h,71K((x - X,)/h) - Semantic Scholar

The Annals of Statistics 1979, Vol. 7, No. 5, 1136-1139

THE Ll CONVERGENCE OF KERNEL DENSITY ESTIMATES 1 BY L. P . DEVROYE AND T. J. WAGNER

University of Texas Let X 1 , X, be a sequence of independent random vectors taking values in Rd with a common probability density f . If fn(x) = (1/h) In d ~7_ 1 K((x - Xi)/hn) is the kernel estimate of f from X 1 , • • • , X, then conditions on K and {h,} are given which insure that f ( fn (x) - f(x)j dx -+„0 in probability or with probability one. No continuity conditions are imposed on f. Let X1 , . . • , Xn be a sequence of independent random vectors taking values in with a common probability density f. The kernel estimate of f from X 1 ,' • • , Xn is given by Rd

fn( x)

d ( 1 /n)h,71K((x ~=

-

X,)/h)

where the kernel K is a bounded probability density on Rd and {h} n is a sequence of positive numbers. We are concerned here with the conditions of f, K and {h} n which insure the L 1 convergence of fn to f, namely, (1)

f Rdl fn (x) - f(x)I dx - n 0

in probability (or w.p.1) .

This concern is motivated by the observation (Schef fe (1947)) that 2

supae~I µn(B) - r(B)I

a f R° I

fn\x) - J ( x)I dx

where is the class of Borel sets in R d and µn and µ are the measures on corresponding to fn and f respectively . Consequently, whenever (1) holds (2)

supB E

(B) - µ(B) I - O

I

in probability (or w .p. 1) .

Of course, (2) reminds one of the Glivenko*Cantelli theorem and its extensions (Winter (1973), Glick (1974)), namely, sup B

v (B) - v (B) I -

E el n

O

in probability (or w .p . 1),

where X1 , • • • , Xn are independent, identically distributed with an arbitrary probability measure v on , vn is the empirical measure for X1, . • • , Xn and e is a strict subclass of Borel sets (Rao (1962), Vapnik and Chervonenkis (1971)) . For our case it is easy to see that µ must be absolutely continuous if (2) is to hold for µn which correspond to kernel estimates . Glick (1974) has shown that whenever fn is a probability density on Rd which is a measurable function of x and X 1 , • • • , Xn , then (1) follows from (3)

fn (x) -nf(x)

in probability (or w .p.1) almost everywhere in

Received May 1977 ; revised January 1978 . 1 Research supported by AFOSR Grant 77-3385 . AMS 1970 subject classifications. 60F15, 62005. Key words and phrases. Density estimation, integral convergence, kernel estimates . 1136

x.



1 13 7

KERNEL DENSITY ESTIMATES

Whenever f is almost everywhere continuous on R", (3) then follows immediately from the known pointwise consistency conditions for kernel estimates . (See, for example, Rosenblatt (1957), Parzen (1962), Cacouilos (1965), Nadaraya (1965), Van Ryzin (1969), Deheuvels (1974) .) This argument fails for those densities on ~d which do not have an almost everywhere continuous version . The main result of this note is that (1) follows without any continuity requirements on f and, consequently, (2) holds for all absolutely continuous probability measures . For comparison, we note that the nearest neighbor density estimate of f (Lof tsgaarden and Quesenberry (1965), Moore and Yackel (1977)) will never satisfy (1) or (2) since its integral over R d is always infinite . Abou-Jaoude (1976a, 1976b) has shown, however, that (1) holds for different types of histogram estimates with no assumptions on f. THEOREM .

Let K be a bounded probability density on R'1 with L(u) = sup11xii> K(x)

for u > 0. If { hn } is a sequence of positive numbers then (1) follows whenever

(4)

hn - n 0

(5)

nh '1 ~n oo ~ (re- an h,,' < oo for all a > 0)

and one of the following conditions holds :

(6)

lixll dK(x) -~ 0 as lixii - oo and f is almost everywhere continuous,

(7)

f is bounded,

(8)

fou a- 'L(u) du < oo .

REMARK .

The condition in (6) imposed on K is equivalent to u dL(u) - 0

as u -~ 00

which is only slightly weaker than (8). PROOF .

Starting, as usual, with n- Efn (x)l + IEf(x) n - f( x)l If(x) n - J(x)l < If(x)

we first show that (9)

Ef„(x) -„f(x)

almost everywhere in x.

The usual argument shows that (9) is implied by (4) and (6) (e .g ., use the d-dimensional version of Theorem lA of Parzen (1962)) . Next (10)

IEf(x) - f(x)I < I11y11<s,,If(x - Y) - f(x)Ihn dK(Y/h„) dY 'Ff11Y11>Slr,,l,l( x

.y) - f( x )l hn dK(Y/hn) "!' •

If X(B) denotes the Lebesgue measure of the Borel set B C_ Rd and if S(x, r) denotes the closed sphere of radius r centered at x then the first term of the right-hand side of (10) is bounded by supy K(Y)X( 5 (0 , s))Is(X, sh~){ I f( .v) -

f(x)I/X(s(x, oh))}

dy



1 13 8

L . P . DEVROYE AND T. J. WAGNER

which tends to 0 for almost every x and every S > 0 if hn -n 0 (see, for example, Zygmund (1959, 1969)) . 1f f is bounded the second term of (10) is bounded by 2 supyf(y)f„yi,>8K(y) dy which can be made arbitrarily small for all n by taking S large enough . Thus (4) and (7) imply (9). Using a theorem of Stein ((1970), pages 62-63) we see that (4) and (8) imply (9) . Looking at fn (x) - Efn (x) we see that it equals 1 Y n1( Ynr - EYnt) n

where Yni hn "K((x - Xj/hn)

Letting supy K(y)

M,

•

we have 0 < Yni

c

d M/hn ,

and EYni2 < (M/h")Ef(x), nn

so that, by Bennett's inequality (Bennett (1962)), P{I fn(x) - Efn(x)I > e} < exp(-2ne 2hnd/

(2MEf(x) n

+ Me)) .

n remains bounded At each point x for which Efn(x) -n f(x) the sequence {Ef(x)) so that, almost everywhere in x, fn (x) - Efn (x) ->n 0 in probability or w .p. l

depending on whether nh d - n oo or ~ re -"no' < oo for all a > 0. Since (3) follows from the conditions of the theorem, (1) now follows from Glick's result . REMARK. The proof also yields the strong pointwise consistency of K is a bounded probability density and (i) hn -n0,an (ii) ~°e -

fn

whenever

d

< oo for a > 0, and (iii) I I x I I "K(x) ->0 as I I x I I-> 00 or f is bounded . This result is similar to the one obtained by Deheuvels (1974) .

Acknowledgment . We wish to thank the referee for pointing out a nice improvement in an earlier version of the theorem given here . REFERENCES Aeon-JAOUDE, S . (1976a) . Conditions necessaires et suffisantes de convergence L 1 en probabilite de l'histogramme pour une densite . Ann. Inst . H . Poincare Sect . B 12213-231 . ABOU-JAOUDE, S. (1976b). Sur la convergence L1 et L of l'estimateur de la partition aleatoire pour une

densite. Ann . Inst. H. Poincare Sect . B 12 299-317. BErm rr, G . (1962) . Probability inequalities for the sum of independent random variables . J. Amer. Statist . Assoc. 57 33-45 . CACOULLOS, T. (1965) . Estimation of a multivariate density. Ann . Inst . Statist . Math . 18 179-190.



KERNEL DENSITY ESTIMATES

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