-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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'1~13 AUPPLEMENTARY NOTES

,,,: k OCT i 8tEll

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,

"° " ", 4%.,•. , '"" % '"% "% ''"% %,'- -% '" " ',- """ %"%-" - "*%"% - , , '% "% """ """,

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

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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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