BIOMATHEMATICS TRAINING PROGRAM
.e
ON THE DISTRI BIJrION OF QUANTI LES OF
RESIDUALS IN A LINEAR MODEL
by
Raymond J. Carroll*
University of North Carolina at Chapel Hill
ABSTRACT
The representation for quantiles discussed by Bahadur (1966) and
Ghosh (1971) is extended to quantiles from the residuals in a linear
model. As an application, a robust test for symmetry is proposed.
AMS 1970 Subject Classifications:
Key Words and Phrases:
Primary 62G30; Secondary 62JOS.
Quanti1es, Order Statistics, Tests for Symmetry.
*This work was supported by the Air Force Office of Scientific Research
under contract AFOSR-7S-2796.
-e
.e
I.
Introduction
Let Yl""'Yn
be independent and identically distributed with
common distribution
of integers
k
n
= np
sample quantile.
Fn
F,
+
0
and let F(~) = p. Let k be a sequence
n
l 2
(n / log n), and let V be the kn th
n
Bahadur (1966) and Ghosh (1971)
is the empirical distribution function of
l 2
n / (Vn -~-(k n /n - Fn (~))/F~(~))
(1)
have shown that if
Y1 ""'Yn '
then
p
-.- O.
Actually, Bahadur's result is an almost sure result, but (1) suffices
for many statistical applications.
Our goal is to extend (1) to
the quantiles of the residuals in a simple linear model, using this
to conduct a quick test for symmetry.
.e
We consider the simple linear model
z.1
=
6
x~
--].-
+ y.
1
where
Yl""'Y
are i.i.d. with distribution F continuous and
n
strictly increasing in a neighborhood of ~, with F(~) = p.
If
6
is an estimate of 6 (possibly the least squares estimate
.o.-n
or one of the new robust estimates), the residuals are
r. = z. - x~ 6 = y. -x~ (6 - 6). Let E (F) be the empirical
1n
1 -l.o.-n
1 -1 .o.-n
n n
distribution function of the residuals (errors), and choose
2
Let k be as above, let
bn = (log n) .
n
I
and define V to be the k th order statistic
n
n
n
of the residuals. We make the following assumptions:
an
= n -1/2 log n ,
= (~-an ,~+a),
n
(AI)
F has two bounded derivatives in a neighborhood of
~
·e
(A2)
, and
F~(~)
a -1 (6 -6 )
n
.o.-n .o.-n
-+
= f(~)
0
>
(a.s.)
O.
2
(A3)
For some
6
(A4)
There exists
>
0,
n
1
x finite with (log n)n- l Lnl (x.
- ~
x ) .... 0
-1
~
Theorem
Under conditions (AI) - (A4),
Remarks
Condition (AI)
quite reasonable.
is the same as Bahadur's, while (A2) is
Condition (A3) guarantees that the design elements
The usual condition on the design matrix
will not vary too wildly.
X is that n-lX~X .... r
(Drygas (1976)), and this implies that if
!o
there is an intercept term there is an
so that
e.
a 1/2 - 6 max{ Ix. I'
: 1 =1, ... ,n} .... 0 .
with
n -l~n
II
(~i
-
~)
.... 0,
(A4) is only a slight strengthening of a standard condition,
which in fact is reduced to
n -l~n
II
Section 2.
and (A4)
(A3)
Note that
(~i
-
~)
.... 0
at the end of
are guaranteed in the problem
considered by Bahadur and Ghosh.
2.
Proof of the Theorem
We first consider simple linear regression without an intercept
z.1
= x.S
+ y ••
11
The problems of simple and multiple linear regression
with an intercept are easy consequences of the proof presented here.
Define the following processes:
G (x)
n
H
n
= n-
= nl / 2
1 n
L {I(r.
'I
1
1=
s x)-I(y. s
1
sup{IG (x) I: x
= n- l / 2
n
~)
- (F(x+x. (6 -6))-F(x))}
In
I }
€
n
n
L
~
I(YI s
i=l
-F(~+a
n
+ an s+an tx.)-I(YI
1
s+an tx.)+F(~)
1
e-
3
·e
where
I (A)
is the indicator of the event A.
We note that from
(AI) - (A4),
n-
(3)
n
I
L {F(~+sa
)-F(~+sa
. Inn
1=
uniformly for
Lemma I
+ta x.)+ta x f(t)}
n 1
n 0
=
O(n- l / 2 )
oslsl,ltls 1
Under (AI) - (A4),
H
O(a.s.)
-+
n
Proof of Lemma I
to show that sup {lwn(s,t)1 :
0
x.1
For this, we may consider only
W
n
x.1
.e
into two parts, one with
<
o.
By (AI)
points s,t
x.
1
~
(a.s.)
by breaking the sum defining
0
~ 0
-+ 0
and the other over the terms
as in Bahadur's proof, it also suffices to consider
= ~ r,n = ribn
(r
= o, ... ,b).
n
Is-~ r,n I S bn , It-n p,n I S b,
n
In-
s,t, s I}
S
I n
L {F(~+sa +ta x·l
. Inn 1
To see this, note that if
then
- F(~+n
1=
a +n
a x ·)}1
r,n n
p,n n 1
= O(anbn -1 ) = O(n -1/2 ).
Now, by Bernstein's inequality (Hoeffding, eq. 2.13)),
co exp{-c,n 1/4 }
r {IWn (s,t)l>d
P
for some constants
.·e
co'~
depending on
E.
Thus,
4
eo
This completes the proof of the Lemma.
Lemma 2
Almost surely as
n +
00.
VEl
n
n
Proof of Lemma 2 We have
Pr{V ~ t + a }
n
n
n
= Pr{ 11ICy·~
1
~ + a +x.CB -B)) ~ k }
n1n
It suffices to show the existence of an
~ Cn)
-+ 0,
where
= Pr{
n
tL ICY1' ~ t+an +tan x.)
1 ~ kn
l
o
n
>
n
0 for which
for some
OStSn,
n~n
o
}
e.
By Lemma I and using equation (3).
+ 0(1)
for some OstSn.
Choosing
n > 0 small,
Cn)
~
-+
0
n~
o
Lemma 2.
Since
o
as in the proof of Bahadur's
o
Proof of the Theorem
.
E (k )
n n
=
k In,
n
Lemmas 1 and 2 may
be applied to show that
1FCVn )-FC~)-Ck n In-E n C~)I-+ 0 Ca.s.) .
1/2 ) Ca.s.)
Further, FCVn)-FCt) = CVn-t)fCt) + DCnn
l/2
EnCt)-FC() =
FnCt)-FC~) + WnCo,an-ICBn-B)) + n-
1
and
n
1
{FCt+x.CB -B))-FC~)}
i=l
n
1
5
·'e
Since
A -1(6 -6) ~
n
n
0
(a.s.),
the proof is complete by Lemma
0
1 and (A4).
3.
An Application
Hinkley (1975) suggested a method based on order statistics
for estimating the parameter of a power transformation; his method
is most applicable in'the location parameter case.
The Theorem
can be used to construct a quick test for symmetry in more complicated models.
order residual.
.e
(4)
Let
and define
p.e 1/2
Flor
F(~p)
V (p)
to be the
n
= p,
n l / 2 (V (p)+ V (l-p)_ 2V (l/2) - (~P+~I_p-2~1/2))
n
n
n
has a limiting normal distribution (with mean zero and variance
independent of the limit distribution of
hypothesis of symmetry,
~P+~I-P
= 2~1/2'
6 •
n
Under the
so one rejects the
hypothesis if
(5)
n l / 2 lv (p) + V (l-p)_2V (1/2)\/0
n
n
n
is too large.
To estimate
0 one needs estimates of the density at
the quantiles
~p' ~l-p' ~1/2'
which can be accomplished as in
Bloch apd Gastwirth (1968) by using the Theorem.
·e
References
1.
Bahadur, R. R. (1966). A note on quanti1es in large samples.
Ann. Math. Statist. 37, 577-580.
Bloch, D. A. and Gastwirth, J. L. (1968). On a simple estimate
of the reciprocal of the density function. Ann. Math. Statist.
39, 1083-1085.
Drygas, H. (1976). Weak and strong consistency of the least
squares estimators in regression models.
Z. Wahrschenin1ichkeitsteorie Verw. Gebiete 34, 119-127.
Ghosh, J. K. (1971). A new proof of the Bahadur representation
of quanti1es and an application. Ann. Math. Statist. 42,
1957-1961.
Hinkley, D. V. (1975). On power transformations to symmetry.
Biometrika 62, 101-111.
e.
e·
UNCLASSIFIED
READ INSTRUCTIONS
BEFORE COMPLETING FORM
REPORT DOCUMENTATION PAGE
·e
1.
R.~ORT
••
TITL.E (_d
Z. 30VT ACCEIiION NO. I.
NUMBI:R
Iub''''.)
RECI~IIENT'1 CATAL.OO
TY~E
I.
On the Distribution of Quantiles of Residuals
in a Linear Model
OP' REPORT 6
NUMBER
~ERIOD
COVERED
rrECHNICAL
•. PERP'ORMING ORG. REPORT NUMBER
Mimeo Series No. 1161
7. AUTHOR(.)
~ONTRACT OR
I.
Raymond J. Carroll
GRANT NUMBER(_)
AFOSR-7S-2796
1O. PROGRAM EL.EMENT, PROJECT, TASK
AREA 6 WORK UNIT NUMBERS
•• PERP'ORMING ORGANIZATION NAME AND ADDRUI
Department of Statistics
University of North Carolina
Chape1 Hi 11 , North Carolina 27514
la.
I I. CONTROLLING O'P',CI NAME AND AOORIlIi
Air Force Office of Scientific Research
Bolling AFB, Washington, D.C.
".
RE~~i-gt~· 1978
NUMBER 0' PAGES
7
II. I.CURITY CLASS. (0' 'hi_ ,.pO")
,'''' MOMITOIittMO AGIINCY NAMI: • AOOR.II(" dill.,.., ,.... C",.,..",,,, 01110.)
UNCLASSIFIED
II.. DECk ASSIP'ICATIONI DOWNORADING
ICH DULE
.
I•• DlSTAI.UTlON ITATEMENT (., "',. """,)
Approved for Public Release: Distribution Unlimited
17. DISTRIBUTION STATEMENT (01 Ut. . . .froo' .'.roll'n 8'.01r :10, "d"'.,., " - It."o,,)
II. IUPPL.IIMENTARY NOTU
I•. KEY WORDI (C""""u. on ,.,..,•• • 'd. ""oo•••.,,
_fllfI.,,"" by .'oole " .....
r)
Qua.ntiles, Order Statistics, Tests for Symmetry
10. ABSTRACT (Con"nu. on ,n.,•••'110 II " •••••. " ..fI
,deft"" ., .'oolr " .....
r)
The representation for quantiles discussed by Bahadur (1966) and Ghosh
(1971) is extended to quantiles from the residuals in a linear model. As
an application, a robust test for symmetry is proposed.
·e
DD
1473
EDITIO.. O~ I NOV I~ II O.IOI.IITE
UNCLASSIFIED
UNCLASSIFIED
SlECURITY Cl.ASS"ICATION 0," TN"
"AG.""'.
Daf. . . . .cI)
UNCLASSIFIED
llECU"ITy CLAIIII'ICATION 01' THIS PAGE(""'.n D.,. Bn'.r.d)