North-Holland A UNIVERSAL LOWER BOUND FOR THE KERNEL ESTIMATE.pdf

Statistics Probability Letters 8 (1989) 419-423 North-Holland October 1989 A UNIVERSAL LOWER BOUND FOR THE KERNEL ESTIMATE Luc DEVROYE School of Computer Science, McGill University, Montreal, Canada H3A 2K6 Received May 1988 Revised September 1988 Abstract: Let f,, be the kernel density estimate with arbitrary smoothing factor h and arbitrary (absolutely integrable) kernel K, based upon an i.i.d. sample of size n drawn from a density f. It is shown that and that Keywords: density estimation, L, error, inequalities, empirical characteristic function, kernel estimate. 1. Inttoduction In this paper, we give a short proof of a lower bound for the expected Li error for the kernel density estimate where K is an absolutely integrable function (the kernel), h 0 is a smoothing factor, K,,(x) p (l/h)K(x/h), and Xi,. . . , X,, are i.i.d. random variables with common density f on the real line (Rosenblatt, 1956; Parzen, 1962). The main result is Theorem 1. Theorem 1 states that even if we are allowed to choose f, K and h, we can’t possibly have an expected error that is roughly speaking smaller than about l/(m). The lower bound is the price we have to pay for the use of the kernel estimate. This result could be used to determine if n is large enough for someone to be able to use the kernel estimate. It also states that under no circumstances can the kernel estimate compare favorably with an estimate which, for a given f, has an expected error 0(1/h). The latter estimates are usually “parametric”. Research of the author was supported by NSERC grant A3456 and FCAR grant EQ-1679. 0167-7152/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland) 419 Volume 8, Number 5 STATISTICS PROBABILITY LETTERS October 1989 There are two kinds of universal lower bounds one can study. First, one might consider lower bounds for jjy( JIM) The bound depends upon n and f only, and is in the spirit of a celebrated L, lower bound obtained by Watson and Lead