AD-AlOG 146 OLD DOMINION UNIV NORFOLK VA DEPT OF MATHEMATICAL SC-ETC P/9 12/1MINIMAX RIDGE REGRESSION.(U)MAY 80 L C PEELE, T.F RYAN F49620-79-C-0125

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* Old Dominion University 61102F 2320 XA5Norfolk, VA

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20 ABSTRACT (Continue on rererse side If necesspry and Identify h&' bj~kik"Tumber).

This work examined minimax linear estimation in multiple linear regression.The application of minimax estimation to regression led to the development of

ridge regression estimators with stochastic ridge parameters. These estimator:,

WU not been established for other ridge estimators. These minimaix-motivated

estimators were examined in several simulation studies. In particlar, flaws

LA. in other simulation studies of ridge estimators were depicted.>(Continued)

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20. Cont.

Consequently, an improved simulation procedure was'used. It was observedfrom these studies that, contrary to published statements, a ridge estimatorcan be considerably superior to the ordinary least squares estimator,

*especially when high pairwise correlations exist among the regression variableF.Robustness considerations were used to suggest a requirement that a %good 4

* generalized ridge regression estimator should satisfy.

Accessionl For

NTSGR&

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S~cuRITY CLSSIFICATIO OFTIjAEe f lt icateredr

-Dsrbuin

AFOSR-11- 8 0 - 10 92

MINIMAX RIDGE REGRESSION

(Final Report)

ABSTRACT

This work examined minimax linear estimation in multiple

linear regression. The application of minimax estimation to

regression led to the development of ridge regression estimators

with stochastic ridge parameters. These estimators were seen

to be invariant under linear transformation; a property which

has not been established for other ridge estimators. These

minimax-motivated estimators were examined in several simulation

studies. In particular, flaws in other simulation studies of

ridge estimators were depicted. Consequently, an improved

simulation procedure was used. It was observed from these

studies that, contrary to published statements, a ridge estimator

can be considerably superior to the ordinary least squares

estimator, especially when high pairwise correlations exist

among the regression variables. Robustness considerations were

used to suggest a requirement that a "good" generalized ridge

regression estimator should satisfy.

Activities partialy supported by AFOSR contract:

I. "The Merits of some I, idgv Regression Est imators, " rescarch byLawrence Peele and Thomas P. Ryan presented by Dr. Ryan atthe 1979 joint ASA, IMS, Biometric Society meetings.

rFV 1* P0VAA or publjo pv els, eg

distribution "T1t0d.

S6 11 108

V -

2. "Minimax Linear Regression Estimators with Application to

Ridge Regression, " by Lawrence Peele and Thomas P. Ryan,

to appear in Technometrics.

3. "Cormnent" (.n invited ridge regression paper) by Lawrence

Peele and Thomas P. Ryan, J. Amer. Statist. Assoc. 75 96-97.

4. "Most-robust Ridge Regression Estimators," oy Lawrence

Peele, submitted for publication.

fi

I.J

AIR FORCE OFFIC( S F SCIENTIFIC RESEARCH (AFSC)NOTICE OF TFANSMIT7AL TO : DC

Th1is toc.,'zAc~tL report has boon rovie,*id ~:,d is-S. approved fo " public release IAW AITE 1J0-12 (7b).

Distribution is unlimited.A. D. BLOSETeholoal IrormtOU O tc . .

-7

V S

MINIMAX LINEAR REGRESSION ESTIMATORS

WITH APPLICATION TO RIDGE REGRESSION2

by

Lawrence Peele and Thomas P. Ryan

AFOSR Technical Report No.i

May, 1980

Old Dominion University

Depa.-tment of Mathematical Sciences

Norfolk, Virginia

I. To appear in Tuchnometrics. Research partially supported

by AFSC under Contract F49620-79-C-0125.

. .

ABSTRACT

This paper considers minimax linear estimation of the para-

meters in a multiple linear regression model. Recent results

are summarized, and some new results, including a transformation

invariance property of minimax estimation, are given. These

minimax estimators of the parameter vector can also be classified

as ridge regression estimators with nonstochastic ridge para-

meters. Some ridge regression estimators with stochastic ridge

parameters can be motivated by minimax estimation considerations.

These minimax-motivated estimators are examined in several

simulation studies and some observations are made based on these

simulations and minimax theory.

A7

)

".A

Tl-o us-- * _Ll_, linear rogrussion imodcl is

I S n Vrk Cs 0 U,-,o n o r-,2ia (2r r no: i, varialas .. ~t:-;.can

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<. '1 u Ll :r c c .aur fo0r s ti:'a" .. ,. t "A.,osr r

.e t '2d. It i; wcii o.' tha"Lt th"is i:10thloU is cqu'ivo'lent to

no~ ~ jsa ci...- .. j I:.rnsa ven lI.Iar C c.I . 1:. 0 n

w~eethe iig :r>.e - s c2iLller nuns _,LoehLic (basckL on

a.,-- - v

the rgcia rc~rs:01o t ?o r (i n K) otprrlb thu,.:nzndn c Icc

.. quaro2sC fs::ma ±: ZS~t U: fJ 3 1~ :-nl a - a c,'t r t o -

Ito s1 - D fO~ v ~ I&t~t ac snw (orwyin 11. 'L ) o

ul 11.

idgo ~ ~ ~ ~ rer otr,t 0' ohkonan c rrr

at minma:: ~uC roross on esitor r 5IC'K 4

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0 f 0 tIlK

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1, - -. .. ~ ..>2 22< fldGfltjCy.tjL

- . , .... . .,.j: L>.cC<02 0.. fla1\'luUaI CO.> u?<fflL.;

N.. .. .- * z:.uL Li>.o :..I2IW.LIM IIV.<..> O.....

.~C IU to L:w ;:.rfli:..~x Iif.cUZ CJ?....,.j....,.. 04....

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Auxt2CarV :.:;2ostx- t.nat IL ':LS an.

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'La

-. V." ~ ~7 n-----w.- -

* . tt-[= ' - .... , . . . i: L C i to t .C Eld.2 1-J,, s "o

* -k ; . ~ . L. "' " . ... . ..'.". " . :' ' ILGCJ aLC :1 A. L otu rL'

-. . ' .. " .. " . - , ',. . u.< w " r " &. ."., ; J . ''._s

.... : ' - "- . ,. - >J :, : t .... A.... .t " zll~ .. AK., Q3 . . . - . . . .

" i_

'( 2; ). .

ts..nsatCi dl:3CUS Sioi; Of Lh rSUILt.i 0.

* . ~ s:; L2 LhiL "Or an~y c

g , n &J z n r

1 Li ar. c{121 .

~~~ o. ,. ii 'A 2nc on tlc2 a ccu.ac. L;1... 2 .

f) : i in ovs a rn

~c oII&;irj ~nariocc ropcrtvl of nn:.xI.n.r i.. o

.2 1.1., L.jtL~.. .A.Ll . ...x.

..... ~~~~ U I Lz u z..c. Li o i .t .,

- - ... . ..... 2 ' :20a0. . U.,..r:

-- ~~ . 2

~aini~uiz~cibv .. is rui;uIL ;;-,s bcc pkvi)rl'. ir

inl L7 _ - .-,)r ~ by Licran~j 2 iilct:ceos

iI6J zr- a c:-scu-,zijicc of rCot i:ia O ?o~3

One2 c-,:- ... SI'* cuk b riqroIt Lrj~ rc

t± io n o L C 0c o:. a; :'1 , f2 u r c t tnma-CC, 70

U:: I, i or. 1:~c a5:~ tL -,Ii08

byacrOL..cr. i~ILL;~ -~. r

W~ni'.~pastL ive ,;-ars, i'.any ridge estini ors have' appecarocc

iL !.o h lea"rea..vo 2)en co:roaprea in. subseqCLr.Ln si;-UI'Iat~u:;

z2tuaies. GOlb I and 1:0 L L 5] state that L%-: doDzen is

: roaolzv a cosra 0v St_: a.3e of the nu:~rof tr

~~~_-V ratonrnaeu-sed uerid-icrrtacu L.;rrach to

select tlie. v-1,lue of zh rd'.nontat This apranis hnichiy

subecrveho;evor, a: e- also bee:n cr4 ticizec o)n other croaunu ;S

(see, e.i.S. Fian -14]).C:s~ctv,~s~:. as

though", m ore foml e asor selecting thne riduge L cn stan t

n.eed -o bLe 1'rcse. ':2 sttr'i in Crc:

where2 . s stchrtc n ecual to <7',has Th-e'. th'e uiO.3i

'o Ir.t i r ou 1. ' z Iat Ia s t ua0s. SC::',C Oflteer sr

* preentedlaLcer in 0h. on. e hVe as tc12

shrinkage, t~s _±mat:or prLu .cnt2LdinLx::l 2, bti i o

tare as'.l as the ru~ u;tr

Befcc .eran ai lacnStueV Of reesi: sia~s

an cxperi wf1L' t us LI.A.ct_ an appropriate 1e f unctik:.

i!A~ ~ ~ ~ IIf-JLF .-

wheru dutzc tn atoorOf ~3(soc, 0.gj.Gjon,'j

have usedi (u) in aaoijLion to telos u.,ix

loss fu:;ctian ""a -a :av goal of a :j.sinsuys

csti:Ijatlin of- the-, wa:.. 10 wwr,2s (7 S ,2

if th-ersa .a os sdfrpe~tor, s~O~

f or e Ithe-r -,0::ao or prediction Lsincc the oss is m S I n

whn i S th 1es sl ,1uare - siatr C.s wa ssr~ t00 :>5

* Section, 2.

£I Ce ao t e2 1h (', ) or (7 or 2 eLC , v;e then21 ::'.U S L: . ue

what i nd Sto u.; - wela h :e f or sss5atn;X\.

k to e f i. odu ,2''))i xzd for ~-I, when ais

the c~vee rete>K~ ot ese otssn

~a~sc 0> 'i~'1 ' ' -

LISO t hC:-;t t.. C ".0 Cu..:Lr n the r- lu I it it on .:Tvi Iy

*..utix, ad cds~ u1. CL' to dc.pondon voialoac 1;:

tc'10to a:sL

w ine r e i s .. ~r) sovor11l valuos Of 0 arc hc

junction ewirn of~ soveral '1" -1aticos 1hJ -

csti:a!Lor- is jrc '7o sn r:e: ) o

Thuis ccncra' u'srarc LS to ;na or sot

we vlou an:cK-u aoccno A o-~orl'

actua-l dLtza. ci-.- w;o)h h ~. nico2 L'

to tho c ta; to *0.tCa L~ i~

(in tcrT-.ns of ;:.a L rror),W W Ih ,.

sn-.allcst of:vl c.s.Tllo rconctal Le I'~

* Tiblc I. Ir.na~i~~r 2sol not-ico WL

in tho- ui1 rst.% Lar o rh tol, to tho, i~~~ 1

mlatrix '. tho,. botto):i' Part oi- th"O tClc Oo :. oi, cvc

* in Tzablo. i) tha-t ~ il bO p r i:atl C f 0 1 .i,- h '1-11,2 ~

3. ~ ~ ~ ~ .1:nau ~s v:Cr:7oort Of~ '::; i:.v:1

-NEEL5 .lc .~.> >:' o

WO ue 1rc thu U.S i L t s ce 0 w Poo0r I v Q r idJtLe (2 st 1;:::r- p L?:-foC)r;.

* r~tl~cto as cur for racalevalles o- 1~ Lr. cc

* ~ ~ ~ ~ ~ ~ ~ t rogesv l-oaioigtrds to caus SQ 1 for the s:nalies-u

lev.uc~ .= )e:uh different froi R- for k3he lr _-t

cienalecase, C~hn smnalle3r so as to prociuce Ijhc P. e

v'alues ,DZ t ue s~~etecevlCcase .. udt~ Za::a:

~±~et exct>:1.0 i:,. the lrgetc :evaa --

For tlnese revors e have usca. a ciffceren, S .. a atiou

pr oc .u r e. xe nave al so generated; oba-crvatLons on. Y s Inr (8S)

but , for taca tralI, a vucter is nra re "io .

aisriatir.on th.-e collctlon of all nr-o-ne q-e:cnt vcos

nor;-.ali(0,r j' crr,:or vcctor CiLS tunLa~ua na is

co~ouedas 3 S: Lo. cni u.c t 10:tio. cn. te

om r anu r ,: c irn s i:,,L it ic) fu S a. tu -- ; u 1h c :.la :f 1ru la Ss

of estinrators. We.L scu nothin- rnhwvr,.;U .m:

vectors that arc unimo~ or. the cui lc-ct -,n of -all no rmi-one vectors

w~~ru (2) cp rnest:.to ditorn a particula wa su-. ;onas

trvt

*ridge. cons-tant. or al :ultiple of- our ridge constlant. would !Pc

a~A. erpla

0 I If ve u:Oc ta rt C0 Ipoaf r. U. G.> iC f. C~ 1- :.,2 F 1

0h r es u Iz -,I L3'D C, I majoCr s i I-,'l a Lof 1 t0' a*u *-L.

Tablie 2. 'no ) I'ie, 1 c an beo d is cerI:nd CfI:n o!- 1 z)h

*table. IFor a ' la tekjreeO of i I ecL; 1 12 S -u

the ae7S~fd Zrc n uniu~onini;

ve.1rsla. S, ce. ., iv ixeR esize of, ti, i

co.nstan.t \ eJeiOfil-orat i

qr~at~rI - co it a-, iirnll ies -heneed f or a l arger r auCe,,

cons taZnt .

IL folio'11 Iro L So utc s1 eo Cn Scti: .1 C C j, I:.

ness n'Ifo ae t5o CL 0ntOSL]C rIas CO:.".,.

an th r1C;c Csa:Ae Cupromn least sona:r. -1 t~r.

A _ .~A A A

q seu r nlN, 5:. - :?/' ( a ddition t- .. uvc r. ;ixa:;

1) sieoldbe ac,)-: <ta,-chas'tic) chiefor -

be .ti n Vthe res ults ir. ThULcef lo...

fact that shou.z n lu be less thn .. 0.,.~c .. s;,n orl

A A-1* teegnitv) C)'2 v'+ lAlwhre .. ,A ar

values of X'X". ana- C is.. an unbiased (sir~o of sA A A\-

i S9 1., -, - '

conrd1 tioncci in theur t E, ls t ~t r ix inr T',L)i 0 2 1is st

the avorugaj valuk? aof s.-ou la be a p r o,:i::a~c t e I f 3,

a:), na to

Ta 1l- 2, ospoialy o thc fsconrd 1055s fun~ction 71fC)-

A-A

0 c c S:;512 ro I f tI-, erC rU:~2~O o~ f :saf

t~tochthe, v c;t :..L ~',tv o f ,;u r s 1:,-u a t on s hfe er:r

. U .. < S~ i~C~cji Qn I e: t L V

tru 6h abs1ise a:iii Le sial sne k-: 0 t: I L L

L: u S: 0rs S in ccrre~Iat."an LarIu, btcould b1- ;iu ua0

a fter conver irng baktora.f,.

-15-

* ?-,I,'.' I ,-

,-.) , , i,. ",xI: (. , .7793, , = *•.',:17)

C 0+

9 0, ("1- 6h

, O l .9753 .00- 349 •.00433" .004-295

• ,VQ .. x4 .5 .. 76 . I

3I I* 9').,. ...)1 0 C A ' . X0 + "1q

1 . 261 .1203v. u034i6 .3., L; S (Iua..

-- +.QI ."97-3 .".0349 .20 ,33" .7 04295

I.36u 0 3t

797G) 3 1 '4 .0 0 C

.1"('10 .02,7 02 ,7 .02371, 95

*' ' . ,- . l f it " ' 3 . O e ..

i 3

• t:..,.2-. j"L") 3 '-i5)" .,070322'

• '.,, .0729207 l.02 77.7- .0297053

L . a i -' -s 12 UV

,. .999iis f0 •02s. 267 027737 .023 9

* 1. 1 ,2 ,5.2263')0 1292 l i .. 9 5'.t'

L~ ui, 11 . 1/,, i 't l; .374' r73 :.27 2i .159

-" :1.111 .Th 'gcn j u9 -, , 1 'qe7~ 1 Iv, . . : .. ,i.1

'.t

.I 4

TABLE 2 - ,'." :"',: ' ; c , _ ''.:2,l. . . .: .

I i .1k ..

d- Avg es"qae - - - '-1

A R-C1 1 ,-: Loss

.00 :,212 1.0002 1 .001C7 1.Oh'7 * ' . "J I ,45 1.0062_ A.0.1 .996 .cOO4u2 i.000 I . , 1( " 7 000

( ..-- .9 ! 0 J g4- U. ' it ,J .37 .0142 A.005 .964 .3u41 40 0 uO, 0 .O .0 9 1 , 5 .123

.o206) .3. 9v1 .95 . 0 . , 5 1 . 3

.005 .&;6 .J1930 0.9959 0.9, I . 1 0 1 .L07 1.0743 3

YX oderaitce1 I1!-condiLionicJ X r~tr'-:

.001831 1.0001 1.0705 1.0024 1.3,)55 1.2499 1.01% A.0001 .993 .0004i9s 1.0002 1C007 1. C21 ..> 5 I.C 76 1 .0117

• .j9 02 9 9Q( 7 i (16 :,i q2!.001 337 . J423) 0.i9 0.99> .O5 77 .6A 3

,71. 0. 55 0. 1.7OO3 . 34 .O-:'O0 9 0 C)q 0 ' 0 ... 0 " " t 5

V e ry !11 -cond it ioned X atix

A 1 1\ - ~ -I ". A- 1 1Avg. R Lcasut S,;arcs U, 4 -i0" Los-

.0326 . 0."7 16.' 1 1 29 1.221 A.006 .i 7 .0O0%d 0.997 L )',5 1.u0 1 1 50 1.079 5

. 0.7 6 ).5 ;2 .0'5 0."3 04.5 -S A.0,1 .932 .00398 0.,?42 912 .lS7 0.j99 0.920 1.000 B

1.631 0.604 0.452 0.31 0.259 0.228 0.210 A.005 .862 .099 0.874 0.824 0.769 0.749 0.739 0.990 3

600 "u' (R2>. 7) ubsurvat io s ;,: gcfnevr.itcd or ac li 1-, .. trix c 1;: a : i uai, .")I e

I r :-.rcs c av,-r L,: ,'USs for Icus L 1kr :-; aC the ratio of L 'v C'd c ouL;s Lor ca !',

" se;Lm or to [t_ dvcrIco 1,):;s -or la:;t su.ires

Lus; A 1:, - .' , . - .,, ,s L '

The i.,a ri.c5 U J |l O . , t e . C . Losc uid arc ,..s v cJ i. I .

, , ,,. I -~ -- 7 .7 , ,- -7---.m rT -

e o l,2. rp ian u 1. 1 ra x crs~~ S.~ o:~c.LO

L per ~v~'S an. C~ 0 a L;o cu -,.: S

1r0s Ul1ts S .' r e p r,2S 0.-,L CG i..c --lu r n r. i''rj 1

lincur rearossin -sInatiors. I1. SuCtion, C3 IuI L I C 1 (2

~ c.a t~s,-;ere also illus:ratca in ~a~ .Accorad-nc'liv,

al n ;c r o s -,c2 ou t: uo 0e in ch :o wr

alt ho uh 1L, ::.avL a >tC; aoico, pciLvI sh;l

of Scijuntif Ic R~erh ~. ura r r. u

*The authors wishi to hnk rcfe-rccs fou L I ' : ai 'I e 117u

su( st~nsC~p&1.iL l cunct~ i 2 tie i L .J

L 1

C, U

u: i iu.t<ur ... ... ,:i u ..L .'.. :."..... . . U 'fl'Y ,z. " s 7 8t '

77

L -AD G012 30:>5. . (I 2c . A > ulatio :" stt ;' Ct. o:: i g

@s r;: >r . ::.v,: ..<Y, QCJ xc,,s o& rcn L :;o at..: ' cs, 'o - ,- c ,

* 1 G .s , . .; . . . . . . . : . . .. .. . . ..'., . 7 ' .' a l x

pa a:, t r "C ..:. .. IZQU-t' iI fi ,I 5 22..' . . . . .

[.6 . [{:::::.: I.; Z, ';. . a~c: ;£,R ',':V I', . F (197 ) . :: i L .,LTD 201 -53, X. Ck..r<ii w .' .i.//X ,.\. -.L~ ~ ~ O] ' .:: .; ''_,, ",'

;: b~i...i szi&$ 1., t..4/ V L o: 10 r ncocoS i ojc:;a . prou 1cp :!2 . ic;;. v ct 5,,i|

CL" 55-67-

" t~~0. 8 'i .:i L ... u. , : : : :,/ ..i), '. 2. , ,:, ... , .1 ,, .> . (1 ;"" .

c 'u LL .. ~lu . . ...

. .< .. .. 66-72.

Li,-'] :2c'............. ... i..... , g ' :' NA£ . D 1 7 :, ,¢ 2n 'i :. c "

r3.ic est. xt r . 1. .. Caor w1~tl F 1'iiok' b P.-7 .

LII] P::pLE . .. (! 7,).' +M:; - o;., iidc < .. r xmzc:c.s_:.n rs.

CII. Dc::.z.:.io': University V ar ::o . ofl 0 ,<t ;: tl a .2~ '.-..:

7t..n-cl ool t c7 Sot r

L;. ... (. C C, hL'D' ' 1 ..

. . . . .

Cr: & 200 Lt rlu.a rcazcssion. .. .Z. I

ft.:

96 Journal of the Amnericant Stat~ticaI Association, March 1980

3. EAnrGPESGE VALUES

X Z- !-I Z XA A.2,_: :. :12 ' : .I i r I'

I I 'r ij ! :.I: AI !IL it-1 _2'

~1111"1 111 11 1 it i li VlI \ -IX ll t~ :11 ,1 ti,

Wol that Eli '11 11'il. LIid k112 0LLIIiL lILLILI II III I :L.*IXCE

A ll L - iilll pt -'L l~ v~it i. - w t I i l' I la Zv' ID tIll- ' t Ill t t o ;;i )l,.I tI

(' Iiilla\II. C' VI Il IIV 0 iljh 1

fo mua t\, - ! b -. L AL-' . Tlo I I \,1 . :o

.4 IML

f'I c I h atL I () 211:121 CII P'7 .~ "L I:

-!LL

I *22L I. , " J , I

1lr LI III2IL ILI ILdL ..*LII L I' ,121 NIIKII:II II.

. .I 1 . .L. 72

t k 11 1 1 o~ IC m enLAWENC C. AL ndTOA P YN

f~.11IiI2I2IX .\I~i\ ". 1i L I TX III I I;,IIIItI 111II~I2I t~l . Jo rna ofthe Ameica Sttisica Asoc7tiT

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