On Bickel's Tests for Heteroscedasticity
by
David Ruppert and Raymond J. Carroll
Institute of
Sta~istics
Mimeo Series #1224
April 1979
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
On Bickel's Tests for Heteroscedasticity
by
David Ruppert* and Raymond J. Carroll**
University of North Carolina at Chapel Hill
Abstract
The asymptotic distribution theory of Bickel's (1978) tests for heteroscedasticity is extended to a wider class of test statistics and distributions,
when the number of regression parameters is fixed.
AMS 1970 Subject Classifications: Primary 62G35; Secondary 62J05, 62JlO
Key Words and Phrases:
Heteroscedasticity, robustness, linear model, regression,
asymptotic distributions.
*This work was supported by the National Science Foundation Grant NSF MCS78-0l240.
**This work was supported by the Air Force Office of Scientific Research under
Contract AFOSR-75-2796.
On Bickel's Tests for Heteroscedasticity
1.
Introduction.
Bickel (1978), generalizing work of Anscombe (1961), considers
the general linear model
Y.1 = T.1n + aCT.1n ,8)e.,
1 T.1 = 8' c.1n
where 8 is an unknown (pXl) vector, c.
(1.1)
for i = l, ... ,n
is an unknown (pXl) vector, and the e.
m
1
are i.i.d. with symmetric distribution function F.
The function aCT. ,8)
1n
expresses the (possible) heteroscedasticity in the model.
It is assumed that
for some function a(·),
a(T,8) = 1
+
8a(T)
+
0(8) as 8
Also a(·,·) is known but 8(= 8 ) is unknown.
n
0 .
+
(1. 2)
Bickel developed tests of the
hypothesis H: 8 = 0 (homoscedasticity) which are robust against gross errors and
Let {t.} be the fitted values from either least squares or a
heavy-tailed F.
1
robust regression method and let b be an even, bounded (for robustness) function.
Bickel's test statistic is
n
l
(a(t.) - a.(t))b(r.)
1
i=l
-
with r. = Y.
1
1
,,2
(1. 3)
1
t. (= ith residual) and
1
ab =
n
l
i=l
(a(t.) - a. (t))
1
2
(n-p)
-1
n
l
i=l
(b (r.)
1 -b.(r))
2
and where for any function g and numbers xl, ... ,xn '
n
g. (x)
= n -1 l
.1
1=
g(x.)
.
1
Let
we
n
2
L
1
~
= 8 (L\ (a(T.) - a (T)) ):Z
E e.b' (e.) (Var b(e )) -~ .
l
o
n i=l
1·
1
1
A
,
(1.4)
2
Bickel's theorem 3.1 shows that
('\ - L\J
v
-+N(O,l)
under assumptions which include:
b is bounded and has two bounded, continuous derivatives,
(1.5)
F has a continuously differentiable density f,
(1.6)
a has two bounded, continuous derivatives.
(1. 7)
(1.8)
Bickel states that (1.5) is "unsatisfactory", and is not satisfied for interesting
choices of b such as Huber's function squared, that is,
(1.9)
Bickel's theorem 3.2 shows that (1.4) holds under considerable weakening of
(1.5), if, in addition to the other assumptions of theorem 3.1, the fitting is by
2
least squares, E e. <
1
00,
and
P is fixed.
Both the assumpt1·on that E e 2. <
1
00
(1.10)
.
t th a t I
an d th e requ1remen
east squares
be used are restrictive and undesirable in situations where robust methods are
deemed necessary, so theorem 3.2 is also unsatisfactory.
In another paper
(Carroll and Ruppert (1979)), we show that in Bickel's theorem 3.1, one can weaken
(1.5) (to include, for example, b given by (1.9)), if (1.8) is strengthened to
e"
3
That paper uses methods of proof very similar to Bickel's.
In this note we
develop alternate methods, and show that assumptions (1.5)-(1.7) of theorem 3.2
can be weakened, if (1.8) is replaced by (1.10).
assumption besides (1.8) be strengthened.
We do not require that any other
In addition, our results are scale invar-
iant, since in the definition of
Ap
we replace b(·) by b(·/a), where
mate of scale.
~s
also obtained in Carroll and Ruppert (1979).
For example,
Scale invariance
&could
2 or the median of
cr is an esti-
be the scale estimate from Huber's (1964, 1973, 1977) proposal
{I r 1 I, ... , Ir n I},
poss ib ly normali zed through divis ion by a
positive constant.
2.
Assumptions and Notations.
and (1.10).
Throughout this paper we assume (1.1), (1.2), (1.3),
Also we assume:
b is bounded, symmetric about 0, Lipschitz continuous, and its
Radon-Nikodym derivative b' is bounded,
(2.1)
b" exists and is bounded except at a finite number of points
(For simplicity, we take the exceptional set to be {~, -~} •
This involves no essential loss of generality),
(2.2)
a is Lipschitz continuous,
n
lim inf n- l
L
i=l
(a(T.)
l. - a • (T))
(2.3)
2
(2.4)
> 0,
F is symmetric and Lipschitz continuous in a neighborhood of
~,
(2.5)
(2.6)
sup sup
.
n
l.:m
IT.l.n I
<
00
(2.7)
(2.8)
where for x = (xl' •.. , xp ),
II x II
= max
R,~p
IxR,l·
4
sup (nn
I
n
2
I II~:1n II )
. 1
<
<
00,
(2.9)
00,
1.=
and for some a > 0,
!,;
A
n 2(a - a)
n
=o
P
(Without loss of generality we take a
=
hold for most robust scale estimates if
(2.10) for most estimates of a even if
Then t.
1.
= c.1.n §.
n
n~ II f3n - sII = 0 P(1).
(1) and
The first relation in (2.10) will
1.)
e _ 0,
e
(2.10)
and assumption (2.9) will imply
*o.
Define ~ and a~ as in (1.3) and (1.4), but with r i
A-I
r ..
n 1.
replaced by a
Main Results.
3.
Theorem 1:
(Compare Bickel (1978), Theorem 3.1).
(~ - L1,J g N (0,1)
As n
+
00,
.
To prove theorem 1, we need a number of preliminary results.
Lemma 1:
l
l
For each i ~ n suppose that gin is a func:tion from RP x R to R ,
bounded uniformly in i and n.
Let Sin' i
~
n, n
=
1,2, ... be non-negative constants
such that
lim sup r
"'in
n+oo i~n
(3.1)
= 0
and
n
sup(
n
I
i=l
2
Sin) <
00
•
Assume that for each R > 0 there exists K(R) such that for all E > 0 and all
e
1
E
R
(in the application of lemma 1, e will be one of the errors),
(3.2)
5
sup ~up suP{\gin(ll,e) - gin(ll',e)I: ll, ll' E RP , !I ll ll,llll'I\.s.R, Illl-ll'll~d
n 1~n
(3.3)
~ K(R)(lel + l)~in c.
Define, for II
E
RP,
n
1
M (~) = n-~ l g. (ll,e.)
n
i=l 1n
~,
and
Nn'Cll) = Mn (~) - Mn lO) .
Then for each R > 0,
sup IN (~) - E N (ll)
Illll~R
n
n
Proof:
I =0
ll) •
P
Without loss of generality we assume R = 1.
Fix II such that I Illi
I ~ 1.
By (2.6), (3.1), and (3.3),
Var
N
n
(ll)
+ 0
as
n
+ 00,
so for each fixed II with Illi ~ 1,
N (ll) - E N (ll)
n
n
~ 0
P
l3.4)
(1).
Define
n
-1,:
H
n
=
n
;z
l (le.1
1
. 1
1=
+ 1)~ . •
1n
Then by l3.2)
H
n
Fix c >
O.
+ E H
By (3.3), if Illlll, I Ill'
n
II
= 0 P (1)
•
.s. 1 and Illl-~'
II
~ c, then
6
(3.5)
For n > 0 and x
= {y
RP define B (nJ
x
€
and a(l), ... ,a(L)
Ily-xll ~ n}. Then choose L
RP;
€
B (l) such that t~l Ba(t) (E)
O
€
0
>
L
c
Then by (3.4) and
BO(l).
(3.5)
sup IN (~J - E N (~)
~1 n
n
I ~ max
t~L
II /::,.11
+
max
t~L
~
~E8a(tJ (E)
+
(Hn
+
n
IN n (~J - Nn (a.)
1 I
sup
0p(l)
IN (a(t)) - E N (a(t))
n
+
I
EINn (~) - Nn (a.)
1
I
E Hn)E .
0
Since H is independent of E, we are done.
n
1
We now introduce additional notation.
A. (M = aCT.
1n
1
B. (0,~,8)
1n
=
~,
c.),
1n
b((l
B"!1n (0,~,8)
and
Lemma 2.
-k
n .2
+
+
=
For 0 E: R and /: ,.
A"! (M = A. (M - A (M ,
1n
1n
·n
-k
-k
n .2 0) (cr(T. ,8)e. - n .2
1
1
~,
c;n)) ,
£
B.1n lo,~,8) - B·n·fo,~,8) .
Define
n
U(0,/::",8)
= n -~ . L1 A"!1n (M B.1n (o,~, 8) and
1=
Vlo,~,8) = Ulo,~,8J
Then for each R > 1,
as n
~
- U(0,0,8) .
k
00
RP define
€
and sup In.2 81 <
00
,
n
sup
IV(o,~, 8) - E V(o,~, EI) 1
I0 I+II~II~R
(1).
= 0
P
7
Proof:
Take R = 1.
We will show that lemma 1 applies with
g.1n ((0,6), e.)
1 = A~1n (6) B.1n (0,6,6) and
1n = 2n
Z;.
-k
2
(1 + Ilc.1n II).
By (2.3),
(2.7), and (2.8)
sup
101+11611~1
sup sup IAin(6)
n i~
I<
00
(3.6)
,
and since b is bounded
sup
sup sup lB. (0,6,6)
10 I+1161 ~1 n i~
1n
I
<
Moreover, by (2.1) and (2.3) there exists K such that 101 + 1161
10'
I
+ 116' II ~ 1, and 10-0' I + 116-6'
lB.1n (0,6,6) - B.1n (0' ,6' ,6)
II
I -<
~
£
(3.7)
00
I
~
1,
implies
1n (Ie.!
1 + 1) and
K £ Z;.
(3.8)
Since
IA~ (6)
1n
B.1n (0,6,6) - A~1n (0) B.1n (0,0,6)!
-< IA!1n (6) (B.1n (0,6,6) - B.1n (0,0,6))!
+
lB.1n (0,0,0) lA!1n (M - A~1n (0))
o
the assumptions of lemma 1 are implied by (3.6)-(3.8).
Lemma 3.
For all R > 0,
IE
sup
I0 1+~611~R
Proof:
I
Take R = 1.
V(0,6,6) 1 = 0(1) .
Then
1
E V(0,6,6) = n-~
+ n-~
n
L
i=l
n
L
i=l
(A~ (6) - A~ (O))E B. (0,0,6)
1n
1n
A~ (6) (E B. (0,6,6) 1n
1n
1n
E
B. (0,0,6)) .
1n
(3.9)
8
Let O(i,n) = {It; - ell < n
_~
k
, It; + ell < n- 4
,
or le11 > n-
'1'1 . = n-~ E b'(a(T ,8)e )(6 a(T p 8)e i - (1 + ni
i
,1
k4
}.
Oefine
1
1
6)6'c in )
'1'2 . = E(B. (6,6,8) - B1·n (O,O,8) - T1 ,i)(l .- IO(i,n)) and
,1
1n
T ,i = E{B (6,6,8) - Bin (O,O,8)}I0(i,n)' Then,
in
3
(3.10)
By (2.2), for some K,
and therefore by (3.6)
(3.11)
By (2.6), E leil IO(i,n)
+
0 as n
+
00.
By (2.1), for some K
1
1'1'3 ,1·1 ~ Kn-':2(E Ie.1 I 1 0 (,1,n ) + 16' c.1n I P(O(i,n)).
1
sup
n-':2
161 + 11611.9
n
IL
A~ (6) '1'3 . (6,6)
i=l
1n
Therefore
I+
(3.12)
O.
,1
Since E b' (a(T.,8)e.) = 0 by (2.1) and (2.5),
1
1
1
'1'1 ,1.(6,6,8) = E b'(a(T.,8)e.)e.
1
1 1 n-':2 6 a(T.,8)
1
(3.13)
Now '1'1 ,1.(6,6,0) is independent of i, so for all 6 and 6
n
L
i=l
A~ (6) '1'1 .(8,6,0) = 0 .
1n
(3.14)
,1
By (1.2) and (3.13),
1
sup
1'1'1.(8,6,8) - '1'1.(8,6,0)1 = o(n-':2) .
181 + I 1611 ~1
'1
,1
(3.15)
By (3.14) and (3.15),
sup
161+11611~1
In-~
n
L
i=1
A~1n (6) '1'1 . (6,6,8) I = 0(1) .
,1
(3.16)
9
8y (3.10), (3.11), (3.12), and (3.16)
sup
101+ 11~11:;;1
1
For all ~, n- Yz
n
L
. 1
n
n -~
Ii=l
L
A~
(~)(E 8.(o,~,8)
1n
1
- E 8.(0,0,8))
1
I
= 0(1). (3.17)
.
(A~ (~) - A~ (0)) E 8.(0,0,0) = 0, and since
1n
1n
1
1=
sup IE(8. (0,0,8) - 8. (0,0,0)) I = 0(1),
i:;;n
1n
1n
1
sup
1~1:;;1
\n- Yz
n
L
1n
1
sup
11~11+lol:;;l
- A~ (O))(E 8. (0,0,8) - E 8. (0,0,0)) I = 0(1) , so that
(A~ (~)
i=l
n- Yz
1n
n
L (A~
i=l
1n
1n
1n
(~) - A~ (O))E 8.(0,0,8)
1n
1
I
= 0(1) , and then by (3.9) and
0
(3.17) the lemma is proven.
Combining Lemmas 2 and 3 we have:
Lemma 4:
For all R > 0,
su~
11~11+lol:;;R
Lemma 5.
IV(o,~, 8)
I =0
(1) .
(3.18)
P
For all R > 0,
In -
sup
101+ II~II:;;R
n
\ (8in(~,O,8)) 2 - Var b(e ) I = 0p(l)
1.-_£1
1
1
(3.19)
and
In- 1 nL
i=l
Proof:
(A~(M)
1
2
-
n- 1 nL
i=l
(aCT.) - a (T))
1
.
2
I = 0(1)
(3.20)
8y the law of large numbers and the boundedness of b
n
-1
n
L (8~ (0,0,0)) 2 +P Var b(e.)
i=l
1
1
Since b is Lipschitz, for some K > 0
sup
101~1~1~1
lB.
1n
(~,o,e) - B.
1n
(0,0,0)
I :;;
n-
k2
K(le·1 + Ic. I).
1
1n
(3.21)
10
Thus, E(
sup
lB. (.6.,6,8) - B. (0,0,0) j) = 0(1), which implies that
I6 1+ 11.6.11 ~1 1n
1n
sup
In
161+11.6.11~1
-1
n
L
(B~
1n (.6.,6,8))
i=1
Now (3.21) and (3.22) imply (3.19).
Lemma 6:
Proof:
2
- (B~ (0,0,0))
2
1fi
= 0 (1).
p
o
It is easy to proof (3.20).
n
1
U(0,0,8) - U(O,O,O) = n-~ E(b'(e 1)e 1)8 L (a(T.) - a.(T))
i=l
1
Since
(3.22)
2
+ 0(1).
1n (0) is bounded uniformly in i and nand 8 -+ 0, we have
E A~1n (O)(B.1n (0,0,8) - B.1n (0,0,0)) = 0(1), uniformly in i and n. Therefore
A~
Var(U(0,0,8) - U(O,O,O)) = 0(1).
By a Taylor expansion and (1.2), (2.9), and
(2.10), for some v.1
IE(B.1n (0,0,8) - B.1n (0,0,8)) - E(b'(e 1)e 1)(a(T.,8)
- 1)
1
.::. IE b"(v i ) (a(T i ,8) - 1)2
= 0(n- 3/ 4 )
where B.1n = {Ie.1 - ~I
Since sup sup
Proof of theorem 1:
A...
-0
1n (0)
I
or
<
00,
1
n
L
i=l
(a(T.) - a (T))
1
.
.
o
the proof is easily finished.
(6n -8)'
First note that since r. = e. a(T.,8) 1
1
1
= U(0,0,8)(n-
for (2.10) allows us to Subst1tute 6 = n
(3.18)-(3.20).
+ O(P(Bin) 1<Y(Ti'8) - 11 )
,
n -~
IA~
i~
n
~
I
~
2-~
Var b(e ))
1
2
c. ,
1n
+ 0 (1)
P
(8-1 - 1) and .6. = n~2 (8 -8) into
A
n
n
By (2.1), (2.4), (2.7), and the Lindeberg Central Limit Theorem,
U(O,O,O) (n-
1
n
L
i=l
(a(T.) - a (T))
Theorem 1 follows from lemma 6.
2
Var b(e,l)
_~
2
V
-+
N(O,l) .
1
o
11
References
Anscombe, F.J. (1961).
Examination of Residuals.
In Proc. Fourth BerkeZey
Symp. Math. Statist. Prob. (J. Neyman, ed.), pp. 1-36.
•
California Press.
Berkeley and Los Angeles, California .
Bickel, Peter J. (1978).
Nonlinearity.
University of
Using Residuals Robustly I: Tests for Heteroscedasticity,
Ann. Statist. 6, 266-291.
Huber, Peter J. (1964).
Robust Estimation of a Location Parameter.
Ann. Math.
Statist. 35, 73-101.
Huber, Peter J. (1973).
Carlo.
Robust Regression: Asymptotics, Conjectures and Monte
Ann. Statist. 5, 799-821.
Huber, Peter J. (1977).
Robust Statistical Procedures.
SIAM, Philadelphia, PA.