Second Order Asymptotic Distributional Representations
of M-Estimators of General Parameters
Jana Jureckova
and
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1828
July 1987
SECOND ORDER ASYMPTOTIC DISTRIBUTIONAL REPRESENTATIONS
OF M-ESTIMATORS OF GENERAL PARAMETERS
v
~
BY JANA JURECKOVA AND PRANAB KUMAR SEN
Charles
University~ Prague~
1
and University of North Carolina at Chapel Hill
Second order asymptotic distributional representations of M-estimators of
general parameters are considered under appropriate ( smoothness) conditions on
the score function and the underlying distribution function. Parallel results
for the one-step version of M-estimators are also studied. These results apply
to the classical maximum likelihood estimators as well.
AMS Subject Classifications. 60F17, 62E20, 62F35, 62G05
Key words and phrases . Influence function; M-estimator, one-step version of
M-estimator; random change of time; score function; weak convergence of M-processes.
1) Work of this author was partially supported by the Office of Naval Research,
Contract No. N00014-83-K-0387.
11. Introduction. Let {X.;i
> I} be a sequence of independent and identically dis-
_
tributed random variables
eo
where x E Rand
E
e
absolutely continuous in
(i.i.d.r.v.) with a distribution function (d.f.) F(x,e )
o
,an open interval in R. Let
e
e=
to t E 8
: R
e
x
~
R be a function,
satisfying some other regularity conditions ( to be
specified in Section 2 ). We assume that E
unique zero at
~
e
~(Xl,e)
exists for all e E
e
and has a
o
e • A consistent estimator M which is a solution ( with respect
o
n
) of the equation
(1.1)
is termed an M-estimator of
eo
(corresponding to the score function
~
). Then [viz.,
Janssen, Jure~kova and Veraverbeke (1985)] there exists a sequence {M } of roots of
n
(1.1), such that as n
k
n 2( M
n
(1.2)
- e0
~
)
00
,
= 0 (1)
p
and
(1.3)
M
n
e0
= -(nyl(e ))
-1
o
n
L i =1 ~(x.,e
1
0
)+
R
n
R = o (n
n
p
-1
),
where
(1.4)
~(x,e)
The specification of R
n
=
=
(a/ae)~(x,e),
e
E
e
O(n-l), in probability, is known as the second order
representation of M-estimators. For the simple location model, Jure~kova (1985) and
Jure~kova and Sen (1987) have shown that under fairly general regularity conditions,
nR
n
( or n
3/4
R) has asymptotically a nondegenerate distribution ( which is typin
cally non-normal ). This second order asymptotic distributional representation
(SOADR) of M-estimators of location is analogous to the Kiefer (1967) SOADR result
for the sample quantiles, and in (refined) asymptotic statistical inference problems,
these SOADR results playa vital role. The same non-normality feature appears in
the SOADR of a general class of ( von Mises ) differentiable statistical functionals~
although the SOADR results in all these cases do not conform to a common
pattern [ viz., Sen (1987)]. It is also known that usually M-estimators ( of location)
may be characterized as von Mises functionals (but defined implicitly). However,
this may entail quite restrictive regularity conditions on the score functions,
~
3
and, generally, for unbounded score functions, the SOADR results for M-estimators
(of general parameters) may not follow from those on the von Mises functionals.
Given the affinity of the M-estimators to the classical maximum likelihood estimators and their good robustness , asymptotic minimaxity and other properties, there
is considerable interest in the study of SOADR results for such general M-estimators.
These considerations lead us to the query : Under what conditions ( on
and F )
~
there exists an asymptotic distribution of (nR ) in (1.3), and, whenever it exists,
n
what is its form? Looking back at the location model, we may observe that enough
smoothness conditions on the score function may be needed, and for score functions
2
having jump discontinuities, we may need to consider n 3 / R
n
instead of nR
n
for
such a SOADR to hold. The object of the current investigation is to study the
SOADR results for general M-estimators under general smoothness conditions and to
provide an affirmative answer to the query made before.
Along with the preliminary notions, the main SOADR results are presented in
Section 2, and their derivations ar:erele"gaFed t,q,,,the:n.e.xt
of maximum likelihood estimators,
~ection.
Like the case
salu.tion of (1.1) can not be found
.o,_;t~n ,the
explicitly, and iterative methods are used eto derive the same. In this context, an
one-step version of M-estimators is often used, .and these adapted estimators are
generally first-order efficient. Therefore, it is of interest to study parallel
SOADR results for them too, and these will be considered in the last section.
~(x,8)
(2.1)
(2.2)
is absolutely continuous in 8 , and the following conditions hold
A(8) = E
8
~(Xl,8)
exists for all 8
€
8 and has a unique zero at 8
=
o
~(x,8) is absolutely continuous in 8 , and there exist a 0 > 0 and a
positive constant K2 ' such that E
8
where ~ (x,8) = (a/ae) ~(x,e);
0
2
..
~(Xl,80+t) 1 ~
(2.3)
Yl (8 0 ) , defined by (1.4) is non-zero and finite;
(2.4)
E ~ (X ,8)
l
8o
2
<
00
in a neighbourhood of 8
o
K , for It I < 0
2
8
o
4(2.5)
there exist a >
a
and a function H(x,8 )' such that E IH(X ,8 )
o
l o
8
I ~(x,8 +t) - ~'(x,8 ) I <
00-
a
..
I
<
00
and
0
It I
a.e. [ F(x,8 ) ], for It I < 8(> 0).
0
H(x,8)
0
Note that (2.2) ensures that E 1~(Xl,80 +t)
8
I
~
o
Kl ~ K; , for
~
It I <
8. Then, it
is known [ viz., Janssen, Jure~kova and Veraverbeke (1985)] that under (2.1), (2.2),
(2.3) and (2.4), the results stated in (1.2) and (1.3) hold. Under the additional
condition in (2.5), we formulate the following SOADR result.
THEOREM Z.l. Suppose that the conditions in (2.1) through (2.5) hold, and let M
n
be the M-estimator of 8
o
far which (1.3) holds. Then, as n
+
00
(2.6)
where
fI).
(2.7)
y 2 (8)
=
E8~(Xl ,8)
and S = ((s . .J) is a 2
1.-J
X
,
(E;;1,E;;2)
Xi
rv
0, S)
2 matrix with the elements
(2.8)
(2.9)
(2.10)
The proof of the theorem is considered in the next section. We make some remarks
here pertaining to the theorem. First, note that if we let
*=
-1
(2.11)
E;;l
E;;l - s12 s ZZ E;;2
(2.1Z)
sll.2
= sll - slZ s 2Z s Z1
-1
*
and E;;Z
and
E;; *
E;;2
S*
* E;;2* )
= ( E;;l'
= Diag( sll.2' s22) ,
then the right hand side of (2.6) can be written as
(Z y (8 o »
l
(2.13)
-1
*
Y (8 )]( E;;Z )
Z o
Z
where E;;l* and E;;Z* are mutually independent. Note that sZ2 in (Z.lO) is a non-zero
finite constant, and hence, E;;Z* is also a nondegenerate r.v. If, in particular,
a with
sll.Z' defined by (Z.lZ), is equal to 0, then
equal to
probability 1.
In this case, (2.13) reduces to a multiple of
, and hence, upto a multiplicative
factor, it has the chi square distribution with one degree of freedom. However, for
sll.Z >
a ,
the distribution of (Z.13) is not expressible in terms of such a chi
square distribution, and it is different from the type arising in the case of von
Mises' functionals. We may also note that the M-estimators include the classical
~
s
maximum likelihood estimators (MLE) as special cases. For the MLE, we have
(2.14)
1jJ(x, e)
= (0/08) log f(x,8) ,
x
R,
E
8
E
8
so that for (2.1) through (2.5) to hold, we need the third order derivatives of
the log density functions ( with respect to 8 ) along with some other compactness
conditions on them. Often, these conditions may appear to be rather restrictive,
and hence, for M-estimators other than the MLE it may be easier to verify (2.1)
through (2.5). However, dealing with the MLE, we may note that for the density
f(x,8) belonging to an exponential family, we have
(2.15)
1jJ(x,8)
= a(8)g(x) + b(8), for suitable a(8),b(8) and g(.).
As such, using the fact that E1jJ(X , 8) = a , we have Eeg(X ) = -b(8)/a(e), 8 E 8
l
l
Further, in this case, we have
..
2
[ aCe)] var(g(X
(2.16)
2
(2.17)
[aCe)] var(g(X
l
l
»,
»
s12
=
•
[a(8)a(e)]var(g(X
l
»,
and sIlo 2 = O.
Thus, in this case, in (2.13), we have
*=0
~l
with probability 1, and hence, the
SOADR relates to a chi square distribution (with one degree of freedom). If, further
•
(2.18)
.""
a(8)/a(e) + [b(8) -
degenerat~
then (2.13) reduces to a
In this special case, nR
~.
b(e)a(8)/~(e)]/
n
,
II
[b(8)~b(8)a(e)/a(e)]
=
0,
r.v. ( with the entire probability mass at 0).
converges in probability to O.
In passing, we may remark that for the particular case of the location model
(with a fixed scale), if the d.f. F is sywmteric and the score function
'.
'
~
is
SOADR in the location model. In passing, we may remark that if the score function
3 4
ljJ(.) admits of (finitely manny) jump discontinuities, then n / R
n
has a SOADR [ viz.,
v
" and Sen (1987)], so that we have a slower rate of convergence.
Jureckova
3. Proof of Theorem 2.1. For notational simplicity, we denote y.(8
) by y.J , j=1,2,
J 0
~~~~~~~~~~~~~~~~~~~~~
and also suppress the index 8 in E(.), P(.), var(.) and cov(.). Consider then
0
the random process
(3.1)
Y
n
Y = { Y (t), t
n
n
-1 n
Y (t) = {Yl
[-B,B] }, where
£
L:. l[ 1jJ(X.,8 +n
n
~=
~
~
-~
~
t)- 1jJ(X.,8 )] - n 2t}, It I _< B ,
~
0
0
a
< B < 00.
belongs to the space D[-B,B] , and it plays the basic role in the proof of the
theorem. First, we consider the following.
Lemma 3.1. Under the hypothesis of Theorem 2.1, Yn converges in law (in the Skoro-
D[-B,BJ ) to a Gaussian process Y = {Y(t), t
khod J 1 -topo1ogy on
(3.2)
and
= t~l
Y(t)
£
[-B,BJ , B« 00) is fixed,
R, we define
£
-1
n
Z (t) = Yl {L:. l[ 1jJ(X.,8 +n
n
~=
~
0
-~
0
t) - 1jJ(X. ,8 )]}, Z (t) = Z (t) - EZ (t) .
~
0
n
n
n
Note that by (3.3), for an arbitrary ~ = (~l""'~p)' and t
(3.4)
[-B,BJ}, where
sl is defined as in (Z.7).
Proof. For every t
(3.3)
- (ZY1)-lYztZ , for t
£
-2 p
p
var { L:j=l ~jZn(tj)}
(tl, ... ,t )
p
t,
P ~ 1,
p
Yl L:j=lL:k=l ~j ~k{nsn(tj,tk)}'
where
(3.5)
Sn(tj,t ) = cov[ 1jJ(X ,8 0+n
k
1
-~
t-'j),-1jJ(Xl,80),1jJ(~,eo+n
t )-1jJ(X ,8 ) ],
1 0
k
t , t : t , t £ [-B,B]
k
j
j
k
for j,k =l,oo.,p. Next, we show that uniformly in
(3.6)
nsn(tj,t k )
titks
+
ll
, as n
-~
00.
+
We prove (3.6) only for j = 1, k=2 , as a similar proof holds for the other cases.
Denote by
(3.7)
Note that for every t £ [-B,B] , we have
(3.8)
2
E[A (X ,8 +n
n l 0
t/m
fa
= E[
1
~ n-~lt'E[
~
n-~ It
2n
~
I
-1 2
-~
t)] = E[
{~(~,80)
t/m
fa
+
t/In •
fa
1jJ(x ,e +v)dv]
l o
v~(~,80+
·2
(a~h<l)
hv)}dv ]2
•
2
0
{1jJ(x ,e ) + v1/J(X ,e +hv) } dv
l o
l o
E[ 2n-~, t I'1jJ 2 (xl,e o) + 2
t E[ 1jJ
2
(~,eo)]
+ 2n
fat / In
-~I t , fat/m
v 2··2
1jJ (X ,8 +hv)dv
l u
2
"'2
]
I
v E[ 1jJ (X ,8 0+hv) ]dv
l
2n -1 t 2sll + (2/3)n -3/2, t 13K2 ' by (2.2).
Similarly, for every t E [-B,B] ,
(3.9)
Combining (3.8) and (3.9), we arrive at (3.6) directly from the quadratic mean
equivalence of A (Xl,t) and n
n
.
covarlance structure n
-1
-~
•
t~(X,8)
and the fact that n
°
1
-k •
2t~(X
l'
8) has the
°
tjtks ll . Further, (3.3), (3.6), (3.9) and the classical
central limit theorem imply that the finite dimensional distributions of the
process
{zo(t), t
n
t E [-B,B]}
(3.6) ,
as n +
[-B,B]}
E
converge to those of
here, ~l is defined as in (Z.7). Note that by (3.4) and
00
varO::f=l ,\Zn(t j )} +
sll(~'~)Z
, for every
Therefore, for every t l ' t z ' such that -B <
(3.10)
E{ !zo(t) - ZO(t )
n
n
l
II
ZO(t ) - ZO(t)
n Z
n
< [ E{ ZO(t) - ZO(tl)}z
n
+sl1{ [ (t - t )
l
n
Z
+
~
tl
t
E
j
[-B,B], j=l, ... ,p.
t z < B , we have for t E (tl,t )'
Z
}
E{ ZO(t ) - ZO(t)}Z
+ ( t z - t)
n
Z
Z
]/Z}
n
2.
sl1 (t z
Consequently, by a modified version of Theorem 15.6 of Billingsley (1968,p.1Z8)
[viz., Lemma 3.1 of Jure~kova (1973)], we conclude that ZO is tight. Then looking
n
at (3.1) and (3.3), it remains only ,to show th'a-t
(3.11)
k
EZ (t) - n 2 t
+,
n
O,','as:h +
00.,
uniformly in t
E
[-B,B].
For this, it suffices to show that
(3.lZ)
Towards this, we make use of the compactness condition in (Z.5), so that the left
hand side of (3.lZ) can be bounded from above by
=
(3.13)
and this converges to 0 as n +
00.
-a/Z
D(n " , ) , t E [-B,B],
a. > 0 ,
This completes the proof of the lemma.
The main idea of the proof of Theorem Z.l is to make a random change of time
t +
k
n 2( M - 8 ) in the process Y , defined in (3.1). This will be accomplished
n o n
in several steps. First, we extend Lemma 3.1 and establish the weak convergence of
a two-dimensional process
(3.14)
Y*
n
= { Y*(t) = ( Y (t), n k2( M - 8 »
~n
n
n
0
,t
E
[-B,B] },
where we may note that the second component of (3.14) is independent of t .
Y*n
Lemma 3.2. Under the hypothesis of Theorem 2.1,
* ={
khod topology) to a Gaussian function Y
converges in law (in the Skoro-
(tt,;l + (Y2/2Yl)t
2 ,t,;2) , t E: [-B,B]}
I
where t,;l' t,;2 are defined in (2.7).
Proof. By (1.3) and Lemma 3.1, Y* is convergent equivalent (in probability) to
n
(3.15 )
yo*
n
= {
= ( Y ()
-~,-l n
) ),tE:[-B,B].
,
}
n t , - n y l L.1=1 4J(x.,8
1
0
yO*(t)
n
Now, the tightness of Y
(proved in Lemma 3.1 via that of ZO ) and (1.2) imply
n
n
0*
the tightness of Yn . Further, the convergence of the finite dimensional distribu0*
tions of Yn
follows along the same line as in the proof of Lemma 3.1, since the
second component in (3.15) is also adaptable to the central limit theorem. Hence,
the details of the proof of the lemma are omitted.
Returning now to the proof of Theorem 2.1, we define
(3.16)
[alB
=
aI(-B
2
a
2
B)
, for every real a and positive B .
Thus [alB is equal to 0 outside the compact interval [-B,B]. Similarly, we define
(3.17)
* = {Iy * (t)]B =
[Y]B
~n
n
(Y (t), Ink2(M - 8 )]B) , , t E: I-B,B]} .
n
n
0
Then, by Lemma 3.2, we obtain that as n
(3.18)
~
[Y:]B
+
00
, t E: I -B,B] },
{(tt,;l - (2y ) -lyz .t,z, [t,;2]B)
l
for every fixed B( > 0); the right hand side of (3.18) is Gaussian and has continuous (a.e.) sample paths. At this stage, we refer to Section 17 of Billingsley
(1968, pp.144-l45), and conclude that by (3.18) and the random change of time:
M -8)]
, we have for every (fixed) B >
k
:i:S
noB
(3.19)
y ( ([n 2(M -8) ]B»
n
n
0
0,
-+
Now, (t,;1,t,;2) , has a bivariate normal distribution with a finite dispersion matrix
S, defined by (2.8)-(2.10). Hence, for every E: >
0, there exists a B ( > 0), such
o
that for every B > B ,
-
0
Similarly, by virtue of (1.2), there exists an n , such that
o
(3.21)
p{
1
n ~I M
n
- eo I
> B
} < E:
, for every B > B
-
Combining (3.19), (3.20) and (3.21), we obtain that
0
and n >
-
n0
9
(3.22)
lim
!<:
n~
--
!:2
< lim
F{ Y ([n (M
n
n~
= p{ ~1[~2]B - (2y l )
~
~1~2
p{
e0
p{ Y (n 2 ( M n'
n
n
-1
- (2y )-lY
2
l
-
»
<
e0 )]B)
-
<
}
< y } +
Y2([~2]B)
~~
y
2
y}
€
~ y } +
+ 3£
€
, for every real y .
Similarly,
(3.23)
The proof of the theorem is complete.
4. SOADR results for one-step M-estimators. Often, it may be difficult to find an
~---------------------------------------
explicit and consistent solution of equation (1.1), and it may be easier to employ
an iterative procedure I viz., Dzhaparidze (1983)]. Star.ting with an initial
'
,
.
M(o)
(consistent ) est1mator
we may conS1'd er t h e succeSS1ve
step est1mators
as
if
A(k_l) = 0
n M(k-l)
M(k)
n'
Yl n
(4.1)
n
=
ljJ(X , Mn(k-l»
, if
-I 0,
{
n
,n
1 = 1,
,n
M(k-l)_(n~l(~k;"'l<~-l'L:~;'l
~l(k-l)
for k =1,2, ... , where by virtue of theassumption~ in (2.2)-(2.5), we may take
(4.2)
It follows from the general results of Jansseni,'Jure'c"kova and Veraverbeke (1985)
that under the assumed regularity conditions ( of Section 2),
M ) = 0 (1) and n(M(k) - M ) = 0 (1) , for k > 2,
n
p
n
n
p
' . . 1 est1mator
.
M(o) 1S
, square-root n consistent, i.e.,
provided t h e 1n1t1a
(4.3)
n( M(l) n
n
M(o) _
(4.4)
n
Thus, for k
~
e
0
= 0p (1).
2, there is no need to study the SOADR results, as they would agree
with that of M in Section 2. The case of M(l) is , however, different, and we
n
n
shall study the same in this section. We introduce the following notations. Let
(4.6)
.
!<:
-1 n
= n 2( M(o) n L:i=l ljJ(X 1.. e0 ) - Y1 ) , Un,2
n
,
-!:2
n
= n L:i=l ljJ(X. , e ) and U = (U l' U
U
n,3
n,2' Un ,3)
1 0
n,
1
(4.5)
U
n,l
n"2(
We assume that as n increases,
~n
e0 )
.
10
dj
(4.7)
U
--*
U
n
,...j
X(3
, r
0
r
) ;
has finite elements .
~
~
I t may be remarked that whenever M( 0) admits a first order expansion, i.e. ,
n
-1 n
8 )
n L:i=l ¢( X.,
1
0
M(o) - 8
0
n
(4.8)
for some suitable score function
¢(x,8), x
-~
+
0
€
R, 8
p
(n
~
),
G , then, we may readily use
the multivariate central limit theorem for the verification of (4.7). Also recall
that the r.v. 's
~l'
~2
.
in Theorem 2.1 relate to U and U (upto the scalar factor
3
l
-1
Y ). Then, we have the following.
l
THEOREM 4.1. Under
(4.7) and the regularity conditions of Theorem 2.1~ for the
.
(1)
one-step M-est1.-mator M
n
(4.9) nR(l)
n( M(l)
n
~
7__.f'
.J
we rIJ.A-ve
or n .....
00
~
U*
n
where
Proof. Before we sketch the outline of the proof, we may note that the first term
on the right hand side of (4.10) agrees with the right hand side of (2.6). Also,
i f the initial estimator M(o) is square root n equivalent to M
n
P
-+
a , as n
-+ 00,
n
ji. e. , n\M(o) - M )
n
n
-1
then (4.8) holds with ¢(x, 8 ) = -Y lj!(x,8 )' so that U + Y U
l 2
o
' ,0
l
3
= 0 with probability one. Hence, the second term vanishes for this special choice
of the initial estimator M(o). Otherwise, the second term on the right hand side
n
,
k
of (4.10) reflects the contribution of n 2 (M
(o)
n
- M ) in the SOADR result, and it
n
emphasizes the importance of an efficient initial estimator in the asymptotic study
of one-step M-estimators.
Returning to the proof of the theorem, we note that'by
v~rtue
of the assumptions
made in Section 2 and in (4.7), we have
1
/'<
n~( Y
l ,n
k2
l
(4.11)
A_
n
(4.12)
n
(4.13)
(
-~
Yl • n
n
U 1 + y U 2 + 0 (1),
n,
p
2 n,
2
-1
ylUn,l - Y2Yl Un ,2 + 0p(l) ;
Y )
l
-1
Yl )
(0)
L:i=l lj!(Xi,Mn
) = Un ,3 + y l un ,2 + n
-~
(U n ,lU n ,2 +
2
Un • 2 Y2 ) +
We may note that in this context, (2.5) and the stochastic convergence of n
..
lj!(x.,8)
1
0
to Y playa basic role
2
0
p
(n
-~
).
-1 n
L:i=l
for brevity, we omit the details of the manipu-
lations leading to these asymptotic expansions. Next, we note that
e
)
11.
nR(l) = n(M(o) _ 8 ) - ~-l En, ,''(X, ,M(o»
n
n
0
Yl,n ~=l~ ~ n
(4.1 4)
+
-lEn ,II(X 8)
Yl i=l~ i' 0 '
so that by (4.11), (4.12), (4.13) and (4.14), we obtain by some routine steps that
(1)
)-1
U2
-2
-2
nRn
= (2Y l
Y2 n,2 + Yl y 2 un ,2 un,3 + Yl Un ,lUn ,3
( 4.15)
+ 0p(l)
The rest of the proof follows by using (4.7) in conjunction with (4.15). Q.E.D.
We conclude this section with the following SOADR result. Looking at (4.3),
we may observe that the second order difference between the one-step M-estimator
M(l) and the
original estimator M is given by
n
n
Z(l)
= n( M(l) - M )
(4.16)
n
n
n
I f for the initial estimator M(o) , we use M , then,
by noting that M is a
n
n
n
solution to (1.1), we may as well claim that the one-step version corresponding
.
to this initial choice is also equal to M
n
U
n,
2 by
As such, using (4.15) I but replacing
1
ri2 (M
-1
- 8 )
n o ' and hence, by (1.3), by -Yl Un ,3 ], we obtain by the same
approach as in the proof of Theorem 4.1 that under the assumed regularity conditions.
nR(l) - nR
(4.17)
n
n
3 -1
(2y ) Y ( U 3 + Y U
)2 +
l
2
n,
. ~J1,.2
Therefore, we conclude that
..{~
(4.18)
(2 yi >,-1
}
~2 ~j
0
(1).
p
..
)
U + Y U )2,
1 2
J
has the multinormal law in (4.7). Note that U + Y U has a
1 2
3
normal distribution with 0 mean, so that, apart from a multiplicative factor, the
where U = (Ul,UZ,U J )
right hand side of (4.18) has the
freedom. Note that
(2y~)-lY2
centra~c~i-s~uare distribution
with 1 degree of
is a constant, and hence, the distribution of
z~l)
is confined to the positive or negative part of the real line according as Y /Y
2 l
is positive or not. For differentiable statistical functionals, a similar but more
complicated form of the asymptotic distribution for jackknifed residuals has been
obtained in Sen (1987); (4.18) corresponds to a special case when only one of the
eigenvalues is non-zero. Finally, we may remark that (4.18) conveys two advantageous
points in favor of the one-step M-estimators. First, if the score function \jJ(x, 8 )
o
is such that Y2
as n
+
00,
h
O, ten,
Z(l) ~
-,
n
o(
and hence, converges in probability to 0 )
l
so that M(l) and M are stochastically equivalent upto the order n- .
n
n
12
This point merits particular attention in the usual location model where
~(x-t).
so that if
~(.)
~(x.t) =
is skew-symmetric (about 0) and the d.f. F is symmetric.
= O. Secondly. if the initial estimator M(o)
( which may not be an M-estin
Qo
mator) admits of a representation as in (4.8). then one should choose M ) in
n
then Y
2
such a way that
~(x.80)
and y ¢(x.8 ) are close to each other, i.e., the influence
l
o
•
function of M(o) and that of M should be close to each other. This explains the
n
n
role of influence functions in the choice of suitable initial estimators from the
f
point of SOADR results.
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