A Note on Estimability in Linear Models

Transcription

1 Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): DOI: /j.statstcs A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs, Fdral Polytchnc, Nkd, Owrr, Imo Stat, Ngra 2 Dpartmnt of Statstcs, Imo Stat Unvrsty, Owrr, Imo Stat, , Ngra Abstract Estmabl functons of th paramtrs ar charactrzd n trms of gnralzd nvrss. Th concpt of stmablty s appld to data from a dsgnd xprmnt on vartal trals. W dmonstrat n ths not that ths tchnqu of solvng th normal quatons s quvalnt to th narst nghbour mthod for th analyss of unbalancd randomzd dsgn. Kywords BLUE, Estmabl Functons, Estmablty, Gnralzd Invrss, Lss than Full Rank, Lnar Combnaton, Lnar Modls 1. Introducton Lnar modls ar gnrally of th form (whr y s an n 1 y = Xβ + ε (1) obsrvaton vctor, X s an n p dsgn matrx of fxd constants havng rank r ( r k), β s an p 1 vctor of unknown paramtrs, ε s an n 1 vctor of unknown random rrors havng zro mans) and E ( y) = Xβ. Th Ordnary Last Squar (OLS) soluton of (1) s ˆ β = ( XX ) 1 Xy, a unqu soluton. In practc, not all lnar modls of th form n (1) ar of full rank. Whn X s not of full rank, thn X X s sngular and th normal quatons ( X X ) b = X y do not hav a unqu soluton. Howvr, thr ar varous approachs of obtanng th nvrs of sngular matrcs, for whch th row chlon form gvn by Elswck t al (1991), Moor Pnros and th gnralzd nvrs, Sarl (1977) ar popular n th ltratur. Th gnralzd nvrs s th approach w apply n ths papr. 2. Form of Estmablty Wth X lss than full rank and X X sngular,.. r k, thr s an nfnt numbr of solutons of ˆβ to th normal quatons. Attnton s thrfor drctd not to th solutons thmslvs but to lnar functons of thr lmnts. Consdr a lnar functon q β of th paramtrs n β, whr q s a known vctor. Ths lnar functon s dfnd * Corrspondng author: sammyadymo@gmal.com (S. O. Adymo) Publshd onln at Copyrght 2014 Scntfc & Acadmc Publshng. All Rghts Rsrvd as bng an stmabl functon f thr xsts som lnar combnatons of th obsrvatons y 1, y2,..., yn whos xpctd valu s q β,.. f thr xsts a vctor t such that th xpctd valu of t y s q β, thn q β s sad to b stmabl. Consdr th followng thorm gvn n Graybll, 1976: Thorm 1. (Graybll, 1976) Assumng a lnar modl n (1), q β s an stmabl functon f and only f thr xst an n 1 vctor t such that q = t X Proof. If thr xst a vctor t such that, thn, ( ) ( ) q = t X E ty = te y = tx β = q β. Only f: Convrsly, f q β s stmabl, thn, E( ty ) = qβ Thus tx β = q β tx = q In addton, Elswck t al (1991) argus that f X s of full matrx rank, ( XX) 1 ( XX) 1 1 {( ) } xsts and th rows of p n X srv as th ncssary st of vctors bcaus XX X Xβ = β. 3. Illustraton W dmonstrat ths dscusson by consdrng th data from a study to compar classcal and narst nghbour mthods n th analyss of vartal trals (S,.g. Nwob, 2000). In th xprmnt, nn (9) dffrnt varts of cassava crop wr trd, sx at a tm ovr a maxmum of fv yars n such a way that ths varts wr not rplcatd

2 Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): qually. Th modl (wthout ntracton) s gvn by whr y = µ + τ + ε = 1,2,...,9; j = r (3) j j y j s th yld from th j th tral of th th varty, µ s th gnral man, th random rror assocatd wth τ s th ffct of th th varty, y j. Equaton (3) s wrttn n matrx form as ε j s y = Xβ + ε. (4) Basd on th modl n (2), th paramtr vctor β s gvn by Th componnts of th modl (4) ar from whr w obtan µ t t t t 9.4 = t t t t t

3 214 S. O. Adymo t al.: A Not on Estmablty n Lnar Modls XX = ; Xy = A gnralzd nvrs of XX wrttn as G such that wth G = s,.g. Sarl (1977) s 1 XXG XX XX = H = G XX = =. and w ( w w w w w w w w w w ) Th functon ( ) q b = w Hb = w + w + w w + w µ + w t + w t w t + w t s stmabl for any gvn valus to th ws. Wth ths w obtan th soluton to th normal quaton as ( ) b ˆ = G Xy = Thrfor, th Bst Lnar Unbasd Estmator (BLUE) of qb s qb ˆ = wb ˆ = 15.9w w 5.2w w + 4.7w + 4.1w + 2.0w 4.3w + 1.9w 4.9w

4 Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): To s f β whr = 0,1,2,3,...,9 s stmabl, w β = β whr, n ths cas, w wrt th paramtr as t dfn T ( t 0, t 1,..., t 9) =, a p p of dmnson 1 p, so that and matrx; T = t s TG ( XX ) = Snc TG ( XX ) T β β consdrng, for = 1 and 2, β s not stmabl. Howvr, <, ths functon may b wrttn = as β1 β2 = T β whr T = [ ] so that TG ( XX ) = ( ) = T. Ths mpls that β1 β2 s stmabl. Smlarly, snc thr ar 9 (nn) paramtrs, takng two (contrast) at a tm gvs 9 C 2 = 36 stmabl functons T = TG XX = T. Thus, w can say that a Thrfor, ( ) lnar combnaton of stmabl functons s stmabl. 4. Conclusons W hav shown that for any arbtrary vctor w, q b = w Hb s stmabl wth BLUE qb ˆ = wb ˆ 0. Th soluton of th normal quaton, ˆb, confrms that ths approach s quvalnt to th Narst Nghbour mthod of analyss of dsgnd xprmnts. Both mthods agr on th slcton of varts though th valu of ths stmats ar

5 216 S. O. Adymo t al.: A Not on Estmablty n Lnar Modls not unqu du to th applcaton of gnralzd nvrss. Furthrmor, w vrfd that th lnar combnaton of stmabl functons s stmabl. Lnar Modl. Th Amrcan Statstcan, Vol 45, No 1. Fb pp [2] Sarl, S. Lnar Modls. Wly, Nw York. (1977). [3] Graybll, F.A. Thory and Applcatons of th Lnar Modl. Blmont, C.A: Wadsworth. (1976). REFERENCES [1] Elswck, R.K., Gnnngs, C., Chnchl, V.M., Dawson, K.S. A Smpl Approach for Fndng Estmabl Functons n [4] Nwob, F.N. A Comparson of Classcal and Narst Nghbour Mthods n th Analyss of Vartal Trals. Journal of Sustanabl Agrcultur and Envronmnt. Vol. 2 No.1. pp (2000).