Robust Small Area Estimation Using Penalized Spline Mixed Models

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1 Scton on Sry Rsarch thods JS 9 Robst Small Ara Estmaton Usng Pnalzd Spln xd odls J. N.. Rao, S.. Snha and. Ronossadat School of athmatcs and Statstcs, Carlton Unrsty, Ottawa, Canada S 5B6 Abstract Small ara stmaton has bn xtnsly stdd ndr lnar mxd modls. In partclar, mprcal bst lnar nbasd prdcton (EBLUP stmators of small ara mans and assocatd stmators of man sqard prdcton rror (SPE that ar narly nbasd ha bn dlopd. Howr, EBLUP stmators can b snst to otlrs. Snha and Rao (9 dlopd a robst EBLUP mthod and dmonstratd ts adantags or th EBLUP ndr a nt ll lnar mxd modl n th prsnc of otlrs n th random small ara ffcts and/or nt ll rrors. A bootstrap mthod of stmatng SPE of th robst EBLUP stmator was also proposd. In ths papr, w rlax th assmpton of lnar rgrsson for th fxd part of th modl and rplac t by a war assmpton of a pnalzd spln rgrsson and dlop robst EBLUP stmators. Bootstrap stmators of SPE ar also dlopd. Rslts of a lmtd smlaton stdy ar smmarzd. y words: Bootstrap, man sqard prdcton rror, otlrs, random ffcts, small ara man, nt ll modl. Introdcton Ara ll and nt ll lnar mxd modls ha bn xtnsly sd n small ara stmaton. In ths papr, w focs on nt ll mxd modls basd on a sngl, contnos axlary arabl, x, rlatd to th arabl of ntrst, y. A basc nt ll lnar mxd modl, calld nstd rror lnar rgrsson modl (Batts t al 988, s gn by y β β,..., N ( j xj j,,..., m; j whr m s th nmbr of small aras, N s th nmbr of poplaton nts n ara, ~ d N(, dnot random small ara ffcts that accont for araton not xpland by th axlary arabls x j, and s ndpndnt of th nt rrors j ~ d N(,. A sampl of n ( nts s drawn from ach ara, and samplng s assmd to b gnorabl n th sns that th poplaton modl ( also holds for th sampl. Hr or ntrst s n stmatng th small ara man Y of th N poplaton als y j. If th 45

2 Scton on Sry Rsarch thods JS 9 samplng fracton n / N s nglgbl, thn Y β βx whr X s th nown poplaton man of th xj for ara. An mprcal bst (or mprcal Bays stmator, abbratd EB, of s obtand as a wghtd sm of a sampl rgrsson stmator y ˆ β ( X x and a synthtc stmator ˆ β ˆ X β, whr ( y, x ar th sampl mans for ara and β ( ˆ β, ˆ β ar consstnt stmators of th rgrsson paramtrs n (. Th optmal wghts dpnd on th aranc componnts θ (, whch ar rplacd by consstnt stmators, for xampl maxmm llhood (L or rstrctd L (REL stmators. Th EB stmator s also an mprcal bst lnar nbasd prdcton (EBLUP stmator wthot assmng normalty. Rao (3, chaptr 7 gs a dtald accont of EB and EBLUP stmaton, assocatd man sqard prdcton rror (SPE approxmaton and a narly nbasd stmator of SPE. Th EBLUP stmator can b snst to otlrs n and j. Snha and Rao (9 stdd robst EBLUP (REBLUP stmaton of, sng som gnral rslts of Fllnr(986 who stdd robst stmaton of random ffcts n lnar mxd modls. Fllnr obtand robst mxd modl qatons to stmat β and (,..., m for gn θ, sng Hbr s ψ fncton and proposd a two-stp trat procdr for gttng a robst stmator of θ and n trn robst stmators of β and. Snha and Rao (9 sd an altrnat mthod of stmatng β and θ by solng robst scor qatons for β and θ and th rsltng robst L (RL stmators ar thn sbstttd n th mxd modl qaton for to gt REBLUP of. Thr smlaton rslts sggstd that th proposd mthod of stmatng θ can b sgnfcantly mor ffcnt than Fllnr s mthod, bt th ffcncy gans wr small whn stmatng th small ara man. Th assmpton of lnar rgrsson n ( may b rstrct n practc. To gt arond ths dffclty, β β xj n ( s rplacd by an nnown smooth fncton f ( x j whch s assmd to b approxmatd sffcntly wll by a pnalzd spln (P-spln fncton (Rprt t al 3, Opsomr t al 8 and Ugart t al 9. Usng a mxd modl rprsntaton of th P-spln n (, an EBLUP stmator of small ara man may b obtand. Th stmaton of SPE of th EBLUP stmator prsnts som dffclts bcas th mxd modl dos not ha a bloc dagonal coaranc strctr, nl (. Opsomr t al (8 stdd ths problm and also proposd a bootstrap stmator of SPE. Rbn-Blr t al (9 stdd P-spln ara ll modls and obtand an EBLUP stmator of small ara man and assocatd bootstrap stmator of SPE. Th EBLUP stmator ndr th P-spln mxd modl can b snst to otlrs n and j, as n th cas of EBLUP ndr th nstd rror lnar rgrsson modl (. In ths papr, w propos to obtan REBLUP stmator of small ara man ndr th P- spln rson of ( sng Fllnr s (986 gnral rslts on robst mxd modl qatons and hs two stp trat mthod; th approach of Snha and Rao (9 for 46

3 Scton on Sry Rsarch thods JS 9 robst stmaton of β and θ rns nto dffclty n th contxt of P-spln mxd modls. W also propos a bootstrap stmator of SPE of th REBLUP stmator. Rslts of a smlaton stdy ar smmarzd.. P-spln xd odl W assm that th tr man spcfcaton f x s wll approxmatd by a P-spln approxmaton sp( x j, basd on th mxd modl formlaton. Th rsltng P-spln mxd modl for th sampl s gn by j j j ( j y sp( x,,,..., m; j,..., n ( whr sp( x j, β β x ( x q (3 j j wth th q dnotng th nots Rgardng th choc of and q,...,q, ( x q max(, x q and ~ N(,. q, w ha followd Rprt t al (3 n or smlaton stdy: ( On nds nogh nots to nsr sffcnt flxblty to ft th data, bt aftr that addtonal nots do not chang th ft mch. ( Plac th nots at th sampl qantls of th nq x als whch gs qal or narly qal nmbr of x als btwn nots. Not that N f ( x j j whch s approxmatd by P N sp( x,. j j In th smlaton stdy, w gnratd xj ~ d N(, X n whch cas w ha E { f ( X } (4 d and E } (5 P β β ( X E{( X q W xprss th P-spln mxd modl ( n matrx form as y Xβ W Z (6 whr ~ N(, I, ~ N(, I m and ~ N(, I n wth n n. Hr s th ctor of th spln ffcts, s th m ctor of th small ara random ffcts 47

4 Scton on Sry Rsarch thods JS 9 and s th n ctor of th nt rrors j. Th matrx form of ( s gn by (6 wthot th trmw.not that (6 dos not ha a bloc dagonal strctr, nl th matrx rson of (, bcas of th addtonal trmw. 3. P-spln EBLUP Estmators W frst consdr EBLUP stmaton of th random ffcts n th P-spln mxd modl (6, followng Fllnr (986, and thn modfy th qatons to gt robst EBLUP stmators n th prsnc of otlrs. For fxd θ (,,, th BLUP stmators ~ ~ β β ( θ, ~ ~ ( θ and ~ ~ ( θ of β, and ar obtand by solng th followng mxd modl qatons of Hndrson (963: X ( y Xβ W Z W ( y Xβ W Z Z ( y Xβ W Z (7 Now rplacng θ by th REL stmator θˆ w gt th EBLUP stmators ˆ ~ β β ( ˆ, θ ˆ ~ ( ˆ θ and ˆ ~ ( ˆ θ. Fllnr (986, followng Harll (977, obtand REL qatons whch ar sold tratly n conjncton wth (7 to gt th REL stmator of θ. Fllnr s REL qatons for th P-spln mxd modl (6 may b wrttn as ~ m /( t ~ /( m t ~ /{( n ( t ( m t } j j (8 whr ~ ~ ~ β ~ ~, t j yj βxj ( xj q tr( T / wth Tand T tr( T / and t dnotng th dagonal blocs of a parttond matrx T whch s th nrs of th parttond matrx wth dagonal blocs gn by W W I and Z Z I m and off-dagonals gn by W Z and ts transpos. An EBLUP stmator, ˆ P, of th P-spln approxmaton to ara man s thn obtand from (5 by rplacng β, β, and by th stmators ˆ β, ˆ β,û and ˆ : P ˆ β ˆ E X ˆ β ( E( X q ˆ ˆ (9 48

5 Scton on Sry Rsarch thods JS 9 Th man sqard prdcton rror (SPE of th P-spln EBLUP stmator ˆ P s gn by SPE( ˆ ( ˆ P E P whr th xpctaton s wth rspct to th tr ndrlyng modl. 4. Robst P-spln EBLUP Estmators W now obtan robst P-spln EBLUP stmators, sng Hbr s (973 robst - stmaton approach wth ψ fncton gn by ψ b ( mn(, b /, whr b > s a tnng constant, commonly chosn asb Robst stmators of β, and for ~ ~ fxd θ, dnotd β (, ~ ~ β θ ( θ and ~ ~ ( θ, ar obtand by solng robst mxd modl qatons. Th lattr qatons ar obtand from (7 by rplacng y Xβ W Z by ψ { ( y Xβ W Z}, by ( ψ and by ψ (. Robst REL qatons for stmatng th aranc componnts n th P-spln mxd modl ar obtand from Fllnr s REL qatons (8 by mang th followng changs: Rplac ~ by ( ~ ψ, ~ by ( ~ ψ and ~ j by ( ~ ψ j, whr ~ ~ ~ ~ ( θ y β β x ~ ( x q ~ j j j j j Now solng th modfd qatons corrspondng to (7 and (8 tratly as dsrbd n Scton 3, w gt a robst stmator of th aranc componnt ctor θ dnotd by ˆ θ ( ˆ, ˆ, ˆ and robst EBLUP stmators of β, and as ˆ ~ β ( ˆ, ˆ ~ ( ˆ β θ θ and ˆ ~ ( ˆ θ. Now sbstttng th robst EBLUP stmators for β, and n th P-spln approxmaton P gn by (5, w gt th robst EBLUP (REBLUP stmator of P as P ˆ β ˆ E X ˆ β ( E{( X q } ˆ ˆ ( 5. Bootstrap Estmaton of SPE It sms dffclt to gt an analytcal formla for an SPE stmator that s narly nbasd. Thrfor, n ths papr w focs on bootstrap stmaton of SPE of th robst P-spln EBLUP stmator ˆ P. Th basc da s to mmc th SPE by sng smlatd sampls gnratd from an stmatd modl, followng Snha and Rao (9 49

6 Scton on Sry Rsarch thods JS 9 who consdrd REBLUP stmaton and assocatd bootstrap SPE stmaton for th nstd rror lnar rgrsson modl. Th stps n mplmntng bootstrap stmaton ar as follows:. Gnrat ndpndntly ~ N(, ˆ, ~ N(, ˆ and ~ N(, ˆ. d ˆ ˆ j β E( X ( E X q ˆ ˆ β β E ( X ( E X q Lt y β, j,..., n ;,..., m and,,..., m. d j j d. Calclat robst P-spln EBLUP {( y, x : j,..., n ;,..., m}. j j ˆ P from th bootstrap data 3. Thortcal bootstrap stmator of SPE of ˆ P s thn gn by msp ˆ ˆ ( B ( P E ( P whr E dnots bootstrap xpctaton. 4. In practc, w gnrat a larg nmbr, B, of bootstrap sampls b (,..., B and ( calclat ˆ b (b P and from ach sampl b. W thn approxmat ( by msp B ( b ( b ˆ ˆ B( a ( P B ( b P. ( In th smlaton stdy (Scton 6 w gnratd a larg nmbr of sampls r,..., R from th assmd tr modl wth spcfd paramtrs and from ach sampl w gnratd B bootstrap sampls and calclatd (. Th als of ˆ P and for ( sampl r ar dnotd by ˆ r ( r ( r ( r and β β E ( x E{( X q } P (r (r whr and ar th als of and for th smlaton rn r gnratd from spcfd dstrbtons, for xampl contamnatd normal dstrbtons, s Scton 6. Th SPE of ˆ s approxmatd by P SPE( ˆ P R R r ( ˆ ( r P ( r (3 Smlarly, th bootstrap SPE stmator ( s calclatd for ach smlaton rn r and thn aragd or r to gt an approxmaton to th xpctaton of th bootstrap SPE stmator. Th smlatd rlat bas (RB of th SPE stmator s thn calclatd from th lattr qantty and th approxmaton (3 sng th formla RB { E( msp ( SPE}/( SPE. (4 B a 5

7 Scton on Sry Rsarch thods JS 9 6. Smlaton Stdy In ths scton w rport som smlaton rslts on th prformanc of th proposd robst P-spln EBLUP stmator and th assocatd bootstrap stmator of SPE. Th followng fnctons f (x wr sd n gnratng th sampls: odl : f ( x x (lnar, odl : f ( x x x (qadratc and odl 3: f ( x ( x xp{ 4( x } (bmp fncton, Brdt t al 5. W assmd that X ~ N(, so that w ha E{ f ( X } qal to for modl, 4 for modl and 3 for modl 3 rspctly. W gnratd { x j : j,...,4;,...,4} from N(, and hld thm fxd for gnratng th sampl rsponss y j from th assmd tr modl y j f ( xj j, whr th random ffcts and th nt rrors j ar drawn thr from contamnatd normal dstrbtons or from t dstrbtons wth 3 dgrs of frdom to rflct otlrs thr n or n j or n both. For th contamnatd dstrbtons w assmd that ~ d ( γ N(, γ N(,, ~ ( γ N(, γ N(, j nd whr and 5. For combnatons of dstrbtons for and, dnotd (,,(,,(, and (,, wr stdd, whr (, ndcats no contamnaton ( γ γ, (, ndcats contamnaton n only ( γ., γ, (, ndcats contamnaton n only ( γ, γ. and (, ndcats contamnaton n both and ( γ., γ.. For spcfd dstrbtons of and j w gnratd R 5 sampls ( r ( r {, : j,...,4;,...,4} and thn th assocatd rsponss j ( { y r j ; j,...,4;,...,4} from th tr modl ( r,..., R. Thn th P-spln EBLUP and robst EBLUP stmators ˆ P and ˆ P wr comptd from ach smlatd sampl ( r {( yj, xj : j,..,4;,...,4} for spcfd nmbr of nots ( q,, 3. Th smlatd SPE of th stmators wr thn comptd sng th gnratd stmats ( ˆ r ( P and ˆ r (r P, and th small ara mans ( r,..., R 5, sng (3 for ˆ P and a smlar xprsson for ˆ P. It may b notd that q corrsponds to th standard nstd rror lnar rgrsson modl (. W smmarz th smlaton rslts basd on arag SPE or th aras. Dtald rslts wll b rportd n a sparat papr aftr mplmntng frthr smlatons basd 5

8 Scton on Sry Rsarch thods JS 9 on a largr nmbr of smlaton rns and othr scnaros. Som broad rslts from th smlaton stdy ar th followng: ( SPE for th P-spln stmators s not affctd by th choc of q, whn w compard th als for q to th corrspondng als for q 3. Ths rslt sggsts that q s a good choc. ( In th cas of contamnaton n (modl (, or n both and (modl (, robst EBLUP lads to sgnfcant rdcton of SPE rlat to EBLUP. For xampl, for q and qadratc tr modl, arag SPE (% for robst EBLUP s 3. compard to for EBLUP ndr (, and 38. compard to 68. ndr (,. On th othr hand, EBLUP s qt robst across th thr modls ndr (,. Snha and Rao (9 obsrd a smlar rslt for th lnar cas (modl and q. (3 In th lnar cas (modl, th ncras n SPE of th P-spln EBLUP or th EBLUP s mnmal across th for contamnaton combnatons. On th othr hand, EBLUP (wth q lads to larg ncras n arag SPE (% rlat to P-spln EBLUP whn th tr modl s qadratc (modl. For xampl, arag SPE (% of EBLUP s 45.7 compard to. for th P-spln EBLUP wth q n th cas of (,, and smlarly for th robst EBLUP rss robst P-spln EBLUP and th thr contamnaton combnatons. In th cas of bmp fncton (modl 3, howr, th ncras n SPE of th EBLUP stmator rlat to th P-spln EBLUP stmator s small (smlarly for th robst EBLUP rss robst P-spln EBLUP. Ths s prhaps d to th fact that th bmp fncton s closr to lnarty. W nd to confrm ths rslt by frthr stdy. (4 Rslts for th t dstrbton cas ar smlar to th abo rslts for th contamnatd dstrbtons. W also comptd th bootstrap stmats of SPE for ach smlaton rn sng th approxmaton ( wth B and thn sd (4 to obtan an approxmaton to th arag absolt rlat bas of th bootstrap SPE stmator for th contamnaton cass. Or rslts sggst that th bootstrap SPE stmator prforms qt wll n trms of arag absolt rlat bas: lss than % n most cass. Bt frthr stdy s ndd to confrm ths rslts. All n all, or lmtd smlaton stdy ndcats that th proposd robst P-spln EBLUP stmator wth nogh nots (say q prforms wll n trms of SPE whn th tr man fncton s not lnar. Also, th proposd bootstrap SPE stmator sms to trac th SPE qt wll. Rfrncs Batts, G. E., Hartr, R.. and Fllr, W. A. (988. An rror componnt modl for prdcton of conty crop aras sng sry and satllt data. Jornal of th Amrcan Statstcal Assocaton, 83, Brdt, F. J., Clasns, G. and Opsomr, J. D. (5. odl-assstd stmaton for complx srys sng pnalsd splns. Bomtra, 9,

9 Scton on Sry Rsarch thods JS 9 Fllnr, W. H. (986. Robst stmaton of aranc componnts. Tchnomtrcs, 8, 5-6. Harll, D. A. (977. axmm llhood approachs to aranc componnt stmaton and to rlatd problms. Jornal of th Amrcan Statstcal Assocaton, 7, Hndrson, C. R. (963. Slcton ndx and xpctd gntc adanc. In Statstcal Gntcs and Plant Brdng, Natonal Rsarch Concl Pblcaton 98, Natonal Acadmy of Scnc, pp Opsomr, J. P., Clasns, G., Ranall,. G., amann, G. and Brdt, F. J. (8. Non-paramtrc small ara stmaton sng pnalzd spln rgrsson. Jornal of th Royal Statstcal Socty, Srs B, 7, Rao, J. N.. (3. Small Ara Estmaton. Wly, Hobon, Nw Jrsy. Rbn-Blr, S., Dochto, C. and Rao, J. N.. (9. Bootstrap stmaton of th man -sqard prdcton rror n small ara modls for bsnss data. Papr prsntd at SAE 9 Confrnc, Elch, Span. Rprt, D., Wand,. P. and Carroll, R. J. (3. Smparamtrc Rgrsson. Cambrdg Unrsty Prss, Nw Yor. Snha, S.. and Rao, J. N.. (9. Robst small ara stmaton. Th Canadan Jornal of Statstcs, 37, Ugart,. D., Gocoa, T., ltno, A. F. and Drban,. (9. Spln smoothng n small ara trnd stmaton and forcastng. Comptatonal Statstcs and Data Analyss, 53,

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