For a proof of (1.8), fix m such that S is positive definite. For any. -m. ~ .... we have Y ni = eni. For any ... depending only on x go"'Xni , not all zero, such that l p+l.
JAN~rJIux~jKx 1988UE 2W 7U~
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Asymptotic normality of minimum Ll-norm estimates
inlinear models P.
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AUTON~aJI.
Z.D. Bal, X.R. Chen, Y. Wu and L.C. Zhao 9.
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September 1987 MUMOIER or PAGES 34
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D MARO0 41 1988
III. SUPPLEMENTARY NOTES
~
Key woords and phrases:
linear model, Minimum L1-Norm estimate, consistency,
asymptotic normality.
0ASI IIAICT fCarstinue an reoee
aido Itnecessary and Identify by bloCh nlnb~f)
Consider the standard linear model Y, = !is+ ei, iIs 1... ,n.... where x1 1 X2,
...
are assumed to be known p-vectors, 8 the unknown p-vector of
regression coefficients, and ell e2 sequence each having a median zero. DD 1JAN 73 1473
..
the independent random error
Define the Minimum L1-Norm estimator Unclassified0 SECURITY CLASSIFICATION Of THIS PAGE (WhenaVole Enla'sel)
Unclassified SLCU94ITY CLASS&UICATION OF THIS PAGK(Whona Does Enloted)
20.
(continued) 0as the solution of the minimization problem n 1 Y
inft '...jY1 --xi'J: 0 e RP1.
-~n
It is proved in this paper that ~nis
asymptotically normal under very weak conditions.
In particular, the
condition imposed on [y1 isexactly the same which ensuring the asymptotic normality of Least Squares estimate: lim
n
~max
x ~n.i(Xjyl-1j)i0.
Unclassi fied 6SECURITY CLAISOPICATION OF THIS PAOUPh.A 000. E'I..D.4
----------
Center for Multivariate Analysis University of Pittsburgh
ASYMPTOTIC NORMALITY OF MINIMUM L1 -NORM ESTIMATES IN LINEAR MODELS* Z.D. Bai, X.R. Chen, Y. Wu and L.C. Zhao Center for Multivariate Analysis University of Pittsburgh
40
0
,/
Technical Report No. 87-35 NTIS DTIC
September 1987
Septembe 1987.
J; By
CRA&I TAB
Fj
t :+ ; ; . ............ ................
.......... .
Dist
Center for Multivariate Analysis Fifth Floor Thackeray Hall University of Pittsburgh Pittsburgh, PA 15260
This work is supported by Contract NOOOl4-85-K-0292 of the Office of Naval Research and Contract F49620-85-C-0008 of the Air Force Office of Scientific Research. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.
ASYMPTOTIC NORMALITY OF MINIMUM L -NORM ESTIMATES IN LINEAR MODELS Z.D. Bai, X.R. Chen, Y. Wu and L.C. Zhao
ABSTRACT Consider the standard linear model Yi =
+
e i , i = I,... ,n,... where
xl, x2, ...are assumed to be known p-vectors, B the unknown p-vector of regression coefficients, and e, e2 , ... the independent random error sequence each having a median zero.
Define the Minimum LI-Norm estimator
n as the solution of the minimization problem J =IYi
inf{Z=llY i -xjBI: o e RPl.
" -
n~n=
It isproved in this paper that n s
asymptotically normal under very weak conditions.
In particular, the
condition imposed on {xi} is exactly the same which ensuring the asymptotic normality of Least Squares estimate: limri max 1 0, we have n2
P( il i/nj L c)
1 for n large. nd2 >1
]i n'
rm
m = I,...,M,
-n 0 is an arbitrarily given constant. from the fact that vn
(2:39)
The existence of such M follows
=. Also, the partitioning of 0 after the m-th round
+
will be denoted by Gmo A typical interval belonging to Gm will be denoted by B or B
and a point selected from it by b or b.
Vm
Jl
°
'
j m l
Define
Ev2/3) >)I
Rni
P(L
,J
i
Ri) 3 '"Jm
Rni~b
°"
>Jlo
2/3m+l 3
//3) m = 1,2,.oo,M;
S=
3 1"Im 3
I
nm 'n su I m .(Rni(8)Rni( P n..jm =l eB... 12.oMl bjl'" . )/
nn/
Note that for any B e GM+l and B e B we have, in view of (2.39), n
n
-
VWI§bI
'np/n-
n
0, attains its maximum at
x = b(6p/a) 2 /m, we have 2 1Pmexp{.i6 -1
n
Pm P3 pmexp{-16-1 (nl9g)m/2} n 'n n
,/9)m/2, n
P Hence, noting that pn
.
9nmgpm96p/c)6Pe-6
< (P/729)Pm(96p/,)6P
we obtain
M m=l
(Um +Vm)
0,
-
n)In1
"n
Hence by (1.19)
)x.II(
i- sgn(e =i
,,ldn
..
j n
0.
-,
At one time it was expected that if condition 10 is replaced by 1': el, .e2' "' are i.i.d. and e1 has a unique median 0, the conclusion of Theorem 1 is still true.
The motivation behind this conjecture is the
simple case of estimating a population median by the sample median, in which the uniqueness of the population median is enough for consistency. Yet the following example shows that this is not true:
Exanple.
In model (1.1) take p = 1 (a is one-dimensional), xn
logn/-in, n = 1,2,3,..., el, e2 , ...
f(u) = juII(jul 0, are met.
In the course of proving Theorem l' we have already shown this (see
(3.15)).
(Note that in proving (3.15) we made no use of f(O)
121n pn/
lsgn(edx
n -"= 21
>
0.)
iXion xi
f(u)du I(IanI 0, it follows that there exist two positive constants h1 and h2 not depending on t, such that
h C1
1)
