Universal smoothing factor selection in density estimation: theory and practice

In earlier work with Gabor Lugosi, we introduced a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the L1 error of the corresponding kernel estimate is not larger than 3 + e times the error of the estimate with the optimal smoothing factor plus a constant times Ov~--~-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 Yatracos. We present a practical implementation of this estimate, report on some comparative results, and highlight some key properties of the new method. K e y W o r d s : Density estimation, kernel estimate, convergence, smoothing factor, minimum distance estimate, asymptotic optimality, simulation study. A M S s u b j e c t c l a s s i f i c a t i o n : 62G05. 1 I n t r o d u c t i o n W e a r e g i v e n an i . i .d , s a m p l e X 1 , . . . , X ~ d r a w n f r o m a n u n k n o w n d e n s i t y f on ]R d, a n d c o n s i d e r t h e A k a i k e P a r z e n R o s e n b l a t t d e n s i t y e s t i m a t e f h(x) = a ' h ( X i= l where K : IR d --+ lR is a fixed kernel with f K = l, Kh(x) = (1/hd)K(x/h) , and h > 0 is the smoothing factor (Akaike, 1954; Parzen, 1962; Rosenblatt, 1956). In this paper, we focus on density estimation without restrictions on the densities. The fundamental problem in kernel density estimation is that of the joint choice of h and K in the absence of a priori information regarding f . Watson and Leadbetter (1963) show that the choice of h and K should not be split into two independent subproblems. Also, the choice The author 's work was supported by NSERC Grant A3456 and by FCAR Grant 90-ER-0291.