THE EQUIVALENCE OF WEAK , STRONG AND COMPLETE CONVERGENCE IN Ll FOR KERNEL DENSITY ESTIMATES ' BY LUC DEVROYE McGill

Let f be a density on R", and let f, be the kernel estimate off, fn(x) _ (nh d)-' ~= 1 K((x-X1)lh) where h = h n is a sequence of positive numbers, and K is an absolutely integrable function with f K(x) dx =1. Let J, = f l f ,(x)-f (x) (dx. We show that when limn h = 0 and limnnh d = oo, then for every e > 0 there exist constants r, no > 0 such that P(Jn > e) <_ exp(-rn), n ? no. Also, when J,-p 0 in probability as n-p oo and K is a density, then limn h = 0 and limnnhd = oo.

[1]  H. Scheffé.  A Useful Convergence Theorem for Probability Distributions , 1947 .

[2]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[3]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[4]  T. Broadbent.  Measure and Integral , 1957, Nature.

[5]  W. Hoeffding.  Probability inequalities for sum of bounded random variables , 1963 .

[6]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[7]  N. Glick.  Sample-Based Classification Procedures Derived from Density Estimators , 1972 .

[8]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[9]  Ned Glick,et al.  Sample-based classification procedures related to empiric distributions , 1976, IEEE Trans. Inf. Theory.

[10]  Saab Abou-Jaoudé.  Conditions nécessaires et suffisantes de convergence L1 en probabilité de l'histogramme pour une densité , 1976 .

[11]  Saab Abou-Jaoudé.  Sur la convergence L1 et L∞ de l'estimateur de la partition aléatoire pour une densité , 1976 .

[12]  E. Slud.  Distribution Inequalities for the Binomial Law , 1977 .

[13]  L. Devroye,et al.  The $L_1$ Convergence of Kernel Density Estimates , 1979 .

[14]  L. Devroye,et al.  Distribution-Free Consistency Results in Nonparametric Discrimination and Regression Function Estimation , 1980 .

[15]  C. Spiegelman,et al.  Consistent Window Estimation in Nonparametric Regression , 1980 .

[16]  L. Devroye,et al.  On the L1 convergence of kernel estimators of regression functions with applications in discrimination , 1980 .