M-ESTlMATES FOR THE HETEROSCEDASTIC
LINEAR MODEL
David Ruppert*
and
Raymond J. Carroll**
University of North Carolina at Chapel Hill
We treat the linear model Y.=
C~ 6
1
-1 -
+
Z.1 where -1
C. is a known vector, _6 is an
unknown parameter, and Var Z.1 is a function of Ic~
-1
for a parameter
= 0
p
~.
For simultaneous M-estimates,
1\
!
61
-
and
which is known, except
A
~,
we show that
(N~~), and find the limit distribution of N~(i-E0. For the special
case of least squares estimation, this limit distribution is the same as the
limit distribution of the weighted least squares using the weights,
.e
w.= (Var Z.)
1
1
-1
, and in general the distribution is that of a "weighted
M-estimate" using these weights.
Moreover, the covariance matrix of the
limit distribution can be consistently estimated, so large sample confidence
ellipsoids' and tests of hypotheses concerning
AMS 1970 subject classifications
!
are feasible.
Primary 62J05; Secondary 62G35.
Keywords and phrases. Weighted M-estimates, weighted least squares, estimated weights, linear regression in scale, robustness, asymptotic representation, asymptotically normal, confidence ellipsoids.
*Supported by National Science Foundation Grant NSF MCS78-0l240
**Supported by Air Force Office of Scientific Research under Grant AFOSR-75-2796
page 2
1.
Introduction.
Consider the linear model
Y.= T. + a.e
111
i
and
T.=
C!S, for i = 1, .•. ,N,
1
-:1.where e , ... ,eN are LLd. with distribution FJ -1-:1.
C. (=C. ) are known elements in
l
N
RP, ~ is a vector parameter in RP , and al, ... ,aNare constants, which express
the possible heteroscedasticity of the model.
function and the a.1 were known
If F were a normal distribution
then the minimum variance unbiased estimator
A
of
~
would be the weighted least squares estimator
N
which is the solution to
A
P((Yi-fi ~)/ai) = minimum,
i=l
(1.1)
2
where p(x) = x.
.e
T
L
~,
If F has heavy tails compared with the Gaussian distribution
or gross errors are possibly present in the data, then one may wish to replace
p(x) = x
2
with a funotion such that 1/J = pI is bounded.
Such estimators have
been called M-estimators by Huber (1964, 1973), because they are generali:zations of maximum likelihood estimators.
still be possible to estimate them.
If the
a.1. are not known it may
For example, Fuller and Rao (1978) con-
sider the case where the Y. occur in groups for which a. is constant, and
1
1
Box and Hill (1974) assume that
8
a.1
(8) = a. = a IT. I
11
(1.2)
for an unknown parameter 8 and estimate both 8
and~
by Bayesian methods.
Both Fuller and Rao and Box and Hill treat only Gaussian errors.
Anscombe (1961) has proposed tests of heteroscedasticity, and Bickel
(1978) has developed robust versions of these, but neither has considered
modifying the estimate of
~
if the null hypothesis of homoscedasticity is rejected.
In many empirical studies, one finds that the dispersion of the residuals
page 3
increases with the magnitude of the fitted values, so that it is reasonable
to assume that o. = O(IT.
1
1
I)
for some nondecreasing function o.
While the
power family (1.2) may be sufficiently rich to model this phenomenon (although
it has obvious difficulties in practice when T.= 0), we prefer to study the
1
general class,
o. = exp(h(T.)
(1. 3)
where
1
-
T
1
8),
-
~ is a parameter in Rq and h is a known function from Rl to Rq .
Notice that choosing hl(T ) - 1 (hI is the first coordinate of
i
becomes a scale parameter
as a result, our estimate of
equivariant regardless of the choice of p.
~
~),
exp(8 l )
will be scale
Two choices of h which have no
practiual difficulties when T = 0 are ~(T) = (1, 10g(1+ITI)) a~d ~(T) = (1, ITI).
i
To motivate our method, suppose F is standard normal so that the loglikelihood is
If
~
1 N
2
2
- - -2
{log 2TI + log o. + ((Y.-T.)/O.) }
i=l
1
1
1
1
L
were known, (1.1) would yield the MLE for
~,
while if 13 were known, the
MLE of 8 would solve
N
L {((Y.-T.)/O.
(8))
i=l
1
1
1
(1.4 )
2
- I} h(T.) ; O.
-
1
A reasonable computational alternative to solving (1.1) and (1.4) simultaneously might
~onsist
of (i) obtaining a preliminary estimate of
squares estimate) and hence estimates for Tl, ..• ,L
n
~
(such as the least
(ii) soive (1.4) using
these estimates, thus obtaining estimates of 0l, .•. ,on' which (iii) are used
to solve (1.1).
We thus suggest the following procedure.
1\
~
of 13 is calculated and assumed to satisfy
(1. 5)
First, a preliminary estimate
e_
page 4
Examples of estimates satisfying (1. 5) are given in Section 5.
stage, define t
i
=
T
A
~i ~
and obtain robustified estimates of
At the second
~
by solving the
following analogue of (1.4):
r {~
i=l
N
(1. 6)
2
((Y.-t.) exp(-h(t.)
1
1
-
T
1
A
e)) - ~} h(t.)
-
-
where ~ is monotone nondecreasing and ~
= E~2'c~),
under the standard normal distribution.
Clearly,
= 0,
1
the expectation being taken
~(x)
= x leads to (1.4).
At the third stage, we now solve a robust version of (1.1):
N
(1. 7)
where
~ ( (Y.
1
T
1\
A
C. = 0,
- -1
C. -S) 10 1. ) -1
1\
o.1 = exp (- h (t 1. ) T
(1.8)
.e
r
i=l
"
e).
-
Our main result (Theorem 4.1) lists conditions under which the limit distribution of
1\
~
defined by (1.5)-(1.8) is the same as that of the estimate which
could be found by solving (1.7) when
0
introduce notation and assumptions.
Section 3 demonstrates that
1
""
'ON
are known.
In Section 2 we
(1.9)
In Section 4 we state and prove the main result.
Remarks.
A.
A If one assumes homoscedasticity (a.
l
= 0),
then h(T.)
-
1
=1.
1\
The estimate
~
be-
comes an ordinary robust regression estimate with preliminary estimate 'of
scale given by (1.5)-(1.6).
See Maronna and Yohai (1979) for further details.
B We do not know i f iterating (1.5)- (1.8) will lead to convergence.
Further, we
do not know whether simultaneous solution of (1.6)- (1. 7) is possible.
page 5
C
Throughout the paper, we will use the following convention.
f on a space X, we say that
o solves
X
f(x)
=0
For areal function
~
if
(1.10)
With this convention, all our estimators will exist but need not be unique.
However, our asymptotic results hold for every appropriate sequence of estimators.
D
The Box and Cox (1964) transformations form an alternative method for dealing
with heteroscedasticity, as well as other deviations from the normal linear
model.
Defining
= log
Y
(A
# 0)
(A
= 0),
they postulate that for some A, yeA) satisfies a homoscedastic normal linear
model.
Their methodology is based upon an entirely different model from ours,
but a practitioner might consider using both on a
given data set.
Carroll
(1978) and Bickel and Doksum (1978) independently studied both Box-Cox and
robustified Box-Cox methods and concluded that variances of the estimated coefficients in the linear model are often much larger when A is estimated than
when it is known.
Therefore, confidence intervals and tests of hn>otheses which
are constructed as if A were known and not estimated are invalid.
In contrast, we show in Section 4 that, if the errors, e., are symmetrically
1
distributed or
ficients when
~(x) =
e
x, then for our method, the variances of estimated coef-
is estimated are similar to those when
e
is known, and confidence
intervals and tests can be validly constructed as if ft were known.
e.
page 6
2.
Le~ ~
Assumptions and notations.
be .a nondecreasing function from
l
l
R to R which satisfies
B1.
E~(el) = 0,
o < E~2(el)
B2.
=
~
<
00,
for some a > 0
as r
+
0 with
> 0,
A(~)
E~2((el+r)(1+S)) - ~ = A(~2)
B4.
as r
B5.
0 and s
+
.
0 and s
+
+
0 with
A(~
2
s+
l
o(lrl +a +
Isl l +a)
) > 0,
there exists Co such that for all 0 <: 1
lim sup E{supl¢((e 1 +r) (l+s)) - ¢((e1+r')(1+s'))I:
k+O
'
Irl, Ir'
.e
I, /51, Is' I
for both ¢
B6.
=~
~ k and Ir-r'l, 15-5'
and ¢
= ~2 ,
= ~
and ¢
~ ko}
and
lim E(¢((el+r)(l+s))
r, S-r()
for both ¢
I
¢(e l ))
2
- 0
2
= ~ •
l
T
The function h from R to Rq, -1
C. in RP and T.1 = -1
C. S, satisfy
lim (sup
N+O i~N
B7.
+ h (T. ) )N-J.?) = 0,
-1
1 N
sup (N-): II~i II
N
1=1
88.
B9.
(lie. II
Letting \
n
1
2
2
+ ~(T) ) <
.
00
be the minimum eigenvalue of
-1 N
H = N
N
then
):
1=1
~(Ti) ~(Ti)
T
lim inf AN = Aoo > 0
N~
BlO.
h is Lipschitz continuous on an interval (possibly infinite), I, such
that T. is in I for all i and N,
1
page 7
Bll:
and
inf inf h(T.)T 6>
N i~N 1
-
_00
letting -1
d. = C. exp(-h(T.) T 8),
-1
1
-
B12.
N
Remark.
N
l.
-l \'
i=l
T
d.d.
-1-:1.
= SN
S
-+
(positive definite).
The monotonicity of W is needed only to prove
i\
IN - consistency
"
of .§. and 6.
The conditions Bl-B6 are notationally
complex but widely applicable.
.
. Eel4 <
They clearly hold for p(x) = x 2/ 2 (W(x)
= x) if Ee = 0,
l
00,
where
A(W) = 1, A(~2) = 2Eei . If F is symmetric and W is odd, BI-B6 can be verified if (i)
~
is constant outside an interval (ii)
~
is Lipschitz continuous
and twice boundedly differentiable except possibly at a finite number of points,
al, ... ,a
k
(iii) F is Lipschitz continuous in neighborhoods of a l ,.· .,ak and
2
(iv) Ew (e ), E~' (e l ) and Eelw(el) ~'(el) are all positive.
l
and
A(~
2
) =
Then A(~)= E~' (e l )
2Eel~(el) ~'(el)'
For the power model (1.2), ~(x) = (1, log Ixl),~ = (log a,8) and BlO, BII
hold ·if
inf inf IT.
1
N l$i$N
(2.1)
1
> O.
Since the power model (1.2) makes little sense when T.= 0, (2.1) is reasonable.
1
To avoid this difficulty, in Section 1 we suggested ~(x) = (1, 10g(1+lxl)), for
which BI0, B11 hold.
The other example ~(x) = (1, Ixl) will satisfy BlO, B11
as long as the {T.} are uniformly bounded.
1
e.
page 8
3.
Estimation of
Theorem 3.1.
~.
/\
Suppose Bl-B8 hold,
(~-~)
=
any solution (see 1.10) to
N
I {~2((Y1·-t.)
1
(3.1)
i=l
exp(-h(t.)T 6)) - 1 -
~} h(t.) = O.
-
1
Then
(3.2)
Proof.
For ""1 in
P
R ,
""2 in
q
R
and
(3.3)
(3.4)
.e
Then let ¢.1 (x,y,z) =
WN(~)
~
2
((x+z)(l+Y)) n
=
_N--1?
I
i=l
¢ (e.
~
and define the process
,a~l) (""),a~2) ("")) h. ("").
1],
_.
1
-
-1-
Note that (3.1) can be rewritten as
k
1\
k/\
WN(N 2 (~~~), N2(~_~)) = O.
By Bl
(3.5)
and Chebyshev's inequality,
W (0)
N -
= aP (1),
so that by our convention (1.10))
"
k/\
WN(~2(~-f),
N2(~_~)) = Opel).
We can therefore prove (3.2) by showing that for each Ml>O, £>0, Q>O
there exists M2>0 such that
page 9
(3.6)
p[
inf
II L1 11 ::;M
1
1
II L1211 ~M2
We will prove (3.6) by modifying the proof of Juretkov~'s (1977) Lenuna 5.2.
We first apply Theorem A.l of the appendix, with x.
-1
a~3)(L1)
= -1
h. (L1)
1
_.
-
- h.1(0).
-
1
sup
IIWN(L1) - WN(O) -
=
-
-
= A(~2),
A(~2)N-~
so that for all M>O,
f
h(T.)a~l)(L1)1 I
i=l -
1
0p (1).
By a Taylor series expansion,
Thus, by 89, setting
Now fix £>0, Ml>O, Q>O.
Use (3.5) to choose y such that
Define
Then D<oo (G
N
and
Then Bl-B8, BlO, Bll imply that the assumptions of
Theorem A.l are met with g.= 1, A(¢,i)
IIQ.II::;M
= -h.1(0)
-
depends only on L1 l ).
Define M2 by
Using B9 and (3.7), find NO such that AN
~
Aoo/2 and
1
-
page 10
e
If 11~211
= M2, II ~lll
~ Ml , and N ~ NO' then with probability at least l-E,
~2WN(~)
T
2
~ -M2 11 W CQ) II + ~z H~ZA(ljI ) - M2D - M2y/Z
N
~ [A(ljI2) Aoo M2/2 - Y - D]M 2 = QM 2 .
Since ljI is nondecreasing,
A2WN(~1'~Zs)
is a nondecreasing function of s.
Thus,
11~211 ~ M2 implies
~2WN(Q.) ~ ~2WN(Al,MZ~21111zll-l)
~(111l211/M2)(M21121111211-lWN(1l1,M21121111211-l))
~ Illlzll Q
Thus,
o
which with the Cauchy-Schwarz inequality proves (3.2).
4.
The limiting distribution of f" is a simple consequence
Estimation of Jh
of the following representation.
Suppose Bl toB12 hold and A(ljI) > O.
Theorem 4.1.
tion (see
(1.10)~
(4.1)
L
i=l
1
(4.2)
T
1\
Til
ljI((Y.-C. 8) exp(-h(t.)
= ~i~'
f" is
any solu-
to
N·
with t.
Then if
1 -1 -
-
1\
(§O-~) = OpeN
TA
8)C. exp{-h(t.)
1
-k
-
-1
i=l
8)
I-
=0
"
2), and ~ satisfying (3.2), then
%1 \ . = N-~.~L S-1 d.ljI(e.)
.
-1
(A(ljI)) .
N (8-8)
- -
TA
1
1
+
0
.p
(1).
This result, strengthened so that the remainder is 0(1) almost surely, has been
established in the homoscedastic case by Carroll and Ruppert (1979).
page 11
Proof.
q
For ~l and ~3 in RP , ~2 in R and ~ = (~1'~2'~3)' define
a~I)(6)
1
h. (6'\
-1 :::J
= N-~
= h (T.1
d: ~
-1
-1
T
+ C. 6
- 1 -3
-~
N
),
and
Define the process
N
\
A.) = N-~.L
UN(=:
1=1
(1)
1
~((e.+a.
1
(2)
(~))(1+a1'
(3)
(6)) (d.+a.
-1
-:I.
(~)).
Note that.(4.l) can be rewritten as
(4.3)
Letting g.=
(e.,r,s)
1 l,¢.
11
= ~((e.+
1
r)(1 + s)), and
A(~,i)
= A(~),
the conditions
of Theorem A.l are implied by Bl, B3, B5 to B8, Bll, and B12, so for all M>O
(4.4)
su~
11611 :s;M
IIU (6) - UN(O) - A(~) s~111 =
N -
Now by Chebyshev's theorem and Bl,
UN (Q)
= .0p (1) .
Therefore, if we set
6*
=0
P
(1)
and therefore, by (4.4)
U(_6*) =
0 P (1)
•
.(1).
0
p
page 12
Consequently, by (4.3) and our convention (1.10),
UN(N~(i-.@), N~&-~), N~(k-.@)) = opel).
(4.5)
By Theorem 3.1,
/\
(~-~)
= OpeN -~),
so we need only establish that
(4.6)
to conclude from (4.3) and (4.4) that (4.2) holds.
But by (4.5), (4.6) holds
if for each n>O, £>0, and M there exists M satisfying
l
2
(4.7)
v
,
Now (4.7) follows from (4.4) in a manner quite similar to Jureckova's (1977)
o
proof of her Lemma 5.2.
2
If EljJ (e1) <
Corollary 4.2 .
.e
and the assumptions of Theorem 4.1 are met,
00
then
Proof.
\
.
Use (4.2) and Loeve's (1963, p 316) Normal Convergence Criterion
kAT
to show that N:? (.§.-..@) ~
D
T -1
xS
-+ N (0,
2
x EljJ (e l ) A (ljJ)
-2
)
for each x in RP •
Since B7, Bll and B12 show that
N
1 im (
N-+00
it
l
. 1
1=
II~.
1
11 2) -111 d. ,,2 = 0,
1
o
is easy to see that the criterion is met.
1\
1\
(one could use.§. instead of
(S -s)
.!::()-
= 0p (N-~),
~).
Then when
B7 and BID imply
sup
'<N
1-
A
10.-0.1 =
11
0
P
(1).
'"
.
(~-~)
page 13
~
Suppose for both
~
=
and
~
~'
=
lim E sup{ I~((l+x) (el+y)) - ~C.el)
I: Ixl
~
£
and Iyl ~ d =
o.
£+0
Then it is easy to prove that
IN-
"
l N
L ~(a~l
(y.-c~ S)) - ~(e.)1 = opel),
i=l
1
1 -1 1
and by the strong law of large numbers,
N-
1 N
L
~(e.) + E~(e.).
'1
1=
1
1
Moreover,
IIN-
N
l
L
i=l
~ sup
i~N
(with
I IAI I
1 N
II
2
la.-cr.1 (N- L Ilc·11
) = 0 P (1)
1
1
i=l
1
equal to, say, the Euclidean norm of the matrix A),
e.
so
N
-1 'i'
N
T P
A-2
L
i=l
a.
C.C.
1
-1-1
+
S.
Therefore, E~2(el) A(~)-2 S can be consistently estimated, and large sample
confidence ellipsoids and tests
5.
ofhypothe~es
IN - consistency of preliminary estimators.
for
a can
be constructed.
Since we require that our pre-
liminary estimate satisfy (1.5), we now give conditions which insure that
1\
an M-estimate
solving
~
N.
L
. 1
T /.
~(Y.-C. Sn) C.
1 -1 -v
1=
-1
=0
1\
satisfy (1.5).
This
~
is not scale equivariant, but this does not affect
/\
the limit distribution of
. 813.
~
is odd
~.
page 14
~
B14.
~ is Lipschitz continuous with Lipschitz constant L~
B15.
the Radon-Nikodyn derivative ~'satisfies E(~(x(el+Y)) l+a
~'(xel)xy) ~ KIxy I
for some K and a>O and all x and y
B16.
N-~(cri+ 1) Ilgill
B17.
sup(N- 2 (cr.+ 1) lid. II ) <
N
i=l
1
-1
B18.
the minimum eigenvalue, AN' of
1 N
~(xel)
-
-+ 0
2
2
00
-1 N I T
N 2 cr. E~ (cr.e l ) d.d.
. 1 1
1
-1-1
1=
satisfies lim inf AN > O.
N-+oo
Theorem 5.1.
Under B13 to B18, (1.9) holds.
We will apply Theorem A.l with ¢i (el,r,s) = ~(cri (e -r))(so ¢i does
1
not depend on s ), g.= cr.,
k.= N-~I Ie. II, A(~,i) = cr. E~l(cr.e ), a~3)= 0
Proof.
.e
1
1
1
and a~2) left undefined.
-1
.
By B13 and B14, (A.3) holds.
1
so that (A.4) and (A.5) hold.
II ~ II ~M
+ N-
k N
1IN- P
T
2
~(cr. (e.-d. ~)
i=l
1
1
L
. 1
1
1
'
By B15, for all r and r'
cr. E~ (cr.e ) d.d. ~I
1
1 1
-1-:1
v
Therefore, for all M>O,
.~ N
- N-
-1
1 NIT
1=
1
Also, (A.5) is implied by B15, and (A. 1), (A.2),
(A.7), (A.8), and (A.9) are easily checked.
sup
1
I
I
i=l
~(cr.e.)
1
1
= 0 (1) •
P
I
Now (5.1) follows exactly as Jureckova's (1977) Lemma 5.2, since
TAT A
Yi - f i ~ = cri (ei - f i (~-.@) ) •
o
page 15
APPENDIX
The following general theorem will be used when studying
Theorem A.l.
A"
~,
8 and
/\
~.
Let g.I
andI
k. I
(=g.N and k.1 N) be sequences of positive constants
such that
lim (sup k.+ k.g.)) = 0
(A. 1)
~
i~N
sup
N
N
(li=l
1
1
1
and
(A.2)
~
Let~.
1
2
2 2
k. + k.g.)
1
1
= c 1<
1
00
.
b e a f unctIon
.
f rom R3 to Rl satls
. f Ylng
.
(A.3)
E</>i(el,O,O) = 0
(A.4)
lim sup
for all i,
E{supl</>i(el,r,s )
k~
Ir I, Ir'
I, Is I, Is' I
~
-
</>.(e.,r',s')I:
1
1
k and Ir-r'
I,
Is-s I
I
~
ko}
~
COog i
for some Co and all 0<1 and all i,
(A.5)
sup
i~N
g~l E(</>(el,r, s)
- </>(el,O,O) - A(</>,i)r) = o(lrll+o. + Is 11+0.)
for positive constants A(</>,i) and 0.>0, and
lim
r,s-+O
(A.6)
Let o.~l)
1
'
sup
• <N.
E(</>. Cel,r, s ) - </>i (el,O,O))
2
= o.
1
1-
l
n
o.~2)
and o.~3) be functions from Rm to R , Rl and R , respectively,
l'
-1
such that
(A.7)
o.~£)(O) = 0,
(A.8)
II o.y) (~)
1
and
- o.y) (y)
II ~
ki
II~-I.II
m
for all x and y in R , i=l, ... ,N, and £=1,2, or 3.
n
of R satisfying
(A.9)
Let ~i (=~iN) be elements
e-
page 16
~
For ~
E
m
R , define the process
1;-N
UN(~) = N- P
.
-
I
. 1
()
()
(
¢. (e.,a. 1 (~), a. 2 (~»(x.+a.3)(~».
1
1=
1
1
-
1
-1
1
-
Then, for all M>O
(A.10)
Proof of Theorem A.1.
(A.Il)
E(UN(~)
Fix M>O.
- UN· (Q»
=
We will show that
N-~ i=I
I A(¢,i)a~I) (~)
1
-
X.+
-1
0
P
(1)
and
for each fixed
~,
and that there exists K depending upon M but-not 0 such
that for all 0>0 and all N
Since for any 0, we can cover a ball of radius M in Rm with a finite number
of balls of radius 0, (A.Il), (A.I2), and (A.I3) pFove that for each 0>0,
N
N--l'. I
1=1
.
A(¢, i)a?)
(Q)~i II ~ KO) = 1,
which proves the theorem.
To prove (A. 11), note that by (A.3),
N
N-~.I
E (UN (Q.) - UN (Q»
=
1=1
-~ ~L E(¢.(e.,a.(1)
N
i=l
1
1
1
(2)
(~),
-
a.
1
(~»
-
-
(3)
¢.(e.,O,O»(x.+~.
1
1
-1
1
by (A.I) and (A.S).
(A.I4)
Ilx.
+ -1
a~3) (~)
-1
II ~ 211_x~... II
E(¢. (e.,a~I)(~), a~2)(~»
for all large N, and by (A.S),
- ¢. (e ,0,0»
1
1
A(¢,i)a~l)(~) + O(g.k. I I~I l)l+a
1
1
1
1
-
-
1
-
1
1
=
.
(~»,
-
page 17
uniformly in i.
e
Therefore
N
E((UN(~ - UN(O)) = N-~.L A(~li)ail)(Q)~i
1=1
+ O(N-~
N
L (g.k.)l+a II~ill),
i=l
1
1
and by (A.2) and (A. g),
N-~
N
L (g, k, ) 1+a II x. II
i=l
~ (g, k. )
1
2
1
-1
\
k2 _
L g. . i=I 1 1
1 1
2~ N 2k
(LN g.12k,)
( L k.)
1
2
'1
1=
11
N
k a
-< g..
'11
1=
=
0
(1) .
Thus (A.I) holds.
Next, by (A.14) we have that for N large
Var(UN(Q.) - UN(Q.))
(2NIt
1 N 2
L g·llx;
, 1 1
1=
2
II
)
1
~
-2
(1)
sup g, E(~. (e, ,a. (~)
'<N
1-
1
1
1
1-)
(2)
a • (~)
1
-
~1' (el,O,O))
2
follows from (A.4), (A.5), (A.7), and (A.B) that
sup g,-2
'<N
1-
1
(1)
E(~,(e.,a,
1
1
1
(~),
-
a.(2)
1
(~))
-
-
Therefore, (A.12) is proveq by applying (A.2).
~.(e.,O,O))
1
1
2 = 0(1).
Finally, by (A.14) the RHS of
(A.13) is less than or equal to
N
2N-~iL II~i II
E
sup{
I~i (ei,a?) (~,
~.(e"a~l)(~*),
a~2)(~*)I:
IIQ.II~
111
1-
M,
2
ai ) (~) -
II_~II~
M,
II_~-_~*II~ o}
which by (A.2), (A.4) to (A.9), and the Cauchy-Schwarz inequality does not exceed
N
2 ~ COo = 2C C~ O.
sup 2( L\ (g.k.))
O1
N
i=l 1 1
Therefore (A.13) is verified.
o
e-
REFERENCES
Anscombe, F. J. (1961), "Examination of residuals", Proc Fourth Berkely
Symp Math Statist Prob (J. Neyman, editor), pp 1-36. University of
California Press, Berkeley and Los Angeles, California.
Bickel ,Peter J. (1978), "Using residuals robustly I: Tests for heteroscedasticity, nonlinearity;' Ann Statist 6, pp 266-291.
Bickel, Peter J. and Doksum, Kj ell A. (1978), "An analysis of transformations
revisited", unpublished manuscript. University-of California, Berkeley.
Box, GeorgeE. P.and Cox. David R. (1964). "An" analysis of transformations",
.~ Roy Statist Soc,· Series B 26, pp 211-252
Box, George E. P. and Hill, William J. (1974), "Correcting inhomogeneity of
variance with power transformation weighting", Technometrics 16,
pp 385-389;
Carroll, Raymond J. (1979), "Robust transformations to achieve approximate
normality", to appear in ~ Roy Statist Soc, Series B.
Carroll, Raymond J. and Ruppert, David (1979), '~lmost sure properties of
robust regression estimates", unpublished manuscript.
Fuller, Wayne A. and Rao, J. N. K. (1978), "Estimation for a linear regression
model with unknown diagonal covariance matrix", Ann Statist 6,
pp 1149-1158.
Huber, Peter J. (1964), "Estimation of a location parameter", Ann Math Statist
35, pp 73-101.
Huber, Peter J. (1973), "Robust regression: Asymptotics, conjectures and MonteCarlo", Ann Statist 5, pp 799-821.
Jureckov~, Jana (1977), "Asymptotic relations of M-estimates and R-estimates
in linear regression model", Ann Statist 5, pp 464-472.
Lo~ve, Michel (1963), Probability Theory, 3rdEdition.
New York.
D. Van Nostrand,
Yohai, Victor J. and Maronna, Ricardo A. (1979), "Asymptotic behavior of
M-estimates for the linear model", Ann Statist 7, pp 258-268.
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~.1-Estimates
RECIPIENT'S CATALOG NUMBER
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18. SUPPLEMENTARY NOTES
It. KEY WORDS (Contlnu. on ,.v.,•••Id. If n.c . . . .ry . .d Id.nllty by block numb.r)
Weighted M-estimates, weighted least squares, estimated weights, linear regression in scale, robustness, asymptotic representation, asymptotically normal,
confidence ellipsoids.
20.
•
ABSTRACT (Contlnu. on ,.ve,. . .Id. If n.c. . . .ryt=d Id.nllty by block numb.,)
We treat the linear model Yi : fi~ + Zi where fi is a known vector, f3 is an unknown parameter, and Var Z.1 is a function of IcT~j
which is known, except for a
-:t
parameter 8. For simultaneous M-estimates, ~ and
we show that (a-e) :
1
2
0p(N- / ) ,-and find the limit distribution of N1/2(1_~. Moreover,-the covariance
matrix of the limit distribution can be consistently estimated, so large sample
confidence ellipsoids and tests of hypotheses concerning S are feasible.
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