 0576.62057)].
We present a practical implementation of this estimate, report on some comparative results, and highlight some key properties of the new method.
We present a practical implementation of this estimate, report on some comparative results, and highlight some key properties of the new method.
Keywords:
kernel estimate; convergence; minimum distance estimate; asymptotic optimality; simulation study; bandwidth; cross-validation; smoothing factorSoftware:
KernSmoothReferences:
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