-A1BU 623 STRONG CONSISTENCY OF ESTIMATION OF NMNER OFREGRESSION VARIABLES WHEN T (U) PITTSBURGH UNIV PACENTER FOR MULTIVARIATE ANALYSIS Y WU JUN 87 TR-87-i5
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N' Strong consistency of estimation of number of
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oo Yuehua Wu F49620-85-C-0008
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'1~13 AUPPLEMENTARY NOTES
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1 KEY WORDS (COwtlnu ote#*veee olde II necessary avid Ifletilly by block number)
Linear model, model selection, regression coefficient, strong consistency
0 5O A I RAC I (C et..,en on revetso ad. i necessry and Ide1fiIJI by bl ock 1 2wmbh {)Consider the linear regression model Yi = x!5 + ei i = 1,2,..., where {x is
a sequence of known p-vectors, 5' = ).., is an unknown p-vector, known
as regression coefficients, {e.} is a sequence of random errors. It is ofinterest to test the hypothesis Hk: 8k+1 = . = 0, k = 0,1,...,p. We do
not assume that the random errors are identically distributed and have zero meanssince it is sometimes unrealistic. As a compensation for this relaxation, weassume the errors have a common bounded support [a,,a,]. Under certain
DD o.,- 1473SECURITY C't'1SMItAfiON OF THIS PAGE (Ol'ton Doe Entered)
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unclassified%.CUNITY CLA' IIICATIUN OF TH8i PAC'(Whf' Deld En#8r0d)
conditions, we obtain the strongly consistent estimate of the number of k fdrwhich k 0 and gk+l . op 0, by using the information theoretical criterila
k+
,J
! '
.r.
*-p
.1. AFOSR-TM!h 87-12 4'5
STRONG CONSISTENCY OF ESTIMATION OF',. NUMBER OF REGRESSION VARIABLES WHENTHE ERRORS ARE INDEPENDENT AND THEIR
EXPECTATIONS ARE NOT EQUAL TO EACH OTHER*4.
i' Yuehua WU
Center for Multivariate Analysis
University of Pittsburgh
Center for Multivariate Analysis
University of Pittsburgh
4M. '";-.'
OS.
AFOSR.TK- 8 7- 1 24 5
STRONG CONSISTENCY OF ESTIMATION OFNUMBER OF REGRESSION VARIABLES WHENTHE ERRORS ARE INDEPENDENT AND THEIR
EXPECTATIONS ARE NOT EQUAL TO EACH OTHER*
Yuelua Wu
-r Center for Multivariate AnalysisUniversity of Pittsburgh
I? Accc;i,.i For
"~NTIS CiA&I
W".' -A
June 1987 // " "-.4.", . ... t -, . . ..... .. .. .
Technical Report No. 87-15
-1r
Center for Multivariate Analysis
Fifth Floor, Thackeray HallUniversity of PittsburghPittsburgh, PA 15260
*Research partially supported by the Air Force Office of Scientific Research (AFSC)under contract F49620-85-C-0008. The United States Government is authorized toreproduce and distribute reprints for governmental purposes notwithstanding anycopyright notation heron.
1
Abstract
Consider: the linear regression model y = x: + e, i = 1, 2 .... where
{x I is a sequence of known p-vectors, 1' = (_ . .. 8 I is an unknown p-
vector, known as regression coefficients, {e)I is a sequence of random errors. It is
of interest to test the hypothesis H = = 0, k = 0, 1 p.. ,k +1 .
We do not assume that the random errors are identically distributed and have zero
* means, since it is sometimes unrealistic. As a compensation for this relaxation, we
assume the errors have a common bounded support [a . a 2. Under certain*1 - 12
conditions, we obtain the strongly consistent estimate of the number k for which J
- 0 and ... = = 0, by using the information theoretical criteria.k+1 P
J,.'"
/-
'F 3
, iI,
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2
1. Introduction
Consider the linear model
y. = x.' + e.,i=1,2, n.
(1)
where x s are experiment points, n = Y.... )' is the regression coefficient
vector to be estimated, and e's are random errors. In the usual linear regression
model it is assumed that the random errors have vanishing expectations and common
variance. In this case, the famous least square estimation (LSE) method plays an
important role in making statistical inference upon the regression coefficient vector
a. In the literature, there are a lot of papers concerning with the LSE and many
important results are obtained (a part of work refers to [1],[2] and [3]).
However the unbiasedness and consistency (even the weak one) of LSE strongly
depend on the assumption that the expectations of errors are zero, and this
assumption is not realistic sometimes. It is of interest to find a consistent estimates
of the regression coefficients when the expectations of errors are not equal to
each other In [4] two methods for finding consistent estimates of the regression
coefficient vector a are proposed.
The first method is to use the measureJw.
I
' q () = max (y. - x.' ) Min (y. - x.'Bn, 1<1<n i i <l<11 i i
The estimator B of a is defined as the vector which minimizes Q (a) The estimate
* ~is temporarily called MD estimate of a in [4] (the estimate based on then
Maximum Difference between residuals)
The second method is to use the measure
W. ~nB Max lyi - x.'aln 1<<n I I
Denote by B the value of a which minimizes Q (B) Also. B is temporarily called
MA eatimate of ( (the estimate based on the Maximum Absolute values of residuals)
. .. .
E . . ..ql >..
...B.*- .
1:'
3
. Under certain conditions, both B and B are shown to be strongly consistent
in [4],
Now let us consider the hypotheses
Hk: 6k+1 = Bk+2 O. = 0 and B k # 0
k = 0,1 . ,p-1.
It is of interest to determine the true model H by using the model selection
criteria Denote by B kn = (Bkin' kkn, 0 ... 0)' the vector which minimizes
Q (B) under the restriction B . - = =0 and denoted by B (Bn k+1 p kn kin
B 0 , 0Y the vector which minimizes Q (B) under the restrictionSkkn n
q= .. = =0O. Writek+1
p
:k - Qn{ (kn)
and
Q = Qn (0kn >
Choose a sequence of constants C, satisfying certain conditions which will be'V nspecified later, and define
"" Rk Q + kC
k k n
and
R -Q + kCk k n
Choose
i ... , .V
4
k = ArgMin{R k e {0, p}}
k
and
k= ArgMin{Rk: k c {0, p}}-kwhere ArgMin denote the index which minimizes the quantities following the symbol
ArgMin.
In this paper we shall consider the consistency of k and k to the true model
k.0
2. Consistency of k
In this section, we make the following general assumptions:
Assumption 1. The errors e, i = 1,2. are independent.
Assumption 2. P{e c [a1, a J 0 and there is a positive constant 6 such
that for any c > 0 and any n, we have
P{e c [a , a + CD] > AE
and
P{e n [a2 - c, a2]} > Ae.
Assumption 3. For any a > 0, there exists a positive constant C such that for
any vector ot ft 0 it follows that
#{i < n, I (x.)-9(a) I < a} > Cn
for large n, hereafter (a) = a/1X1
.J%.
5
Assumption 4. There exists a positive constant m such that
lxii > m, for i 1,2,
Now let us estimate Q (s). Definen n
E = i <n, -x!.B -B > 0}n I n
', (2) "E ( i < n, x!n >0}n I n
Split S = { x e RP: l xi 11 into d disjoint parts Z, . such that Vp 1' "' d
x. y ,x'y > 3/4. Lety Z , j = 1, . d. Define EJ = {i < n, Z(x)'y >' J J n -
3/4), j= 1 . .. d. By Assumtion 3, there exists S > 0 such that
# > 6 n, j = 1,2, ,d.
It is easy to see that -( n - a) e Z and i c EJ implies that
n n
and that 9Aj - B) c I and i c EJ implies thatnj n
x - ) > 0, i.e. i E "
Take r satisfying
r - 0 and nr /logn -=.. n n3
we have
.%
=,"
6
P(Q n(n) < a2 - a1 - 2r n
<P(max a. r~ )r + P( lE a. +8 rn n
d --
dE~ P( E "a ) i a 2 - rn' 91(n jJ=1 icE
n
d+ E P( mi " > a I + r n c .
n.nnj=1 iEn
d
Z P( max e. < a - r 9( - IJ=1 i E I 2 n ' nj
ndA
+ " P( m i e. > a + r -
j':' nnoC icE1n
d
< . P( max e. < a - r )1 ileE J i - 2J=l C
nd
+ [ P( min e. >a + r,jl icEJ , n
n
< 2d(I-r ) 1 n< 2de n 1 < 2d/n 2
n
for large n. By Borel-Cantelli Lemma we have
Q (B >a - a - 2r, a.s.
when n is large enough.
Let k be the index of the true model and let 0 be the true parameterThen 0Then obviously we have .for p > k > k
-. , - -
04
7
n ' ' n ' n Q n 'kn
:< Q( ) < Q (B) <a - an n 0-2 1
Thus
-,' )0 < Qn ( () < 2r , p > k > k
If we take C such that C n . C /r n, then for k > kk ' n n n l > 0
01,,.,[ Rk - Rk = k- kO )Cn + Q( n) - n( ) >0O,
f k knn, (2)
for all large n.
Next, we consider the case of k < k0. Denote>00
n =Ik 0~o > 0
and define
E = i < n. 2 (x.) + 9(. - Q), < 1/2}
E n {i < n, Ik (x.) - (k n B 0 < 1/2}
Split S into b disjoint parts 11 11 b such that V x. y c fl Ix -Yp 1 b,
1/4 Let II. j= 1, b DefineJ
* 8
Fj = {i < n,J(x.) -(x < 1/41 j = 1 b.nJ
S.. By Assumption 3. there exists 6 > 0 such that2
#(F )> 52n, j = 1,2 ,b.
A,2
It is easy to see that
4,.-
-9( - O) 13 I. and i c Fj
-n 0 j n
which implies that
I9 x. ) + ( - < 1/2. i e. i c E.
Also,
"n c JI. and i C Fj
0 n
which implies that
.. (X- + -kI < 1/2, i.e. i E- ,kn 0n
For i E ,we haveWA3
,0-B ) xi 1 I kn-aOI 9' (xi) 9- ( n BO
. "kn%'=~
Ol0
.2
"A ..- . .- . . .; , . . , , . . . . . . . . . . : . . .. . .. .. . .. . . . ., . . - . - . , , .. . . . . - - - - . . . . . ." . . - " , n " " " " " " " " " - " - -" - " -" -
9
:S -rT1 - 1/2) = -m/2.
Similarly for i c E , we have
x! ( mn > rTI/2.
Hence
Qn(kn) > ma.e i mine. + mTi E iEE-E
n n
Thus
P (Qn(kn) <a 2 - a, + m'r/2)
< P( ma e. < a2 - mrn/4) + P( min e. > a + mrT/'.)-icE icE
n n
E P( ma. e.<a2 -mn/4, -9 II..)j=1 i c E
n
b )+ Z P( min e > a + mn/4, 9( - B) I.)
-'.'-'.~ i E
n
bAE P( max e. < a - mTl/4, -9( - O) e: I.)
J=1 ie-" - 2 0jn
*OpP[ 4
10
b+ P( min e. > a + mTI/4, A
j= ieFJ I - 1 9Jn
b
E P( may e < a 2 - m/4)j=1 i C FJ
n
b"4 + < Z P( min e. > a + mn/4)
.j 1 i CFI
n
n - AmI 6 n/ 4 2< 2b(I - <m/4) 2 < 2be 2 2b/n
for large n. By Borel-Cantelli Lemma, we have, with probability one,
Qn(a a 2 - 1 + m-l/2, for all large n.
Thus for k < kO, we have
Rk - R 0 = (Bkn) - Qn (ak ) o-k) nOn 0
> mr/2 - (k - k)C > 0,-- n
(3)
for large n, since C -. 0n
(2) and (3) imply that k is strongly consistent. Summarize the above
arguments, we get the following theorem.
Theorem 1. Choose C satisfyingn
%I
0 - 11
,,(i) C ,
'4"'
(ii) nC /lognn
Suppose the four Assumptions given at the beginning of this section are true, then
k -, k, a.s
Proof. Use the arguments given before. We only need to note that for any
sequence of C satisfying (i) and (ii), we can always choose r such thatn n
(i)' r /C -b, 0,
(ii) nr /logn -
SQ. E. D.
3. Consistency of k
, In this section, we shall make the following general assumptions:
Assumptiom 1 The error e, i = 1, 2. ... are independent,
. %
Assumptiom 2 11 a, < a V in => P(e [a, a]) = 0 there is a positive2 n 1 2
constant L such that for any E > 0 and for any n, we have
P(e c [a - c. a ]) > AE
n 2 2-Assumptiom 3. Same as Assumptiom 3 in Section 2.
Assumptiom 4'. There exists a positive constant m such that
V,=,..
IxI > m, for i = 1,2, .[
Now let us estimate (. Define, , .n.
~ ;.,',. ,..y!. ~ . . p .. .... * ** . *,..* L*
12
E { < n, x' > 0}n -nf
Split S into d disjoint parts ,......- such that V x, y c , x'y > 3/4.p 1 d J
Let yJ E 2., j = 1..... d. Define =n ={i < n, 9.(x )Y"J > 3/4}, j =1, . d. By
Assumption 3, there exists 6 > 0 such that
# > S n, j = 1, ,dn - 1'
It is easy to see that
S c E . and i c EJn J n
I nl n
Take r satisfyingn
rn -b 0, nr n/logn -n n"
We have
P(Q ( < a2 - r ) < P( max e < a 2 r nnn-2 n - 2EF nIcE
n
dI P( max e < a - r, 9 C)
J=1 icE J
d
SP( max e < a2 -r, -,(sSJ 2 n n j
n
13
d
P max e. < a -r)-Ej I 2 n
- - n -Ar 6n 2< d (0 Ar ) 1 < de n 1 < d/n
n
f or large n. By Borel-Cantelli Lemma we have
> (~ a - r , a.s.nn -2 no
when n is large enough.
Let k be the index of the true model and let be the true parameter.0 0
Then obviously we have for p > k > k0
n n n pn -n kn
Qn (Bkn) - n 0-< 20
Thus
-n k n n kn -no 0'0
If we take C such that
C -. 0, C /r ccn n n
then for k >k 0
@4
14
R - R (k - k ) Cn +Q( ) -Q( > 0k k 0n 0 kn n k 0
(4)
for all large n.
Next, we consider the case of k < k Denote0
,TIk 0 ~o > 0
and define
E- < n, .(x.) '9( -(B ) < -1/2}
Split S into b disjoint parts 11.....IV, such that V x, y c II, x'y > 1/2.P b
Let j = 1..... b. Define F as F i < n, (x)' > 275/280}, =J ~ n n
1,.... b. By Assumption 3, there exists > 0 such that2
ns- 2 = 1,
It is easy to see that -9(a -k O) 0 fl and i Fn imply thatkn 0 j n
For i c , we haven".
." A
15A'.
X1! 1 5/
ix ( -kn 0) Ixii I kn-Yo l(x i ) ' k I > mT/2
Hence
Q(0 > maxe. + mTj/2i n
Thus
P (Qn (5) < a2 + mn/ 4 )
< P( max e2 < a2 - mTI/4)
2 2
L b __
< P( max e. < a2 - m,'1/ 4, -,2,Bkn- BO) cE I
' )J
-=I i 2kn0En
b -
< P( max e. < a2 - mT/ 4 )
nb -
SPA ma, e. < 2- m kn/4)cn
i F
P( maxme. < a n -bi42
'.
- - n 2<b( 0 mnI/ 4) 2 < b/n
for large n. By Borel-Cantelli Lemma, we have with probability one, when n large
enough
16
Q(8k ) > a 2+ Mn4
Thus for k < k ,we have-~ 0
R -R Q8 )-( )-(k -kCk k n kn' n knk k0 0n 0 n
<mT/4 - (k 0 k)C n> 0,
p (5)
for largh n, since C ~0.In
(4) and (5) proves k is consistent. Summarize the above arguments, we get
the following theorem.
Theorem 2. Choose C nsatisf ying
nn
'U(ii) nC In/logn -
Suppose the four assumptions given at the beginning of this section are true, then
k -* k,as.
Proof. Use the arguments given before, we only need to notice that for any
sequence of C satisfying (i) and 60,) we can always choose r such thatnn
Wi' r /C -0r, n
60i' nr /logn -
Q.ED,
Z .. '~
*17
4. General Case
N In this section we consider the same regression model (1) But the problem
we are going to solve is to determine the subset (or the model) J = {1 < j 1 <
< jk < p} such that B = 0 if and only if j c J We make the same
assumptions as given in previons sections
Of course, we can use the procedure described in section 2 and 3 to
determine the model J as follows: For each permutation Tr of = (
similarly rearranging (x I, .. x P), we get a new model Mt . Under this model,
using the approach given in section 2 and 3, we obtain estimates k = kA = min k-T T T
and k = k - = min k and let J ={t1), ... k)} andJ =J ={T( 1).....7T IT 11 1 1 1
(1k)}, we can easily prove that, by using Theorem 1 and 2, J J, a. s. and J
J, a. s
An alternative method to estimate J is given as follows: Suppose T is a
subset of { 1, - , p} Consider the model T:
y n x (T) (T) + e n, n n n
where x T) = lx c T) and B(T) 1. j c T)'. Letj j1 j
Q (T) min { max (y. x.(T)'B(T)B(T) ]<i<n
m in (y. - x. (T) B (T))}l<i<n
and
Q (T) min max Iy - x (T)' (T)I"ny
B(T) <i<n
Define
.4..
18
SRT - In(T) + #(T)Cn
and
R] Q (T) + # (T) C2 n
Choose J such that2
R3 = minR2 T
and choose J such that2
R- = min R12 T T
We can also prove that J 2~ J, a. s. and *J 2 .J, a. s .However, there would be
too much computation involved when p is relatively large. In the first case. there are
=totally p! permutations whileas in the second there are 2Psubsets of'{ 1,.. p}.
In light of this, we propose another approach to estimate J which only involves p+
* 1 quantities to be computed.
Now let
B (j) - (B 1, - , j l . .. P
and def ine
Qn (j) -min { max (y. S x (j))a (j) i<i<n
-min (y. - X'BU))I<i<n
and
LIS'S * . - - - -'Vp .* ss.
19
Q (j) minn max y - , x (j) I
n U() li<n
WNrite
R (n, j) =Q n(j) - Q - Cn
and
R (n,j)=Q n(j) - Q - C.n
* We choose
i ~ j n [j {jj; R (n, j) >0}
n
and
j n {j I =j {j: R(n~j) > 03n
Then we have the following theorems.
Theorem 3. Under the conditions of theorem 1, we have that
J ,-J , a. s.n
where model J ii .i is the true one.
Proof If c .J, by (3) with the replacement that k 0 p and k p- 1, we
have that with probability one, Rnj) > 0 for all large n. i. e., jc n Hence, when n
*. -**.r*
20
large enough, J J Conversely, if j ' J, using the same argument as proving
theorem 1, we have
R(n,j) = Q (j) - Q - Cn p n
< O(logn/n) - C a. s.=1 n
which together with (ii) implies that
R(n,j) < 0, for large n,
i. e j ' J when n large enough. Therefore J J which completes the proof ofn n
Theorem 3.
Theorem 4. Under the conditions of theorem 2, we have that
J -. J, a. s.n
where model J = j is the true one1 k
Proof. If j c J, by (5) with the replacement that k = p and k p- 1, we
have that with probability one, R(n,j) > 0 for all large n, i. e., j c J . Hence, when n
nn'' =ilarge enough, Jn J. Conversely, if i ;J, using the same argument as proving
theorem 2, we have
R (n,j) = Q (j - Q - Cn p n
< O(logn/n) - Cn a. s.
which together with (ii) implies that
21
R(n,j) < 0, for large n,
e 't J when n large enough. Therefore J J which completes the proof ofn n
Theorem 4.
. N
S.%
:-V
,xS'
22
REFERENCES
1 Drygas, H (1976). Weak and strong consistency of the lease squaresestimators in regression models. Z. Wahrsch. Verw Gebiete, 34,119-127
2 Lai, T. L., Robbins, H and Wei, C. Z. (1979). Stong consistency of least" squares estimates in multiple regression II. Journal of Multivariate
Analysis, 9, 343-361
3, Oberhofer, W (1982) The Consistency of nonlinear regressionminimizing the L -norm. Ann. Statist. 10. 316-319
4. Wu, Y. (1986) On strong consistent estimates of regression
coefficients when the errors are not independently and identicallydistributed. Tech. Report No 86-06, Center for Multivariate Analysis,
*Univ. of Pittsburgh
* W.. o I
..-
"'".".'
,,
unclassifiedL ,t. UI4 It CL A V IIIl Al IOt U L , tI I W A(L (I,.* l- ioi .j
REPORT DOCUMENTATION PAGE HEAD INSR1UCTIONS
I. HEP',RT NUMBER GOVT ACCESSION NO 3. RECIPIEN'S CATALOG NUMBER
87-15
b4. TITL (and Subttle) S. TYPE OF REPORT & PEFI1OU COVERED
Strong consistency of estimation of number ofregression variables when the errors are indep- technical - June 1987endent and their expectations are not equal to 6. PERFORMINGONG.RLPORT NUMULR
RarhothPr_ 87-157. AUTHOR(.) 4. CONTRACT OR GRANT NUMOER(.)
Yuehua Wu F49620-85-C-0008
., ""- - PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMr:NT. PRO iE CT. TASKfor nalsisARE[A & WORK UNIT NUMBERSCenter for Multivariate Analysis
"-I..- University of Pittsburgh, 515 Thackeray HallPittsburgh, PA 15260
I. CQNTROLLING OFFICE NAME AND ADDRESS 12. REPQORT DATE
Air Force Office' of Scientific Research June 1987Department of the Air Force I. NUMBER OF PAGES
Bolling Air Force Base, DC 20332 2214. MONITORING AGENCY NAME & AODRESS(II dllerent froi,,n Cortroillng Ollce) I6. SECURITY CLASS. (of this report)
0 unclassified
SCH %EDULE.,, %' .._%.. . -I-S;.--ECL ASSI'FIC ATIO-NDOWN GR ADIN G'-SH UL
IS. DISTHIBUTION STATEMENT (ol Ihis Report)
Approved for public release; distribution unlimited
17. DISTRIB4JTION STATEMENT (of ithe abltact entered in Buck 20, if different (rom Report)
IIII. !1UPPLEMENTARY NOTES
I2 KEY WORDS (Coflintu ur, 80 ev6 e side i nIec**sawy wed i e(li, by block number)
Linear model, model selection, regression coefficient, strong consistency
* "20 AB3iI RAC T (CoIllnue o, reverse side11 necessry end dJdnerilty by block number)
Consider the linear regression model yi = x'O + e., i = 1,2,..., where {xi} is
a sequence of known p-vectors, 0' = (1,..,8 p ) is an unknown p-vector, known
as regression coefficients, {e.} is a sequence of random errors. It is ofinterest to test the hypothesis Hk: 0k+1 = ... = op = 0, k = 0,1,...,p. We do
not assume that the random errors are identically distributed and have zero meanssince it is sometimes unrealistic. As a compensation for this relaxation, weassume the errors have a common bounded support [a,,a,]. Under certainOH M
DD I FJAN 7 1473SECURITY CLASSIfI AO OF TNS PAGE -(.n Da, a-Ent ,rou
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conditions, we obtain the strongly consistent estimate of the number of k fdrwhich #0 and 5 ... 5p 0, by using the information theoretical criteriakk+lp
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