On Properties of the difference between two modified C p statistics in the nested multivariate linear regression models

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1 Global Journal o Pur Ald Mathatcs. ISSN Volu 1, Nubr 1 (016), Rsarch Inda Publcatons htt:// On Prorts o th drnc btwn two odd C statstcs n th nstd ultvarat lnar rgrsson odls S. Boony 1, J. Jtthavch V. Lorchrachoonkul School o Ald Statstcs, Natonal Insttut o vlont Adnstraton Bangkok 1040, Thal. Abstract Th statstc whch s th drnc o th odd C statstcs n th ull odl n th rducd odl s dvlod or varabl slcton n th contoranous ultvarat lnar rgrsson nstd odl. Th xctaton, th varanc th dstrbuton o th statstc ar drvd va thr thors thr las. Kywords: Modd C statstc, tst statstc or varabl slcton, dstrbuton o th tst statstc. Mathatcs Subct Classcaton: 6H10, 6H15 Introducton In th syst-o-quatons odl wth contoranously but no srally corrlatd dsturbanc th dndnt varabl vctor y can b wrttn n th stackd or o th syst o quatons odl as yxβε, (1) whr y s th n 1 vctor consstng o subvctors y, th n 1 dndnt varabl vctor n quaton, 1,,,, X s th n T dagonal atrx consstng o dagonal subatrcs, X, th n atrx o ndndnt varabls ncludng th constant unt vctor n quaton wth rank( X ), 1,,,, s th nubr 481

2 48 S. Boony t al o aratrs n quaton, T, β ( β1 β β ) s th 1 aratr 1 vctor consstng o subvctors β, th 1 aratr vctor n quaton, 1,,,, n s th nubr o obsrvatons n ach quaton. Th n 1 ro dsturbanc vctor, ε, s assud to b uncorrlatd across obsrvatons n th sa quaton but contoranously corrlatd across obsrvatons n drnt quatons dstrbutd as Nn( 0, Ω ) whr Ω Σ I n th covaranc atrx Σ can b wrttn as Σ. 1 Th ultvarat rgrsson odl s a scal cas o th syst-o-quatons odl whr X X,, 1,,,. Th total nubr o aratrs n th ultvarat rgrsson odl s qual to. In a ultvarat lnar rgrsson odl, th odd T C statstc suggstd by Lorchrachoonkul Jtthavch (01) bcos 1 1 y y M E( εω ε) ( n ) tr( Σ Σ ), () whr ε s an n 1 vctor o ro rrors n th odl dstrbutd as Nn ( 0, Ω ) 1 1, Ωy Σy I n, Σ y s an covaranc atrx o dndnt varabls. Th stat o M n () n th ull odl wth th total nubr o aratrs can b xrssd as whr MC Ωˆ ( n ) tr( Σˆ S ), (3) 1 1 y y s th n 1 vctor consstng o n1subvctor o rsduals n quaton,, 1,,,, n th ull odl, Σ ˆ y s th stat o sal covaranc atrx o th rsdual vctors o Σ y T S s th s, whch s usd as th stat Σ rovdd that th ull odl s a good aroxaton o th unknown tru odl. Th odd C statstc n () s drnt ro th orgnal dnton o Mallows statstc, th xtndd dnton o Mallows C statstc n a sngl quaton odl to a ultvarat rgrsson odl (Sarks t al., 1983; Fukosh Satoh, 1997; Yanaghara Satoh, 010). Th unknown rror covaranc atrx o th tru odl n th orgnal dnton s rlacd by th known covaranc atrx o dndnt varabls as shown n (). Th rst o th ar s organzd as ollows. Th hyothss tstng or varabl slcton undr th odd C statstc n () n th contoranous ultvarat lnar rgrsson nstd odl s ntroducd n th nxt C

3 On Prorts o th drnc btwn two odd C statstcs n th nstd 483 scton. Scton 3 rsnts th drvaton o th statstcal rorts th dstrbuton o th drnc o th odd C statstcs n th ull odl n a rducd odl n th ultvarat lnar rgrsson. Th nal scton s conclusons. Hyothss Tstng In varabl slcton n th ultvarat lnar rgrsson odl n (1) undr th M crtron n (), an xlanatory varabl can b lnatd ro th odl sccaton ts dscard dos not ncras th valu o th statstc sgncantly. Othrws, such xlanatory varabl s rtand n th odl. In othr words, a varabl or a nubr o varabls can b lnatd ro th odl sccaton th null hyothss s not rctd at a sgncant lvl. H M M aganst H : M M, (4) 0 : r a r whr M M r ar th valus o th odd C statstcs o th ull odl th rducd odl rsctvly. In ordr to tst th hyothss (4), t s sucnt to consdr whthr th drnc MC MC, (5) r s sgncantly drnt ro zro. Fro (3), t s obvous that th drnc can b xrssd as ˆ 1 ˆ 1 ( ) Ω ( ) dtr( Σ S ), (6) r y r y whr r ar th rsdual vctors n th ull rducd odls rsctvly, d s th drnc n th nubr o aratrs n th ull rducd odls, d. r Prorts o th Statstc In ths scton, th xctaton, th varanc th dstrbuton o th statstc ar nvstgatd. Th statstc n (6) conssts o two trs. Th rst tr n th quadratc or can b xrssd n trs o a lnar cobnaton o th roducts o two corrlatd noral vctors as ˆ 1 r y r ˆ y r ˆ r y ( ) Ω ( ) ( ) ( ) d d, (7) whr d r d. Slarly, r tr( Σˆ S ) whch s a art o th 1 y scond tr o th statstc n (6) can b wrttn n trs o a lnar cobnaton o th roduct o two corrlatd noral vctors as (7), tr ˆ 1 ( Σy S ) ˆ ˆ,, 1,,...,, y s y ( n ) (8)

4 484 S. Boony t al whr s, 1,,..., ; 1,,...,. ( n ) Notd that both trs n th LHSs o (7) (8) ar n th slar or o wghtd su o th roduct o twocorrlatd vctors whch ay b rrsntd by aqa whr a s a n 1 vctor consstng o corrlatd subvctors o sz n 1, a, wth zro an varanc I, 1,,,, Q s an n n atrx. La 1. Lt a n a n N ( 0, I ) rsctvly, s an n 1 vctor dstrbutd as Thn a b th n 1 corrlatd vctors dstrbutd as a s a corrlaton cocnt btwn a n a n a a (1 ) a a a a N a n n 0 a I n (, ) N ( 0, I ), ndndnt o a, a a z. (9) a a ar dntcally dstrbutd as n a n N ( 0, I ). a, z a s dnd as Proo o La 1. Snc a z ar ndndnt noral ro vctors wth th sa zro an varanc I, th xctaton o a a E( a ) E a a (1 ) a z a a 0, th varanc o a can b xrssd as a a a a n a a n a a a n Var( a ) I (1 ) I I. Thror, a s dstrbutd as a n n a n a s qual to zro, N ( 0, I ), th dntcal dstrbuton as a. La. Lt a b a n 1 vctor consstng o corrlatd subvctors, a s, o sz n 1 dstrbutd as n a n N ( 0, I ), a Q I whr Σ s an atrx. Thus, n a a b a corrlaton cocnt btwn a aqa u aa, (10) 1

5 On Prorts o th drnc btwn two odd C statstcs n th nstd 485 whr u a a a 1, s th th lnt o th atrx Σ. Proo o La. Th quadratc tr a Qa can b xrssd as aqa aa. (11) 1 1 By La 1, th ro vctor a n (11) can b rlacd by th ro vctor n (9). Thn (11) bcos a a a Qa a a (1 ). a a a z 1 1 a 1 1 Snc z s ndndnt o aqa u aa, 1 a a, w gt whr u. (1) a a a 1 By La, t s obvous that (7) (8) can b rsctvly r-wrttn as ˆ 1 r y r w ( ) Ω ( ) d d, (13) tr( Σˆ S ) w, (14) 1 y 1 whr w 1 ˆ y d, d d (15) 1 w ˆ y. (16) n 1 Substtutng (13) (14) nto (6) ylds dd 1 w d w. (17) 1 1 d Thor 1. Lt d b dstrbutd as Nn( 0, d I ) n Nn(, ) n dd w b as dnd n (15) (16) rsctvly. Thn th dstrbuton o w1 1 d R1 1, whr can b aroxatd by a a 0 I, w 1 ar chosn so that th rst two

6 486 S. Boony t al cuulants o th dstrbutons ar qual. Slarly, th dstrbuton o w 1 b aroxatd by w w1 w1 d w 1 1, 1, can R undr th sa crtra, whr w 1 n w w1 w1 d n w 1, w w w 1 s th corrlaton cocnt btwn Proo o Thor 1. Snc dd d, w 1 w w w 1 d s th corrlaton cocnt btwn d dd rdo, t s obvous that th xctaton o w1 1 dd E w1 nw 1. 1 d 1 dd Th varanc o w1 1 d can b wrttn as., d, s dstrbutd as ch-squard wth n dgrs o d can b wrttn as d 1 1 d d d 1 1 ( d d, dd Var w w Var w w Cov ) 1 d 1 1 d d d nw1 w1 w1 d, 1 dd sncvar dd n, 1,, d s th corrlaton cocnt o d d dd (Issrls, 1981). d

7 On Prorts o th drnc btwn two odd C statstcs n th nstd 487 dd Th dstrbuton o th wghtd su o corrlatd ch-squard varabl w1 1 b aroxatd by R d can (Brown, 1975; Makab, 003 Hou, 005) whr th rst two cuulants o th two dstrbutons ar qual. Thus, quatng th rst two dd cuulants o w1 R 1 ylds d n w, (18) Solvng (18) (19) ylds 1 1 nw1 w1 w1 d 1 1 w 1 w 1 1 n w w1 w1 d 1 w1 w1 d w 1 1 Slarly, t can b shown that n w 1 w w w 1 w 1 w w w 1. (19), (0). (1),, whr s th corrlaton cocnt o (Issrls, 1981). La 3. dw1 d W ar ndndnt whr d ar th n 1 vctors consstng o subvctors d, dstrbutd as n d n () (3) N ( 0, I ), subvctors,

8 488 S. Boony t al dstrbutd as n n N ( 0, I ) rsctvly, 1,,...,, W 1 W ar th n n dagonal atrcs wth th th w dagonal lnt 1 Proo o La 3. Th rsdual vctors rducd odl can b rsctvly xrssd n tr o d w rsctvly. r n th ull odl n th y (Graybll, 1976), ( I M ) y, (4) n ( I M ) y, (5) r n r 1 ( ) whr M X X X X 1 r r ( r r ) r M X X X X. Fro (4) (5), th drnc vctor d can b wrttn as d ( M M ) y. (6) r Consdr th tr MM r by arttonng th atrx X nto Xr X d. 1 1 ( XrX r) 0 X M ( ) r rm Xr Xr Xr Xr Xr Xd M r, (7) 1 0 ( XdX d) X d snc X r X d ar ndndnt. Slarly, M Mr M r. (8) Thn, by (7), (8) th dotnt rorty o dw d W can b xrssd n tr o y as 1 w d W d y M M M M y 1 1 ( )( ) r r 1 d w1 y ( ) r 1 d w ( )( ) n n 1 w y ( ) n 1 M M r, th quadratc tr y M M y, (9) W y I M I M y y I M y, (30) whr y s a stardzd or o y wth varanc I n. Fro (9) (30), t can b sad that y ( M Mr ) y y ( In M ) y ar ndndnt, thn dw1 d W ar ndndnt. Agan, by (7) th dotnt rorty o M, t can b concludd that ( M M )( I M ) 0, r n

9 On Prorts o th drnc btwn two odd C statstcs n th nstd 489 whch lads to th concluson that y( M M ) y y( I M ) y ar ndndnt r n (Graybll, 1976). Thor. Th xctaton th varanc o th statstc can b xrssd as d d n ˆ y d ˆ y d d d, 1 n n d n d S d S 1 n d 4 nd, d d S d S d S S 1 n k d k1 y dk dk k ˆ y k k k1. k d y d k dk k1 k k1 y k k whr S ˆ, S ˆ, S ˆ S Proo o Thor. By Thor 1, th xctaton o th statstc n (17) can b xrssd as 1 1 d. (31) Substtutng 1, 1, ro (0)-(3) nto (31) gvs 1 1 n w dw. (3) Substtutng (15) (16) nto (3) gvs n ˆ y d ˆ d d d y n d d n ˆ y d ˆ y d d d. (33) 1 n n By Thor 1 La 3, th varanc o statstc can b wrttn as 1 1 8d. (34) Agan, substtutng 1, 1, ro (0)-(3) nto (34) ylds n w1 4d w w1 w1 d 4d w w 1. (35) Substtutng (15) (16) nto (35) gvs

10 490 S. Boony t al d n d S d S 1 n d 4 nd, d d S d S d S S 1 n whr S d, S d, S S ar as dnd n Thor. Thor 3. Th dstrbuton o th statstc s a gaa dstrbuton wth sha aratr scal aratr. Proo o Thor 3. Snc th statstc n (17) s a lnar cobnaton o two ndndnt Ch-squard varabls, th dstrbuton o can b aroxatd by whr ar chosn so that th rst two cuulants o th two dstrbutons ar qual (Brown, 1975; Makab, 003 Hou, 005). By Thor, t s obvous that ( ). Thror, t can b concludd that th dstrbuton o th statstc s a gaa dstrbuton wth sha aratr (36) scal aratr. By Thor, th stardzd statstc can b wrttn as T th cntral lt thor, convrgs to N(0,1) as n. whch, ro Conclusons In ths ar, thr thors thr las ar rovd to rovd th statstcal rorts o th statstc whch can b usd n varabl slcton n th contoranous ultvarat lnar rgrsson nstd odl. Ths s anothr altrnatv o varabl slcton by tstng th hyothss (4) nstad o drct coarson o th odd C statstcs. Wthout th hyothss tstng, th backward rocdur o varabl lnaton suggsts to rtan any varabl n th odl sccaton ts lnaton rsults n th odd C statstc hghr than MC vn n th cas o slght ncras. But by hyothss tstng, a varabl can b lnatd ro th odl sccaton ts lnaton dos not caus th rcton o th null hyothss. It can b concludd that th nubr o ndndnt varabls n th nal odl by usng th hyothss tstng n varabl lnaton s at ost qual to th nubr o ndndnt varabls n th nal odl by th convntonal varabl lnaton. Th roosd conct can b asly xtndd or varabl slcton by th orward stws rocdurs o varabl slcton.

11 On Prorts o th drnc btwn two odd C statstcs n th nstd 491 Rrncs [1] Brown, M.B., 1975, A thod or cobnng non-ndndnt, on-sdd tsts o sgncanc. Botrcs,31, [] Fukosh, Y., Satoh, K., 1997, Modd AIC C n ultvarat Lnar Rgrsson. Botrka, 84(3), [3] Graybll, F.A., Thor Alcaton o th Lnar odl. uxbury, Canada, [4] Hou, C.., 005, A sl aroxaton or th dstrbuton o th wghtd cobnaton o nonndndnt or ndndnt robablts. Statstcs Probablty Lttrs, 73, [5] Issrls, L., 1981, On a orula or th roduct-ont cocnt o any ordr o a noral rquncy dstrbuton on n any nubr o varabls. Botrka, 1, [6] Lorchrachoonkul, V., Jtthavch, J., 01, A odd C statstc n a systo-quatons odl. Journal o Statstcal Plannng Inrnc,14, [7] Makab, K.H., 003, Wghtd nvrs ch-squar thod corrlatd sgncanc tsts. Journal o Ald Statstcs, 30, [8] Sarks, R.S., Trosk, L., Coutsourds,., 1983, Th ultvarat C. Councatons n Statstcs-Thory Mthods, 1(15), [9] Yanaghara, H., Satoh, K., 010, An unbasd C crtron or ultvarat rdg rgrsson. Journal o Multvarat Analyss, 101,

12 49 S. Boony t al