•
!
ASYMPTOTIC MAXIMAL DEVIATION OF M-SMOOTHERS*
Wolfgang Hardle
Universitat Heidelberg
Sonderforschungsbereich 123
Jm Neuenheimer Feld 293
0-6900 Heidelberg 1
KEY WORDS:
University of North Carolina
Department of Statistics
321 Phillips Hall 039A
Chapel Hill, NC 27514
Robust nonparametric regression, Kernel estimators,
Uniform consistency, Robust smoothing
1970 AMS Subject Classification:
62G05, 62J05, 62F35
*Research partially supported by the "Deutsche Forschungsgemeinschaft"
SFB123, "Stochastische Mathematische Modelle".
Research also partially supported by the Air Force of Scientific
Research Contract AFOSR-F49620 82 C 0009.
ABSTRACT
Let (Xl ,Yl), ... ,(Xn'Y n) be iid rv's with pdf f(x,y) and let m(x)
E(YIX
!yf(x,y)dy/fX(x) be the regression function of Y on X. The
n
function m(x) is estimated by m (x) a solution of (nh)-l L K((x-X.)/h)~(Y.-·)
n
. 1
'
1
1=
for some odd and bounded ~-function making mn(x) a robust estimate of m(x).
=
x)
=
=
Probabilities of maximal deviation of Imn(x) - m(x)
I
are computed in a
similar way as in Bickel and Rosenblatt (1973) for density estimation and
in Johnston (1982) for nonparametric regression function estimation.
=
a
1.
BACKGROUND AND INTRODUCTION
Nadaraya (1964) and Watson (1964) independently proposed the following
kernel estimator
n
n
mn*(x) = (nh )-1 I K((x-X.)/h )V./[(nh )-1 I K((x-XJ.)/h )]
n i=l
1
n 1
n j=l
n
(1. 1)
of the regression function m(x) =
fy
f(x,y)dy/fx(x) where fx(x) denotes the
marginal density of X,K:(o) is a kernel and {h n} is a sequence of positive
Basically this estimator averages the V's around
constants C'bandwidth
ll
).
X = x motivated from the integral formula for m(x) above.
The numerator is
a weighted local average of the VIS while the denominator is a density estimate of fx(x).
It is clear that occasional outliers generated by heavy tailed conditional densities f(ylx) introduce smooth peaks and troughs
curve
m~(x).
Such outliers occur quite often in practice.
1982 Figure 7 or Bussian et a1., 1982).
of
m~(x)
in the estimated
(Ruppert et al.,
To avoid this misleading property
due to spiky V-observations we introduce a robust estimate, the
M-smoothe~
mn(x) as the solution of
(1 .2)
(nh )-1 L K((x-X.)/hn)~(V.- 0) = 0,
n i=l
1
1
n
where
~
denotes a bounded, odd and continuous function.
Note that if
~(u)
=
u, then mn is the Nadaraya-Watson estimator m~. Bias and variance rates for
mn(x) with K as the uniform window where obtained by Stuetzle and Mittal (1979),
robustness properties, consistency and asymptotic normality of mn(x) were
considered by Hardle (1982).
For the case of nonrandom design, i.e. Xi attains
fixed values, we may refer to Hard1e and Gasser (1982). In this paper we show that
-2k
k
P{(26 log n)2[ sup I(m (t) - m(t))or(t)I/A(K)2- d]
Os;t~;l
n
n
(1. 3)
x}
exp(~x))
exp(-2
+
<
where 6, r(t), A(K), dn are suitable scaling parameters.
The result (1.3) improves upon that of Johnston (1982) in a number of
ways.
First, Johnston obtains results like (1.3), but for estimates different
from the Nadaraya-Watson estimator (1.1); our result (1.3) of course applies
to the Nadaraya-Watson estimator as a special case.
a much broader class of estimators.
Secondly, (1.3) holds for
Finally, we obtain (1.3) under assumptions
weaker than those needed by Johnston.
2.
ASSUMPTIONS AND RESULTS
We write h for the bandwidth hn from here on unless there is no need to
do so. We make use of the following assumptions.
(Al)
the kernel K(o) is positive has compact support [-A,A] and is
continuously differentiable.
(A2)
(nh)-2(10g n)
+ 0
(n log n) h
(nh 3)- 1(1og n) 2 s; r~, t,1 a cons tant .
(A3)
3/2
k
h- 3(10g n)
.
~
J f (y)dy = 0(1),
Iyl> a y
n
5/2
+
0
f (y) the mqrginal density
y
of Y, {an}~=l a sequence of constants tending to infinity as n
(A4)
inf Iq(t) I
2
qo > 0, where q(t) =
Os;t~l
(AS)
E(~'(Y-m(t))
+
00.
IX=t)-fX(t)
the regression function m(x) is twice continuously differentiable, the
conditional densities f(ylx) are symmetric for all x,
~
is piecewise
twice continuously differentiable.
We need some more definitions before we discuss the assumptions.
-3-
Define
a
2 (t) =
E(~2(V-m(t))IX=t)
n
Hn(t) = (nh)-l
I K((x-X.)/h)~(V.-m(t))
. 1
1
1
1=
n
D (t) = (nh)-l
n
I K((x-X.)/h)~'(V.-m(t)).
. 1
1
1
1=
We further assume that a 2(t) and fx(t) are differentiable.
Assumption (Al) on the compact support of the kernel could possibly be
relaxed introducing a cutoff technique as Csorgo" and Hall (1982) for density
estimators.
Assumption (A2) has purely technical reasons:
to keep the bias down
and to ensure the vanishing of the nonlinear remainder terms.
Assumption
(A3) appears in a somewhat modified form also in Johnston's paper (1982).
When we want to apply the following theorem to the Nadaraya-Watson estimator
m*(x) we have actually to
n
restate (A2) as h-
1981).
{ y2 f (y)dy (which
Iy >a
y
Assumption (AS) stating the symmetry
densities is common in robustness considerations (Huber,
It guarantees that the only solution of
E(VIX=x).
(log n)
n
is assumption Al in Johnston (1982)).
of the conditional
J
f~(Y-')f(ylx)dy
=
a is
m(x) =
If we had skew distributions then we would no longer estimate the
conditional mean but rather a conditional Quantile such as the median.
Theorem
-8
Let h = n , 1/5
<
dn = (28 log n)~ +
8
<
A2
1/3 and A(K) = fK (u)du
-A
(28 log n)-~{log(cl(K)/rr~)
and
+
~[log 8
2
if cl(K) = K2(A) + K (-A)/[2A(K)] > a
d = (28 log n)~ + (28 log n)-~ {log (C (K)/2rr)}
n
2
A
otherwise with c2(K) = f[K ' (u)]2 dU /[2A(K)] .
-A
Then (1.3) holds with
+ log log nJ} ,
-4-
r(t)
=
(nh)~q(t)[a2(t)fx(t)J-~ .
This theorem can be used to construct uniform confidence intervals for the
regression function as stated in the following corollary.
Corollary:
Assuming the theorem above holds, an approximate (l-a)x 100%
confidence band over [O,lJ is
mn(t)
±
where c(a)
(nh)-~[a2(t)fx(t)A(K)J~ q-l(t)[dn+c(a)(28 log n)-~J • [A(K)J~
log 2 - logllog(l-a) I.
=
The proof is essentially based on a linearization argument due to Taylor series
expansion.
The leading linear term will then be approximated in a similar way
as in Johnston (1982), Bickel and Rosenblatt (1973).
The main idea behind the
proof is a strong approximation of the empirical process of {(Xi 'Yi)}~=l by a
sequence of Brownian bridges (with two dimensional time) as provided Tusnady
4It
(1977).
It follows by Taylor expansions applied to the defining equation (1.2) that
(2.1)
where [Hn(t)-EHn(t)]/q(t) is the leading linear term and
(2.2)
Rn(t) = Hn(t)[q(t)-On(t)]/[On(t).q(t)] + EHn(t)/q(t)
+
~(m
n
I
(t)-m(t))2 • [0 (t)J": l • (nh)-l
K((x-X.)/h)'¥II(Y.-m(t)+r (i)(t)),
n
i=l .
1
1
n
lr~i)(t)1
is the remainder term.
II R II = sup IR (t) I
n
O~t~l
n
=
<
jmn(t)-m(t) I.
In the third section it is shown (Lemma 3.1) that
0p((nh log n)-~).
Furthermore the rescaled linear part
-5-
is approximated by a sequence of Gaussian processes, leading finally to the
following process
Y (t) = h-~ JK((t-x)/h) dW(x),
5 ,n
as in Bickel and Rosenblatt (1973).
We also need the Rosenblatt transformation (Rosenblatt, 1952).
T(x,y) = (FXIy(x Iy), FY(y) )
which transforms (Xi ,Vi) into T(X i ,Y i ) =(Xi ,Vi) mutually independent uniform
rv1s. With the aid of this transformation Theorem 1 of Tusn~dy (1977) may be
applied to obtain the following lemma.
Lemma 2.1:
On a suitable probability space there exists a sequence of
Brownian bridges Bn such that
suplz (x,Y)-B n(T(x,y)
x, Y n
I
= O(n-~(log n)2)
a.s.,
where Zn(x,y) = n~[Fn(X,Y)-F(X,Y)J denotes the empirical process of {(Xi 'Yi)}~=l.
Before we define the different approximating processes let us first rewrite
Yn(t) as a stochastic integral with respect to the empirical process Zn(x,y).
Yn(t) =
1:
L
h-~g'(t)-2ffK((t-x)/h)~(y-m(t))dZn(x,y),
g'(t) =
')
0~(t)
fX(t).
The approximating processes are now
Yo net) = (h9(t))-~JJK((t-X)/h)~(y-m(t))dZn(X,y),
,
r
n
where r n = {Iyl~ an}' g(t) = E(~2(y-m(t))"I(IYI~ an)jX=t)"fX(t)
Yl n(t) =
,
(hg(t))-~JJK((t-X)/h)~(y-m(t))
dB (T(x,y)),
r
n
n
{B n} being the sequence of Brownian bridges from Lemma 2.1.
= (hg(t))-~JJK((t-X)/h)~(y-m(t)) dW (T(x,y))
r
n
n
{W n} being the sequence of Wiener processes satisfying
Bn(x',y') = Wn(x ' , yl) - x'y'Wn(l,l)
-6-
Y3 ,n(t)
Y4 ,n(t)
(hg(t))-~ffK((t-x)/h)~(y-m(x)) dWn(T(x,y))
=
f
n
= (hg(t))-~fg(x)~K((t-x)/h)dW(x)
Ys ,n (t) = h-~fK((t-x)/h)dW(x),
{W(o)} being the Wiener process on (_00,00).
Lemmata 3.2 to 3.7 ensure that all these processes have the same limit distributions.
The results then follows from the following lemma
Lemma 2.2 (Bickel and Rosenblatt (1973)).
Let dn , A(K), 8 as in the theorem.
Let
Then
k
P((28 log n)
4It
3.
I Ys (t) I/[A(K)J~ O~t~l,n
sup
2{
.
d } < x) ~ e- 2e
n
-x
.
PROOFS
We
show first that II Rn II = sup IR (t)
O~t~l n
des i red rate (nh log n)
Lemma 3.1:
I
vanishes asymptotically with the
-k
2.
For the remainder term R (t) defined in (2.2) we have
n
(3.1)
II Rnll
=
0p((nh log n)-~)
First we have by the positivity of the kernel K and 1~1I1 < C
1
1
II Rnll ~ [ inf (IO n(t)loq(t))r {11 H 11 II q-on ll +11 0nlloll EH II}
O~t~l
n
n
Proof:
0
+ C1
·11 mn-ml1 2
0
[
inf IOn(t) IT 1 ·1I f \I ,
O~t~l
n
n
where f n
=
(nh)-l
I
. 1
1=
K((x-X.)/h).
1
The desired result (3.1) will then follow if we prove the following:
(3.2)
(3.2)
-7(3. 3)
(3.4)
(3.5)
II
q- Dn II = 0p(n-~4h -~4(log n)-!<:2 )
II EH n II = O(h 2)
II mn-ml1 2 = Op((nh)-~(109 n)-~).
Define Un(t) = n~h~(log n)~[Hn(t)-EHn(t)].
We first show that Un(t) ~ 0 for all t.
since
This follows from Markov's inequality
n
U (t) = L U. (t),
n
i=l"n
where U. (t) = n- 3/ 4h- 3/ 4 (log n)~[K((t-X.)/h)~(y.-m(t))-EK((t-X)/h)-~(y-m(t))],
, ,n
. "
are iid rv's and thus
P(IUn(t)I>E:):::; €-2n-~h-~(log n)-h-1EK2((t-X)/h)~2(Y-m(t)).
The RHS of this inequality tends to zero since
h-1EK2((t-X)/h)~2(Y-m(t)) = h-l!K2((t-u)/h)E(~2(Y-m(t)) IX=u)fX(u)du
~ a2(t)-f x(t)-!K 2(U)dU
by continuity of a2(t) and fX(t).
Next we show the tightness of Un(t) using the following moment condition
(Billingsley, 1968, Th. 15.6)
E{IUn(t)-Un(t l ) I-!U n(t 2 )-U n(t)!} : :; C2 -(t 2-t l )2
where C2 is a constant.
By the Schwarz inequality,
E{ IUn(t)-Un(t l ) 1-IU n(t 2)-U n(t)
I}
~ {E[Un (t)-U n(t l )]2- E[Un(t2)-Un(t)]2}~ .
It suffices to consider only the term E{U n(t)-U n(t l )]2
Using the Lipschitz continuity of
K,~,m
and assumption (A2) we have
-8-
~ {(log n)(nh)-3/2 " E[A+BJ2}~
~ CA(nh)-~(lOg n)~lt-tl 1+ CB(n-!:ih- 3/ 4(lOg n)~"lt-tll ~ c3"lt-t l l
n
where A =.L K((t-Xi)/h)[~(Yi-m(t))-~(Yi-m(tl))J
1 =1
n
B =iIl~(Yi-m(tl))[K((tl-Xi)/h)-K((t-Xi)/h)J,
and C ' C are Lipschitz bounds for ~, m, K.
A B
Since (3.4) follows from the well-known bias calculation
EHn(t) = h-lfK((t-u)/h)E(~(y-m(t)) IX=u)fX(u)du = O(h 2),
2
.
where O(h ) is independent of t (Parzen, 1962) we have from assumption (A2)
that II EHnl1 = o((nh)-~(log n)-~).
Statement (3.2) thus follows using tightness of Un (t) and the inequality
IIHnll~ II Hn-EHnll+ IIEHnll.
Statement (3.3) follows in the same way as (3.2) using
and the continuity properties of
K,~'
~ssumption
(A2)
,m.
Finally from Hardle and Luckhaus (1982), where uniform continuity of
mn(t)-m(t) is shown, we have
-k
k
II mn-mll = 0p((nh) 2(10g n) 2),
which implies (3.5) "
Now the assertion of the lemma follows since by tightness of Dn(t),
inf ID (t) I --+p q and thus
n
0
O~t~l
II
Rnll = op((nh)-~(109 n)-~)(l + II fnll).
Finally by Theorem 3.1 of Bickel and Rosenblatt (1973)
thus the desired result
case, i.e.
(3.6)
~(u)
II
Rnll = op((nh)-~(109 n)-~) follows.
= u, the remainder term Rh reads
Rn = [m~ - mJ[fx-fnJf~l+ E(mn-mfn)/f x '
n
where m (x) = (nh)-l I K((x-X.)/h)Y ..
n
i=l
1
1
II
fnll = 0p(l),
In the nonrobust
-9-
Johnston (1982) proved that (mn-E mn)/f has the desired asymptotic
distribution as
stated in our Theorem.
So if we apply the recent result of Mack and Silverman (1982)
and Luckhaus (1982) to
Rosenblatt (1973) to
II
II m~-mll
or Hardle
and the well known result from Bickel and
fx-fnll we may conclude that the first term on the RHS
of (3.6) is op((nh)-~(109 n)-~). The second term in (3.6) is
[h-1IK((t-u)/h)om(u)f(u)du - m(t)h-1IK((t-U)/h)f(U)dU]/fx(t)
which is by the same calculations as mentioned above (Parzen, 1962) of the
order O(h 2). This shows that our result generalizes Johnston's paper. Our
theorem says also that the confidence bounds are smaller. Johnston had
s2(t) = E(y 2 !X=t) as a factor for the asymptotic confidence bound, we have
o2(t) = var(YIX=t) which is in general smaller than s2(t).
~
We now begin
with the subsequent approximations of the processes YO,n to YS,n.
Lemma 3.2:
II
Proof:
YO,n-Yl ,nil = O((nh)-~(log n)2)
Let t be fixed and put L(y) =
~(y-m(t))
a.s.
still depending on t.
Use integration by parts and obtain:
IIL(y)K((t-x)/h)dZ (x,y) =
r
n
n
A a
I In L(y)K(u)dZn(t-hou,y) =
u=-A y=-a n
A an
A
= I f Z (t-h ou,y)d[L(y)K(u)] + L(a )f Z (t-hou,a )dK(u)
-A -a n
n -A n
n
=
n
A
-L(-a n ) IZn(t-hou, -an)dK(u) + K(A)[
-A
an
f
Zn(t-hou,y)dL(y)
-a n
+L(an)Z n(t-hoA,an)-L(-an)Zn(t-hoA,-a n)]
an
-K(-A)[ f Z (t+hoA,y)dL(y) + L(an)Zn(t+hoA,a n )
-a
n
n
-L(-an)Zn(t+hoA,-a n)] .
-10-
If we apply the same operations to Y1,n with Bn(T(x,y)) instead of Zn(x,y)
and use Lemma 2.1 we finally obtain
~
k
(t) - Y (t) I = O((nh) _k2(10g n) 2)
, n , 1n
a.s.
sup h g(t) 21Yo
O~t~l
usi n9 the di fferenti abil ity and boundedness of ljJ.
Lemma 3.3:
Proof:
Note that the Jacobi of T(x,y) is f(x,y) hence
!Y 1 n(t) - Y2 n(t) 1=I(q(t)h)-~ffljJ(y-m(t)K((t-x)/h)f(x,Y)dxdYI·IWn(l
,1) I
"
rn
It follows that
h-~II Y1 n- Y2 II ~IWn(l,l) 1·11 g-~II - sup h- 1 fflljJ(y-m(t))K((t-x)/h) If(x,y)dxdy
O~t~l
,n
,
r
n
Since II g-~II is bounded by assumption and ljJ is bounded we have
h-~II Y1 ,n,n
-Y 2 II ~Iw n(1,1)1· C4 -h- 1f(K((t-x)/h))dX = 0p(l).
Lemma 3.4:
Proof:
The difference
!Y 2 ,n (t)-Y 3 ,n (t)1 may be written as
1(9(t)h)-~fJ[ljJ(y-m(t))-ljJ(y-m(x)JK((t-X)/h)dWn(T(X,y))
rn
I
If we use the fact that ljJ,m are uniformly continuous this is smaller than
h-~lq(t)I-~·Op(h)
and the lemma thus follows.
Lemma 3.5:
Proof:
-11-
+ h-~IK(A)W(t-hA){g(~(~1)]~ - 1}1
+ h-~IK(-A)W(t+hA){[g(~(~~A)]~ - 1}1
•
=
Sl ,n (t) + S2 ,n (t) + S3 , n(t) , say.
The second term can be estimated by
by the mean value theorem it follows that
h-~ II S2, n II
=
0p (1 ) .
The first term Sl,n is estimated as follows.
h-1S
l,n
(t)
A
Ih-lf W(t_uh){K'(U)([9(t-Uh) ]~ - l} du
-A
g(t)
=
-kj W(t_uh)K(u)[g(t-Uh)]-~[g'(t-Uh)Jdul
g(t)
g(t)
2_
•
=
A
IT1,n(t) - T2 ,n(t) I
,
say.
A
C5 " f IW(t-hu) Idu
-A
0p(l) by assumption on g(t)
=
To estimate Tl ,n we again use the mean value theorem to conclude that
suph-ll[g(t-Uh)J~_ll <C
Ostsl
g(t)
6
"lui
hence
II Tl
A
nil s C6 " sup f IW(t-hu)K'(u)uldu
,
Ostsl -A
=
0 (1).
P
Since S3,n(t) is estimated as S2,n(t) we finally obtain the desired result.
The next lemma shows that the truncation introduced through {an} does not
affect the limiting distribution.
-12Lemma 3.6:
II
•
Yn
- YO, n II =
ap (( 1og
n) -~) .
We shall only show that g'(t)-~h-~
Proof:
= (log n)~h-~
v (t)
n
~
a
~(y-m(t))K((t-x)/h)dZ (x,y)
JJ
lR-f
n
n
JJ ~(y-m(t))K((t-x)/h)dZn(x,y)
{Iyl>a n}
for all t
and then we show tightness of Vn(t), the result then follows.
k
L
i1
n)2(nh)-~2.L {~(y.-m(t)I{1
Vn(t) = (log
1=1
1
I> }(Yi)K((t-X.)/h)
Y an
1
- E~(Yi-m(t))oI{IYI>an}(Yi)K((t-Xi)/h)}
•
n
=IXn·(t)
i =1
,1
where {Xn,i(t)}~=l are iid for each n with EXn,;(t) = a for all t
E
We have then
EX~,i(t) s (log n)(nh)-lE~2(Yi-m(t))I{IYI>an}(Yi)K2((t-Xi)/h)
s
sup
-A~usA
K2(U)o(10g n)(nh)-lE~2(y._ m(t))I{1 I }(Y.)
1
y >a n 1
hence
n
var{Vn(t)} = E(
L X .(t))2 = no EX 2n,1.(t)
i=l n,1
J f (y)dyoM"1
sup K2 (u)h- 1 (log n)
-AsusA
{!yl>a n} y
~
where M~ denotes an upper bound for ~2.
[0,1].
-13-
This term tends to zero by assumption (A3).
Thus by Markov's inequality we
conclude that
•
Vn(t)
p) 0
for all t
E
[O,lJ .
To prove tightness of {Vn(t)} we refer again to the following moment condition
as stated in Lemma 3.1.
E{IVn(t) - Vn(t l ) loIV n(t 2)-V n (t) I} ~ C1o(t2-tl)2
C' denoting a constant, t E [t l ,t 2J.
We again estimate the left hand side by Schwarz's inequality and estimate each
factor separately.
E[Vn(t)-Vn(tl)J
2
1
n
= (log n)(nh)- E{ I
- E(Ifl (t, t ' X. , Y• ) 01
l 1 1
n
{ Iy I>a}
n
~.
(t,t ,Xi,Yi)ol
l
i =1 n
(Y . ) )} 2 ,
{ Iy I>a }
n
1
where Ifln(t,tl,Xi,Y i ) = ~(Yi-m(t))K((t-Xi)/h)-~(Yi-m(tl))K((tl-Xl)/h)
Since ~,m,K are Lipschitz continuous it follows
{E[Vn(t)-Vn(tl)J2}~
•
~ C7° (log n)~h-3/2It-tllo{
J f (Y)dY}~
{Iyl>a } y
n
If we apply the same estimations to Vn(t 2)-V n(t l ) we finally have
3
E{ IVn(t)-Vn(t l ) loIV n(t 2)-V n(t) I} : ; C~(lOg n)h- It -t l ll t 2-tl
.
J fy(y)dY
{ Iy I>a n }
~ C' °lt2-tlI2 since tdtl,t2J .
Lemma 3.7:
by assumption (A3).
Let A(K) = JK 2(u)du and let {d n} as in the theorem.
(28 log n)~[11 Y3 ,n ll /[A(K)J~ - dnJ
has the same asymptotic distribution as
(20 log
n)~[11
Y ,n ll
4
/[A(K)J~ - dnJ
Then
(Y.)1
-14-
Proof:
Y3 ,n (t) is a Gaussian process with
EY 3 ,n(t)
..
=
0
and covariance function
r 3(t l ,t 2) = EY 3 ,n(t l )Y 3 ,n(t 2)
=
[g(tl)g(t2)]-~h-lff~2(y-m(x))K((tl-x)/h)K((t2-x)/h)f(X,y)dxdy.
f
n
= h-l[g(tl)g(t2)]-~ff~2(y-m(x))f(Ylx)dYK((tl-x)/h)K((t2-x)/h)fX(x)dx
f
=
n
r 4(t l ,t 2) the covariance function of the Gaussian process Y4 ,n(t), which
proves the lemma.
ACKNOWLEDGEMENT
The author wishes to thank R.J. Carroll for helpful remarks and suggestions
•
on a first draft of this paper.
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