ON PRELIME~ARY TEST AND SHRINKAGE
M-ESTIMATION IN LINEAR MODELS
by
Pranab Kumar Sen and
A.K.~.D.
Ehsanes Saleh
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1493
December 1985
ON PRELIMINARY TEST AND SHRINKAGE M-ESTIMATION IN LINEAR MJDELS
PRAl:\JAB KUMAR
sEif
and
A.K.MJ. EHSANES SALEH 2
University of North Carolina, Chapel Hill, and Carleton University, Ottawa, Canada
In
a general univariate linear rrodel, M-esti.nation of a subset of
pararreters is considered when the complementary subset is plausibly
redundant. Both the preliminary test and shrinkage versions of the
usual M-estinB.tors are considered (along with the classical versions),
and, in the light of their asymptotic distributional risks, t.l-}e
relative asymptotic risk-efficiency results
a~e
studied in detail.
'Ihough the shrinkage H-estimators may daninate their classical versions,
they do not , in general, dominate the preliminary test versions.
]-.
J!l.!Eg;lllSj.J-.9E~
COnsider the usual linear rrodel :
B
-n-
= A
(1.1)
where
A
-n
B
is an n
+ e ; e = (e , ••• , e )
-n
-n
n
l
(6 " " , B ) I is a vector of unknown (regression
1
x
p
Pararreters,
p (design) matrix of known regression constants, n > p ::.. 1 ,
and the errors e.
1
are independent and identically distributed
AMS Subject Classification Nos : 62C15, 62FIO, 62G05, 62H12
Key vJords and Phrases: Asymptotic distributional risk; asymptotic distributional
risk-efficiency; James-Stein rule; linear rrodel; local alternatives; M-estimators;
minimaxity; preliminary test; robustness; shrinkage estimator.
1) vJork supported by the National Institutes of Health, COntract No. N01- HV12243-1, •
2) WOrk supported by the NSERC ( Canada), Grant
No. A3088 and by Gr-5 grant
from tlle Faculty of Graduate Studies and Research, Carleton University ,
ottawa, Canada.
-2-
(i'i'd') random variables
(dof·)
(r'~
with a distribution function
F, defined on the real line R.
t'r~lity,
we may assume that A
Without any loss of gen-
is of rank p, and consider the
-n
following partitioning (where p = PI + P2' PI
(~~ =
(1. 2)
(~i
'
~2)
and
A
=
,~n
(A
-n 1
'
~
0, P
2
~
0):
..
A
-n 2)'
n x
PIx I P 2 x I
so that (1.1) may also be expressed as ~n = ~nl~l + ~n2~2 + :n°
We are primarily interested in the estimation of
plausible that
~2is
"close to" O.
~l
when it is
This situation may arise, for
example, in a multi-factor design, vlhere
~l
stands for the main-
effects dnd 02 for the interactions; it may be quite likely (though
can not be taken for granted)
that all the interactions are insig-
nificant and one may then be mainly interested in the estimation of
the main-effects.
models.
Other examples of this type abound in linear
Also, instead of the null pivo!: for
o
I~,
0
if we have ;:my
0
other specified ~2' then working with ~n = ~n-~n2~2' we may reduce
the pivot to O.
Instead of the classical least squares estimators (LSE)
(opti-
mal for normal F) or the maximum likelihood estimators (MLE)
(based
on some assumed form of F), we shall be more interested in some general robust estimators, namely, the M-estimators (which contain both
the LSE and MLE as special cases).
For the global (unrestrained)
model in (1.2), we denote an M-estimator of
that
~ln
is an unrestrained r1-estimator
properties of
~ln'
B by
(~ME)
of
~n =(~in'
~l'
~2n);
so
For various
v
'
we may refer to Jureckova
(1977), Yohai and
Maronna (1979) and Singer and Sen (1985), among others.
for the restrained model'.
X
_n --
A
B
_n 11
_
Secondly,
+ e_n (l" .e. , B
0)
_2 =
_ , let B
_I n
•
-3-
be the corresponding M-estimator of
ed H-estimator
(~)
of ~l·
is 0
than the UME when B
~, 2
This
~l;
~ln
is termed a restrain-
generally performs better
Rl'1E
(or very close to 0) •
~,
But, for ~2 avlay
from the pivot 0, the ro1E may be considerably biased, inefficient,
?nd, even, possibly, inconsistent, while the UME retains its performance characteristics steadily over the variation of §2.
For
this reason, often, to incorporate the rather uncertain prior information on
~2
~l'
in the estimation of
a suitable (M-) test sta-
In a
APT
preliminary test M-estimation (PTl'1E) formulation, the PTME B
is
-In
tistic (for testing
chosen as the ENE or
Ho:~2=~)
mm,
is taken into consideration.
according as this preliminary test leads
to the acceptance or rejection of H.
o
The shrinkage M-estimator
(SME), based on the usual James-Stein (1961) rule, incorporates
the same test statistic in a more smoother manner.
"close to" 0, generally, both the
PT~'E
and
S!,1E
When
~2
is verv
perform better than
the UME, but the EME may still be better than either of them.
the other hand, for
~2
On
away from 0, the RI,m may perform rather
poorly, while both the PTf1E and SHE are robust.
This relative
picture on the performance characteristics of all the four versions
of M-estimators can best be studied in an asymptotic set
to that in Sen (1984) or Sen and Saleh (1985).
UD
similar
Shrinkage M- estima-
tion of the multivariate location has also been studied in the same
vein by Saleh and Sen (1985).
The object of the present study is to
focus mainly on the linear models.
In passing, we may remark that
for the particular case of Pl=O, i.e., P2=P, we have the classical
shrinkage model, while for PI :
1, we have a partial shrinkaoe model,
not treated in this generality in other places.
-4-
The proposed PTME and SME, along with the preliminary notions,
are presented in Section 2.
The notion of asyreptotic distributional
risk (ADR) is considered in Section 3, and, in this light, the ADR
~~-estimiltors
n-'sul ts for the' various ve'rsions of the
in the same section.
~~ncy
arc cons i c!f'n'(l
The main results on the asymptotic risk-effic-
(ARE) of the different versions of M-estimators are presented
in Section 4.
The concluding section deals with some general dis-
cussions (including the asymptotic (distributional) minimax character
of these
2.
estimator~.
The Proposed PTME and SME.
tion ~) = HJ(x), x
~'7e
R} needed for the definition of H-estimators.
E
assume that
(2.1)
where
~j
First, \"e introduce the score func-
til
~l
and
(-x) = -
~2
(x) =
1/) 1
(x) +
(x)
,
X
L
R,
are both nondecreasing and skew-symmetric (i.e.,
~j(x),
YX,
j=1,2);
bounded interval in R and
many jumps.
~2
~2
~l
is absolutely continuous on any
is a step-function having only finitely
Also, we assume that there exists a positive and finite
constant k, such that
lV(X)
=
~(k)
sism x, for Ixl .::: k, and ~) is non-
constilnt on [-k,kl, so that
a
(2.2)
Let then A"
..·n
(2.3)
<
o~2 = J R ~2(x)dF(X)
<
00.
(al~,···,a~),
and for every n(_>l)
~
~n
~n(~)
and b
E
RP , define
= (Mnl(~) , ... ,Mnp(~)) ~
I ~ =1
~ i ~)
(xi - ~ ~ ~ i)'
b
E
P
R .
Also, we assume that the dofo F (of the e.) is symmetric about 0,
1
so that
(2.4)
!R
~(x)dF(x)
=
a
...
-5Further, we let
( 2.5)
=
C
-~n
(
A' A.
~nr~nl
A'
A )
~nl~n2
A' A
A' A
~ri2~nl
( ~nll
~n2-n2
,
~n12)
~n22
-n2l
and assume that as n increases,
(2.6)
n
(2.7)
n
-1
-1
n
):.
1=
Note that (2.6)
1
(i1. i1 :)
{ a. C
~12)
~positive definite),
~21 ~22
0 ( 1)
=
H (b) =
~
a:} =
~1
0 (n
_!?
_.) =
0 (
1), as n -->
00
(6_ ' n ,6 2' n )' of 6 is a solution to
1
~n
~n
-1
~l~n
Now, the U!\1E 6
(2.9)
2
~-1-1
~ll
and (2.7) ensure that
max
l<i<n
(2.8)
(
----> C
C
~n
~
~
o.
~
We also write ~n(~) = (~~(l) (~1'~2)' ~~(2) (~1'~2»; where for the
M and b, we use the same partitioning as in (1.2).
--n
Then the RHE
~ln of §l is a solution of
(2.10 )
For the PTME and SME, we need to introduce a suitable (M-)
statistic for testing the null hypothesis
we proceed as 1n Sen (1982)
(2.11 )
where
Ho:~2
o.
test
Towards this,
and Singer and Sen (1985), and let
~n (2)
~ln,
(2.12) s2
n
~1'
the IDm of
n
-1
),n
~i=l
",2
'¥
is defined bv (2.10).
(X.
~1 -
lUso, let
':i(l)
l
c - ll
C 12·
(2.13) C~ n22 • l = ~n22
- C
.
-n 2l -n
~n
Then, an appropriate (aligned M-)
(2.14)
r
n =
test statistic is
s~2 {&~(2)~n22 1~n(2)}.
Under H ' (n has asymptotically chi-square d·f· with P2degrees of
o
freedom (DF).
ficance a:
following:
Thus, corresponding to a prescribed level of signi-
O<a<l, the preliminary test for H
o
may be based on the
-6>
(2.15 )
LlCccpt
where Xv2
£
2
Jl
0,
dOf· with p. 2 DF
a J.'s th e upper looa% point a f th e c h'J. square.
..
The PTME is then defined by
ApT
2
(2.16 )
~ln = §In r( !n>X P2 ,a) + §In r
2
(J:n~XD2
,a)'
where rCA) stands for the indicator function of the set A.
~
for defining the PTME, it suffices to assume that P2
Note that
1.
For P2 ~ 3 and ~n12 non-null, we may consider the SMLE as followso
First, proceeding as in Singer and Sen (1985), we obtain that for
la.rge n,
(2.17)
where
(2.18 )
n
)2
((3
-
-In
~1102
= ~ll
__
Y = JR
-
(31)
"\.r
"v
-
oJ'
c-
1
C
C
~ 12 _ 2 2 _ 21
(0,
P, -
:2
0\jJ y
-2-1
<:11 2)'
0
and
e'
\jJ (X) {- f ~ (x) / f (x) }dF (x) ,
and it assumed that the dofo P has an absolutely continuous density
function f with a finite Fisher information ref)
dF(x).
=
J
R
(f~(x)/f(x))20
Let then W be a given positive definite (pod o ) matrix
(which we adopt in the definition of the risk, later on), and let
(2.19)
Also,
dn
let c
1
-1
= cllPl (n\I\]C-11
__n 02) = smallest characteristic root of ntA7C
--n11 02 0
0
< c < 2 (P
2
- 2), 2 ::.3, be a posi ti ve shrinkage factor.
Define then
(2.20 )
~ln
-
-1
n
+ (r-PI -cd nwe
-nl l 0 2
-Ix. -1
n )(8- 1 n
8_
1n
).
Note that the Mahalanobis distance of §In from §1 is
(2.21)
With this interpretation of the loss function, it may be quite
-7natural to choose W
d
n
= n
-1
~nll'2
(-C ll • 2 ), in which case, by (2.19),
= 1 ' and hence, (2.20) reduces to
"S
(2.22)
-1
A
.~
A
B
= B1
+ (I - c{' - ) (B 1 - B1 )
-In
- n
-PI
~
- n - n
In the sequel, we shall mainly use the SMLE in (2.22), though in
the last section we shall comment on the general case in (2.20).
that in the PTME, the indicator functions are 0-1 valued r·v"
Note
while
1n (2.20) or (2.22), we have a smoother version for the SME.
We may note that the test for H
o
any (fixed)
~2
~
Q,
based ont
so that both the PTME and
totically equivalent to the mlE B .
-ln
n
S~lli
is consistent for
would be asymp-
Hence, to avoid this asymp-
totic degeneracy, we consider the case when
~2
is "close to"
?
and
where the different versions of the M-estimators have non-equivalent
performance characteristics.
3.
ADR of PTI1E and SME.
In the classical normal theory model, with
a loss function defined as in (2.21), the risk is computed as the
In our case, to retain the simplicity of the assumed
expected loss.
regularity conditions, we shall compute the risk by reference to the
asymptotic distribution, and term the same as the asymptotic distributional risk
(ADR).
Under additional regularity conditions, en-
suring the existence of the neaative moments of(n' the asymptotic
£~~~
may also be computed, and these two would generally yield com-
parable results.
As such, we shall mainly confine ourselves to the
study of the ADR properties of all the versions of M-estimators, and
comment on their asymptotic dominance in the light of the ADR too.
To avoid the limiting degeneracy, we consider a shrinking
neighborhood of the pivot (0) while studying these ADR results.
Specifically, we consider the sequence {K } of alternatives, where
n
-8(3.1)
so that the null hypothesis H reduces to H :
o
0
*
~ln
For a suitable estimator
G* (x)
( 3 • 2)
=
lim
n-+ oo P
{n
~2
-*
(R
of
~l'
-6)
-In -1
*
~1) ~~
(~ln -~l)'
O.
we denote by
< x
-
IK n
}, -XERPI ,
where we assume that G* is non-degenerate.
tion n(Sl
_ *n -
~=
Then, with a loss func-
for a suitable W, the ADR of
R*ln
is
given by
Tr{W J ••• J
( 3 • 3)
-
R'P
xx~dG
* (x)}
*
Tr1WV},
say,
*
where V is the dispersion matrix for the asymototic distribution
Now,
by virtue of (2.17) <:lnc1 (3.2)-(3.3),
for the
lJj\11'~,
we
have
(3.4)
For the RHE, we may use the 1ineari ty resu1 ts of cTureckova (1977)
along with those of Jureckova and Sen (1984)
and Singer and Sen
(1985) , and claim that under {K n} ,
1,
-2 -1
2
( 3 . 5)
n 2 (B,-,In - ~l ) 'V ) ( (C -1 C
y
,0
S:ll) ,
PI _11_12 \jJ
A
so that the ADR of the RME is equal to
(3.6)
( 3 • 7)
R(~l;~) =
-1
M
2 -2
(0\jJ Y
)
-1
= ~21~11~~ll~12'
We may further note that by virtue of the same linearity results on
aliCJned H-statistics, under {K },
n
I
(3. B)
( 3 .9)
f"ln .~ r~lnf- Cll~12~2n + 0p
(n
+
"),
0
1?
(1),
so that the PTME and S}lli may both be expressed in terms of the UME
*
~.
-90.
_n
Recall that under tK n },
n~(B-~
B~ B~
-~
-In - -1'-2n - n
(3.10)
Thus, by virtue of (2.16),
(3.8),
(3.9) and (3.10), we obtain by
some standard steos that for the PTME
(3.11)
/I.)
where H
stands for the noncentral chi square d·f· with
(x~O)
q
J +
q OF and noncentrali ty paraIl'eter 0, and
(3.12 )
Now, by virtue of (2.22),
(3.8) and (3.9), we obtain that
under {K } ,
n
1
(3.13)
A
n'2 (B
-In
V
-
~1 )
->-
[)_1_U +
2
c 01jJ
2
Y
-1
~11~12 (~2~ + U
{
(!?2vU +
0
1
+E, )}
C
-22·1 (~2l! -,
where
(3.14 )
Therefore,
C
WP.
-1
have
(3.15)
0
2 -2
2 tjJ Y
2
c
4 -4
01jJ Y
E
cE
) (n.2~ + ~) ~C21~11~D1U
1
(J?2~
JL (~2U +~) '~21~
~~22.1 (J?2~
1+
+
§)
-1
>1~H~12 (~2~ +0
-1
(D2~ +~) ~~22.1 (1?2~
+0
+
§))
1.
)
-10-
Then, we note that
(3.16)
Also, bv (3.7),
(3.14) and the Stein identity [viz., Appendix B of
Judge and Bock (1978)], the last term on the right hand side of
(3.15 ) is eoual to
0 2 y-2 2
-1
-4
(3.17)
1jJ
c Tr(~1~22.1) E( X p +2(£'1))
2
while the second term reduces to
(3.18)
Thus, we have
(3.19 )
In this context, it may be recalled that
(3.20)
(<
<
Also, note that for the general SME in (2.20), if we let d = ch 1')
-1
(~~11.2)' then the second term on the right hand side of (3.13)
will be
(3.21)
so that for the ADR, we have an expression Darallel to (3.19) where
Tr(~~;~.l)
and
(~~B~)
are to be reD laced by
Tr(~O~;~.l)
and
(~_~o~),
resnectively, and where
(3.22 )
In the light of the ADR, the asymptotic distributional risk-efficiency (ADRE)
results are considered in the next section.
.
-114.
ADRE results.
Note that for
2 -2
a~y
(3.19) all have the common value
these are all ADR equivalent.
assu~ed
that ~12 i
~.
=::'
~12
(3.4),
-1
Tr(~~ll)
(3.6),
(3.11) and
(VC), and hence,
Hence, in the sequel, it will be
-1
-1
~ll
Note that ~11.2
is then p.s.d.
and hence,
bv (3. 4) and (3. 6) ,
(4.1)
and hence,
(4.2)
moves away from 0 i.e.,
remains the same.
~~M~
7
+
~,
"
R(~l;~)
~
+
w,
R(~l;~)
but
Thus, excepting in a neighborhood of
0, the
~=
RtlE has generally higher ADR than the UHE.
Since the l\OH of tho Ut1l: is fixed, we mil" choos"
'-which will lead to R(§l;W) = Pl.
We study the ADR
the PTME and SME under this setup.
treated briefly in the next section.
Therefore, we have
"'PT
R(§l ;~) according as
2
>
a:lj!
y--:z-
-~y
--
2
()
~roperties
-2-C
I!)
11 • 2 '
of
By (3.11) and the above choice
(4.3)
<
.'
The general case of W will be
of W, we have
(4.4)
r,7
-12~=O,
Thus, in a neighborhood of
the
PT~m
has a smaller (larger) ADR
than the UME (ID1E), and this neighborhood is contained in the neighborhood in which the RIm has a smaller ADR than the UME.
However,
both the second and third terms on the right hand side of (4.3) are
bounded functions, each converging to 0 as
L\+CO).
~
moves away from 0
Thus, unlike the case of the :qHE, the ADR of the
not blow up as 6+ 00) ; rather, this ADP has the asymptote
the maximum ADR of the PTME is generally (slightly)
PT~1E
(i.e.,
does
Pi' although
larger than
D
l
.
This later feature deprives the PTME from having the asymptotic
minimax character (in the light of the ADR) .
For the SME, we consider the case of W
y2
Ow -2 ~11.2'
so that
(4.5)
In this content, letting H*
(4.6)
*
< chl(~ )
=
*
hTr(M ), say,
where
(4.7)
o
< /t=dt 1 (~ * ) /Tr (M *
(.::: 1)
AS
Thus, in order that R(~ l;~) < R(~l;~) ,v~,asufficient condition is that
( 4. 8)
2E(X
-2
D2
+2(6))-cE(X
-4
-4
+2(6)-(c+4)h6E(X
4(6))>0, Y6>0.
P2
P2+-
Since, we have the identitv that
-13-
= 6E(X
( 4 .9)
-4
+4(6)),
P2
it suffices to show that
(4.10)
(2-11 (c+4)) E [X
-2
P2
+2
(L\))
-
(P2-2X
-4
P2
+2
(L\)]
+
> 0, VA >
o.
In particular, if we choose (in(2.22) c=p 2-2 ' it follows that under
(/1=0) , ( 4 . 8) is equal to 1/P2 ' and (4.10 ) is Dositive for all
II
0
6>0 whenever h::.2/(P2+ 2 ) .
-
On the other hand, if it > 2/ (0 2 +2) ,
(4.10)
can still be made ?ositive by choosing c smaller than P2-2.
For the classical multinormal mean problem, 11=1/0 (So that
(2+P2))' and hence,
1/<
(4.10) is positive for every O<c<2(o-2).
2/
How-
ever, in the model under consideration, the choice of c in (2.22)
is dependent on it in (4.7), and smaller is the value of
h, the
larger is the range of c for which the SHE dominates the UHE (in
the light of their ADR).
and SME.
In the same vein, we now compare the PTME
First consider the case where
(4.5), we have under
Ho:~=~
and
T:!=Y 2 a~ -2
*)
-cTr(~
*
Then, by (4.3) and
~11.2'
AS
APT
-1
-1
2
R(§li~) - R(~l i~)=Tr(~12~22~2l~11)HD2+2(XP2,aiO)
(4.11)
Tr(M)
~=O.
-1
-1
-1
[ 2P 2 - c/(P2(P2-2))),
-1
-1
-1-1
= Tr(~2l~11(~11-~12~22~2l) ~11~12~22:1)= Tr(~2l~11~12~22.l)-
-1
-1
-1
-1
-1
-1
-1-1
Tr(~2lCll~12~22~2l~11~12~22.1)=Tr(~21~11~12~220l)Tr(~2l~11~12~22·l)
1
C
C- ).
-11 2-11
0
Therefore,
(4.11)
reduces to
-14-
(4.12 )
-1
2
-1
2 (2-c/~2-2)].
(XD ,a'·0) - CPo
C ) }[H O +2'
{Tr(I --11·2-11
C
.
2
2
Now, c (P2-2)
val ue 1 at c
-1
=
(2- C(P2-2)
2- 2.
illf
0
'-I
, for O<c<7. (P2-2)
So that
L
(4.13)
C
E
,
attains the maximwll
{(4.12)}
(0,2 (P2-2))
-1
l Tr (I-~ll. 2~11) ]
[H
2
+
(~2'
rt; 0)
p2 2
-(l-2/P2) ].
Consequently, when a, the level of significance, for the preliminary
2
test is not so large, in the sense that H +2(Xb
aiO) > 1-2/P
2
P2
2'
2
2
(note that H (L
;0) = I-a> H +2(~
a;O)), then (4.13) is
°2 'P2,a
P2
~2'
!.
positive, and hence, the SHE does not dominate the PTf'1E, under H ,
o
unless a is large,
This also shows that f?r any fixed a (o<a<l),
2
inequality holds for II P -+2(X
;0),
n ,et
2
2
and hence, for large P2' the SHE may dominate the PTtIlE, even under
as
H.
a
p
2 increases, the opposite
To complete the picture, we consider the general case where
need not be equal to
o.
There, we have by (4.3)
~
t
and (4.5),
(4.14 )
By using (4.6)
converges to O.
and (4.9), it can be shown that as 6+ +
cr, (4.14)
,
However, for intermediate values of A, the second
term in (4.14) dominates the picture, and hence,
"s
R(~l;~)
<
"PT
R(§i
;~).
Thus, the PTHE generally fails to dominate the 8ME (for all /I. > 0).
This shows that none of the
I)TME
and SME dominates the other unless
•
-15either a is large or 02 is large.
5.~?~~_~~~~~~!_~~~~~~~. It follows from the results of Section 4
that both the PTME and SI''1E are robust from the risk-efficiency point
of view.
Of the two, the SME may have generally the asymptotic mini-
max character (See (4.10), while the PTME generally has the maximum
ADR greater than that of the UME, and hence, is not minimax.
The
relative-risk of the PTME and SME is relatively close to 1 in the
tail where t>.
+
-+-
OJ,
although, SHE mav have smaller ADR than the PTHE
In a neighborhood of
in the tail.
~=O,
generally, the PTME dominates
the SME, although just outside this domain, the PTME may have an ADR
larger than both the UME and
SI'~E.
This suggests that when we have
apriori reasons to suspect that 6 is close to 0, the PTI'1E may be
preferred to the SME; on the contrary, for larger t>., SME is nreferred.
However, for the PTME, we do not need that P2
LO
~
1, while for the SMR
have good risk-efficiency propert v , we need that P2
more than that:
h > 2/(P +2)
2
for c=(P -2)).
2
2
3 (actually,
Finally, we mav remark
that in Section 4, we have mainly considered the SME in (2.22) and
taken
r.:!='Y 2
-2
0ljJ
~11.2.
For the general 8ME in (2.20) with an arbi-
trary W, we have the ADR of the SME given by (3.19) with M replaced
by IV!
o
in (3.22).
With this modification, we can proceed as in Sec-
tion 4 and draw conclusions quite similar to those in the same section.
W,
If, however, we use the SME in (3.22) but use an arbitrary
then the asymDtoti c minimax character (in the 1 iqh t of U1e J\DR)
of (2.22) may not hold.
Also, in that case, the PTME may have a
better performance characteristic than the SME in (2.22).
use of W
=
'Y
2
-2
0ljJ
But, the
<;;11.2 has already been justified earlier by the
use of the Mahalanobis distance, and hence, our general conclusions
of Section 4 stand well.
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