Variance Estimation After Imputation

Transcription

1 Suve Metodolog, Jue Vol 7, o, pp Sttstcs Cd, Ctlogue o 00 Vce Estmto Afte mputto Je Kwg Km Abstct mputto s commol used to compeste fo tem oespose Vce estmto fte mputto s geeted cosdeble dscusso d sevel vce estmtos ve bee poposed We popose vce estmto bsed o pseudo dt set used ol fo vce estmto Stdd complete dt vce estmtos ppled to te pseudo dt set led to cosstet estmtos fo le estmtos ude vous mputto metods, cludg wtout eplcemet ot deck mputto d wt eplcemet ot deck mputto Te smptotc equvlece of te poposed metod d te dusted ckkfe metod of Ro d Stte (995) s llustted Te poposed metod s dectl pplcble to vce estmto fo two pse smplg Ke Wods: Two pse smplg; tem oespose; Detemstc mputto; Rdom mputto toducto mputto, setg vlues fo mssg tems, s commol used fo dlg mssg suve dt A dvtge of mputto s ts coveece Tt s, we c ppl stdd complete dt pogms fo computg pot estmtes to te mputed dt set Rub (996), F (996), d Ro (996) evewed vous ssues o mputto All mputto metods use some tpe of model Afte desgtg model, we c use ete detemstc mputto o dom mputto bsed o te model Ude dom mputto, mssg vlues e mputed b te use of some fom of pobblt smplg We cll ts ddtol dom mecsm te mputto mecsm O te ote d, detemstc mputto does ot toduce ddtol dom mecsm We te set of espodets s vewed s dom smple fom te ogl smple, te selecto mecsm of te espodets s clled te espose mecsm Te espose mecsm s ofte egded s te secod pse of smplg See Sädl d Swesso (987) fo detls Wt sutble mputto model d metod, te bs due to oespose c be getl educed eltve to usg ol te obseved dt Howeve, t s well kow tt vce estmto wc uses te mputed dt s f t wee obseved dt s cosstet Vous metods ve bee poposed fo vce estmto fte mputto Rub d Sceke (986) d Rub (987) dvocte multple mputto Multple mputto cetes multple dt sets d clcultes te complete dt sttstcs fo ec mputed dt set Te vce estmto s clculted b combg two tems, te wtdtset vce tem d te betwee dtset vce tem Multple mputto pples stdd vce estmtos to ec dt set to compute wt dtset vce tems d pples te stdd pot estmtos to compute betwee mputed dtset vce tem Ts metod eques te mputto metod to be pope Tt s, te mputto sould stsf codtos 3 Rub (987, pges 8 9) Tese codtos e ot lws es to ceve (Fo emple, see F 99) Eve te multple mputto metods descbed Scfe (997) e ot sow to be pope te sese of Rub As oted b Ro (996), some commol used mputto metods, cludg ot deck mputto d egesso mputto, e ot pope Ro d So (99) d Ro d Stte (995) poposed dusted ckkfe vce estmto Te suggested pocedue s pplcble to umbe of mputto metods d smple desgs Te ctul clculto usg stdd complete dt softwe s ot es becuse specl computtos e pefomed to dust te mputed vlues fo ec pseudo eplcte Also, Sädl (99) poposed vce estmto metod tt eplctl uses te model cosdeed fo mputto Essetll, Rub s metod geetes sevel pseudo dt sets fo vce estmto d pples te stdd vce estmtos to ec dt set to compute te wtdtset vce tems, wle Ro s metod d Sädl s metod ppl specl vce estmto to te mputed dt set ts ppe, metod to cete sgle pseudo dt set fo vce estmto s poposed secto, te ew metod s toduced two pse smplg set up secto 3, we llustte etesos of te suggested metod to te dom mputto metod secto 4, we eted te suggested metod to comple smplg desgs secto 5, compsos e mde wt te dusted ckkfe vce estmto secto 6, lmted smulto stud s peseted Some cocludg emks e mde secto 7 Outles of some poofs e gve te pped Je Kwg Km, Westt, 650 Resec Boulevd, Rockvlle, Mld, 0850, USA

2 76 Km: Vce Estmto Afte mputto A Vce Estmto Metod We outle vce estmto pocedue pplcble fo two pse smples d fo mputed smples Te pocedue eques septe dt set fo vce estmto ddto to te tbulto dt set To toduce te pocedue d to llustte te cocepts, cosde two pse smple et te secod pse be smple dom smple of sze selected fom te fst pse, wc s smple dom smple of sze selected fom fte populto et te egesso estmto of te me of cctestc be wee (, ) = (, ), = =, µ = + ( ) β, () = = = β = ( ) ( ) ( ) d te secod pse uts e deed fom oe to t s well kow (eg, Coc 977, equto 5) tt te vce of te egesso estmto s, ppomtel, V{ µ } = [ ρ + ( ρ )] σ, () wee ρ s te populto coelto betwee d d σ, s te populto vce of A estmto of te vce s, b clsscl egesso teo, = V { µ } = ( ) ( ) ( ) ( ) = + (3) wee ( ) = + β fo =,,,, d = = Obseve tt s ltetve w of wtg µ () et d Te, / = [ ( ) ( ) ] (4) c,,,, = = c ( ), =,,, = (5) V { µ } = ( ) ( ) (6) wee s te me of te, s well s te me of te, becuse te sum of s zeo Equto (6) s te opetol fom of te suggested estmto Te vce estmto dt set cots te pseudo obsevto To te etet tt te model fo mputto mtces tt of two pse smplg, equto (6) s pplcble to mputed dt set Fo emple, f we ssume tt mssg dt e mssg t dom d use egesso to mpute te mssg vlue wt, te equto (6) s mmedtel pplcble Of couse, egesso mputto o two pse smplg c use vecto 3 Etesos to Rdom mputto A modete eteso of te metod descbed secto ebles us to estmte te vce of smple me usg dom mputto fct, ltetve ppoces e possble As oe ppoc, ssume tt te mputto model s te egesso model = β + e (7) wee te fst elemet of eve s equl to d te e e ucoelted (0, σ e ) dom vbles Assume te model s estmted d tt te mputed vlues e = + e, = +, +,, (8) wee = β wt β = ( = ) = d e s cose t dom fom te set e = { e = ; =,,, } Te estmto of te me of s µ = (9) = wee = f =,,, f te e e cose wt eplcemet wt equl pobblt fom te set e, te te vce µ s, ppomtel, V{ µ } = [ R + ( + m) ( R )] σ, (0) wee m = d R s te squed multple coelto coeffcet betwee d Te cese vce due to usg dom mputto wt e, te t usg e 0, s m( R ) σ Teefoe, estmto of te vce of te mputed smple me s gve b (6) wee te c of (4) s / c = [ ( ) ( + m)( p ) ], () d p s te dmeso of β We ve = V { µ } = ( ) ( ) = + ( + m)( p) ( ) () Sttstcs Cd, Ctlogue o 00

3 Suve Metodolog, Jue wee = = Te estmto of te vce usg c of equto () s estmto of te ucodtol vce, te vege ove ll possble mputed smple Devtos of (0) d () e gve Apped A To cosde ltetve vce estmto ppoc, we ssume tt dom selecto pocedue s used fo mputto but plce o estcto o te pocedue, ote t tt te pobbltes of selecto e vesel popotol to te pobblt tt te vlue espods ddto, we ecod te umbe of tmes e vlue s used s doo te mputto et wt,,, = = c ( ) =,,, (3) / = [ ( ) ( ) ] ( + ) (4) c p d wee d s te umbe of tmes e s used s doo Te / tem [ ( ) ( p ) ] s used to dust fo te effect of estmtg p egesso pmetes Te, te vce estmto (6) c be wtte s = V { µ } = ( ) ( ) = + ( p) ( + d ) ( ) (5) f te mputto metod s smple dom smplg wt eplcemet, te, codtol o te smple d te espodets, m E{( + d ) } = + (6) wee te otto s used ee to deote te epectto wt espect to te mputto mecsm geeted b dom mputto Te eqult (6) estblses te equvlece of () to (5) ude wt eplcemet selecto t s sow Apped B tt V { µ } (5) s lso vld estmto we doos e selected wtout eplcemet Sce te poposed vce estmto metod s te codtol vce gve te elzed mputed smple, t s wde pplcblt 4 Comple Smplg Desgs 4 Detemstc mputto Te suggested metod s pplcble to comple smplg desgs s well s to smple dom smplg Assume tt te full smple estmto of te me of c be wtte s = = w, wee w s te smplg wegt of ut te smple Assume tt w = = f te fst elemets e obseved d te emg elemets e mssg, te te estmto of te me of ude egesso mputto s wee = w + w = = + (7) = β, w w = = β = Hee w s te smplg wegt of ut te secodpse smple d s defed b P ( s te secod pse smple s w = te fst pse smple) w Also, w = = f we ssume tt te secod pse smple s dom smple of sze fom te fst pse smple, te w = w Ude cet codtos we c wte te estmto (7) s = w (8) = Te epesetto (8) olds f ( w ) w s te colum spce of te mt X = (,, ) becuse te we ve * = w ( ) = 0 fom = w ( ) = 0 We ssume sequece of smples d fte popultos suc s tt descbed Fulle (998) Defe = = w d (, ) = w = (, ) We lso dopt te sme ssumptos s Fulle (998) Tt s d E(,, ) = ( µ, µ, µ ), (9) V O (0) {( β β ),,, } = ( ), wee ( µ, µ ) = = (, ) d = Fo =,,,, defe β = ( ) = f ut espods we smpled = 0 otewse, d = (,,, ) Te eteded defto of s dscussed b F (99) d used So d Steel (999) ow, let wee l = = w () w w ( ) () = + Sttstcs Cd, Ctlogue o 00

4 78 Km: Vce Estmto Afte mputto wt = β Te, we ve = l + ( )( β β ) B (9) d (0), we ve = l + Op ( ) d V ( Y ) = V ( Y ) + o( ) ow, l V ( Y ) = V[ E( Y )] + E[ V ( Y ) ] (3) l l l Te fst tem o te gt sde of (3) s 0 becuse E( l Y ) = 0 ude model (7) To estmte te secod tem (3), ote tt codtol o, l s le estmto Hece, te stdd vce estmto metod ppled to te pseudo dt set Y { ; =,,, } wll ubsedl estmte te vce of l = w = Sce te set Y s ot obsevble, we c use te set Y { ; =,,, }, wee = + w w ( ) (4) to get cosstet vce estmto To llustte tt te set Y c be used to ppomte te vce estmto, ssume tt te full smple vce estmto of c be wtte s V = c ( ) = ( ) t wee s te umbe of eplctos, c s te ( ) ( ) eplcto fcto, d = = w M s te t epl ( ) cte of Te tem M s te eplcto multple t ppled to te wegt of ut t te eplcto Fo emple, ude smple dom smplg, te ckkfe multple s ( ) ( ) f M = 0 f = Assume tt te eplcte vce estmto ppled to te set Y to get ( ) = V = c ( ) *( ) ( ) wee = w M = wt beg defed (4) Te, we ve = + ( + )( β β ) (5) ( ) ( ) ( ) ( ) l l wee ( ) ( ) ( ) wm w = w = (, ) (, ) t s sow Apped C tt = ( ) ( l l ) + p ( ) = V V c o (6) Teefoe, te stdd ckkfe vce estmto ppled to te pseudo dt set Y c be used to ppomte te stdd ckkfe vce estmto ppled to te pseudo dt set Y s 4 Rdom mputto Te gumets fo vce estmto wt dom mputto e qute sml to tose fo detemstc mputto descbed te pevous subsecto Fst, defe te mputto dcto fucto f ut s used s doo fo ut d = 0 otewse (7) Te, te estmto of te me of usg dom mputto s wee d = w (8) = = + ( + d ) ( ) (9) = ( ) = d d w w (30) f te ogl smple wegts e te sme, te d s te umbe of tmes tt ut s used s doo We ssume tt E[ ( + d ) F ] = (3) wee F = {(,, ); =,,, } Te epectto (3) s wt espect to te ot dstbuto of te espose mecsm d te mputto mecsm Te, we ve E( F ) = f we ssume equl espose pobblt, te, b (3), te pobblt of selecto of doos sould be popotol to te wegts Ts s te Ro d So (99) setup fo dom mputto ow, let = w [ + ( + d )( )] (3) l = wee = β Te, we lso ve = l + ( d )( β β ) wee d = = w ( + d ) B te ssumpto (3), we ve E( d F ) = 0 Ude mld / codtos, d = Op ( ) d = l + Op ( ) ow, V ( Y ) = V[ E( Y, d)] + E[ V ( Y, d )] wee d = ( d, d,, d ) Codtol o d d, te estmto l s le estmto Hece, te pseudo dt = + ( + d )( ) (33) c be used to estmte te vce of Sttstcs Cd, Ctlogue o 00

5 Suve Metodolog, Jue Compsos wt Adusted Jckkfe Metod Ro d Stte (995) poposed dusted ckkfe vce estmto fo te to mputto poblem Ude te setup descbed secto 4, te to mputed estmto of µ s µ = w [ + ( ) ] = wt = R d R = ( = w ) = w Te Ro d Stte (995) vce estmto s V ( ) = c µ µ = ( ), (34) wee te dusted ckkfe eplcte t te s wee ( ) ( ) ( ) ( ) w M = t eplcto µ = (35) ( ) f R = = R f = 0 (36) ( ) ( ) ( ) wt R = ( = w M ) = w M Te dusted vlues (36) te Ro d Stte (995) metod c lso be egded s pseudo dt fo vce estmto ote tt te clculto of te pseudo dt (36) eques ( ) eclculto of R fo ec wt = We modf te clculto of te pseudo vlues (5) to f = 0 = + c ( ) f =, (37) wee = = w, = = w d c = Te tem ( / ) s seted to mpove te codtol popetes of V J gve te fst pse smple Te esultg vce estmto s ppomtel equvlet to te dusted ckkfe vce estmto (34) To see ts, ote tt te dusted vlues (35) c be wtte te fom ( ) wm ( ) ( ) ( ) = ( ) S wm Z ( ) = ( ) T wm = µ = = :, wee A = : B deotes tt we defe B to be A Also, defe Z = = w, S = = w, d T = = w Te b te fst ode Tlo epso, ( ) ( ) S S ( ) S Z = Z + ( Z Z ) ( ) T T T ( ) Z ( ) ( ) ( ZS + S S T T ) T T ( ) S Z ( ) ( ) S = Z + S T T T T ote tt te gt sde of (38) s ectl equl to = ( ) S Z S w M + T T T (38) Tus, te pseudo dt fo vce estmto c be wtte s S Z S = +, T T T wc educes to (37) Hece, te poposed metod s ectl fst ode Tlo lezto of te Ro d Stte metod te cse of to mputto Teefoe, we c epect ou poposed metod to ve te sme smptotc popetes s te Ro d Stte metod up to te ode of Te vce estmto metod usg te pseudo dt set clculted b (37) s es to mplemet becuse we c dectl use estg softwe, wc s moe dffcult wt te Ro d So (99) o Ro d Stte (995) metod Futemoe, f we clculte te pseudo dt b (3), te te dt set woks fo wtout eplcemet ot deck mputto s well s fo wt eplcemet ot deck mputto 6 A Smulto Stud Te pecedg teo ws tested smulto stud usg tfcl, fte populto, fom wc epeted smples wee dw Te populto s = 3 stt, clustes sttum, d 0 ultmte uts ec cluste Te vlues of te populto pmetes wee cose to coespod to el popultos ecouteed te US tol Assessmet of Eductol Pogess Stud (Hse d Teppg 985) d e lsted Tble Te fte populto uts e wee d = + e, d µ σ = = ~ (, ),,,,,,,,, Sttstcs Cd, Ctlogue o 00

6 80 Km: Vce Estmto Afte mputto d ρ e ~ σ = ρ 0,,,,, 0 So, Ce d Ce (998) lso used te sme populto te smulto stud Te vlue of te t cluste coelto ρ cosdeed te smulto s ρ = 03 Smultos wt ote vlues of ρ poduced sml esults d e ot lsted ee fo bevt Tble Pmetes of te Fte Populto fo Smulto H µ σ µ σ We cosde sttfed cluste smplg desg, wee = clustes e selected wt eplcemet fom sttum wt equl pobblt d ll of te ultmte uts te selected clustes e te smple Te smplg fcto s 64% Fo ec smpled ut, espose dcto vble s geeted fom d ~ Beoull ( p ), d tt s depedet of Te vlue of p cosdeed te smulto e p = 09, 08, 07, 06, d 05 A set of 5,000 smples wee selected usg te sme smplg desg ec of te selected smples, tee mputto metods e cosdeed; [M] Wt eplcemet wegted ot deck mputto cosdeed b Ro d So (99), wee mssg vlue s mputed b vlue doml selected fom te espodets wt eplcemet wt pobblt popotol to te suve wegts [M] Wtout eplcemet wegted ot deck mputto, wc s te sme s [M] epect tt te selecto ws pefomed usg wtout eplcemet smple Te wtout eplcemet selecto of doos s ced out sstemtcll usg te metod descbed b Hse, Huwtz, d Mdow (953, pge 343) fom te espodets soted b dom ode [M3] Ovell me mputto, wee te wegted me of te espodets te smple s mputed Hece, ll te mputto metods use sgle mputto cell tt collpses ll te stt ec mputed dt set we computed tee vce estmtos V, ve vce estmto tetg te mputed dt s f t wee obseved dt, V, te dusted ckkfe vce estmto of Ro d So (99) fo [M] d [M] d of Ro d Stte (995) fo [M3], d V, te ckkfe vce estmto bsed o te pseudo dt Te pseudo dt set s costucted b (9) fo [M] d [M] d b (4) fo [M3] Te complete smple vce estmto used stdd ckkfe fo sttfed cluste smplg, wc cluste s deleted fo ec eplcto ote tt te stdd ckkfe s cosstet estmto of te vce ude te model wt ozeo tcluste coelto Tus, te stdd ckkfe metod bsed o te pseudo dt c be pplcble to te dt set cosdeed Te pot estmtos of te populto me e ubsed ude te tee dffeet mputto scemes d e ot lsted ee Tble pesets te eltve bs of te tee vce estmtos, te stdd eo of te eltve bs of te vce estmtos, d te smple coelto coeffcet betwee te Ro s dusted ckkfe vce estmto d te ew vce estmto bsed o te 5,000 smples Te eltve bs of V s estmto of te vce of s clculted b [V ( )] [ ( B EB V ) V B ( )], wee te subscpt B deotes te dstbuto geeted b te Mote Clo smulto Te coelto coeffcets of te two vce estmtos e computed to gve mesue te eltve let bevo of te two vce estmtos Tble Reltve Bs of te Vce Estmto, Stdd Eo of te Reltve Bs, d Smple Coelto Coeffcet Betwee te Ro s Vce Estmto d te ew Vce Estmto Bsed o 5,000 Smples Respose mputto Rte (p) Rel Bs 00 (SE 00) Co Metod ve Ro ew Coeff M 740 (0) 6 (03) 70 (04) 0967 M 750 (00) 4 (0) 08 (03) 0974 M3 803 (03) 6 (05) 5 (04) 000 M 3445 (0) 065 (03) 049 (05) 0939 M 389 (0) 49 (04) 09 (03) 0947 M (0) 59 (03) 59 (03) 000 M 4896 (0) 0 (99) 04 (04) 09 M 4476 (0) 53 (05) 076 (05) 090 M3 50(0) 53 (05) 5 (04) 000 M 5980 (0) 58 (05) 7 (06) 089 M 5486 (03) 70 (07) 075 (07) 0899 M3 64 (00) 035 (04) 035 (0) 000 M 6975 (99) 084 (03) (03) 0873 M 5990 (0) 507 (07) 7 (06) 087 M (97) 99 (00) 98 (00) 000 Sttstcs Cd, Ctlogue o 00

7 Suve Metodolog, Jue 00 8 Tble suppots ou teo te followg ws As s well kow, te ve vce estmto seousl udeestmtes te tue vce Te dusted ckkfe vce estmto pefoms well fo [M] d [M3], but ot fo [M] Te teo fo te dusted ckkfe metod ssumes tt ot deck mputtos e doe usg te wt eplcemet selecto wc s ot used [M] As te espose te deceses Tble, te eltve bs of te dusted ckkfe becomes lge Te ew metod bsed o te pseudo dt pefoms well eve fo te wtout eplcemet mputto [M] As ws dscussed t te ed of secto 3, sgle fomul (9) c be used s te pseudo dt fo lge clss of mputto metods 3 As s obseved te coelto coeffcets, te bevos of te dusted ckkfe vce estmto d te poposed vce estmto e ve sml fo me mputto [M3] Ts s becuse te two vce estmtos e smptotcll equvlet, s dscussed secto 5 7 Cocludg Remks We ve descbed metods of mkg pseudo dt to be used fo vce estmto Geell spekg, te pseudo dt c be descbed s,,, = = c g ( ) =,,,, (39) wee s te pedcted vlue of ude te model used fo mputto f c g =, te te vce estmto tets te mputed vlues s obsevtos A sutble coce of c g > leds to cosstet vce estmto f te mputto metod s detemstc d te espodets e egded s dom smple fom te ogl smple, te c = > Fo two pse smplg wt comple desg, c, = w w wee w s te smplg wegt of te ut fo te fst pse smple d w s te smplg wegt of te ut fo te secod pse smple Te g (39) s te dustemet mde to mpove te codtol popetes gve te ul vble Fo to mputto, g = ( ) wee = = w d = = w Fo egesso mputto wt scl, = + k k k = g ( ) w ( ) ( ) ete cse, we ve = w g = Wle ts ppe ws ude evew, So d Steel (999) lso povded sml metods te cse of detemstc mputto Ou metod s moe geel te sese tt we lso cosdeed dom mputto d toduced c tem to mpove fte smple popetes Ackowledgemets Te uto tks s tess dvse We A Fulle fo vluble dscussos Te uto lso tks Pmel Abbtt, F J Bedt, ou Rzzo, Rcd Vllt, d te efeees fo elpful commets, wc getl mpoved te ppe Most of ts wok ws doe wle te uto ws gdute studet t ow Stte Uvest d ws fuded pt b coopetve geemet 68 3A75 43 betwee te USDA tul Resouces Cosevto Sevce d ow Stte Uvest d b Coopetve Ageemet 43 3AEU betwee ow Stte Uvest, te tol Agcultul Sttstcs Sevce d te US Bueu of Cesus Apped A Poof of Equto (0) d () Te estmto µ (9) c be wtte s µ = + ( + d ) e = = (A) wee d s te umbe of tmes tt ut s used s doo Ude te equl pobblt d wt eplcemet mputto mecsm, we ve d E ( d ) = m ( ) f = m Cov ( d, d ) = m f wee te subscpt deotes te vto due to te mputto mecsm t follows tt E ( µ ) = = d V ( µ ) = m e Hece, = V ( µ ) = V + E m e = = (A) ow, b sml gumet sml to te oe ledg to (), we ve Sttstcs Cd, Ctlogue o 00

8 8 Km: Vce Estmto Afte mputto V = [ R + ( R )] σ = (A3) Sce = ( ) β + o p (), we ppl clsscl egesso teo to get d E ( p) e = ( R ) σ, = E ( ) ( ) = R σ = (A4) (A5) Teefoe, (0) s poved d te estmto () s cosstet fo te vce (0) B Vldt of (5) Ude te Wtout Replcemet mputto Mecsm We ssume tt m = k + t wee k d t e oegtve teges d t < et te estmto of te me of ve te fom (A) et te mputto be pefomed suc tt t of te espodets e used k + tmes fo mputto d t uts e used k tmes fo mputto Te t of te espodets tt e used k + tmes e cose b smple dom smplg wtout eplcemet Te, d E ( d ) = k + t = m ( ) f = t t Cov ( d, d ) = t f So, b sml gumets s te poof of (A), we ve V ( µ ) = V ( ) + E t e = Hece, usg (A3) d (A4), we ve (B) V{ µ } = [ R + ( + t)( R )] σ (B) ow, codtol o te elzed smple d te espodets, we ve t t E {( + d ) } = + ( { µ }) = ( ) ( ) = E V = + [ + t( t)]( p) ( ) Teefoe, usg (A4) d (A5), we ve te ppomte ubsedess of te V { µ } ude te wtout eplcemet mputto mecsm C Poof of Equto (6) ( ) ( ) ( ) Fst, defe R ( = )( β β ) d R = ( )( β β ) Fom te eqult (5), ( ) = ( ) = + + = V c A B C ( ) wee A c ( ( ) = = l l ), B = = c ( R R ), ( ) ( ) d C = = c ( l l )( R R ) Hece, b te ssumpto (0), (6) follows becuse A = Op ( ), B = op ( ), d C = op ( ) Te lst popet comes fom te Cuc Scwtz eqult, C A B Refeeces Coc, WG (977) Smplg Tecques ew Yok: Jo Wle & Sos, c F, RE (99) A desg bsed pespectve o mssg dt vce Poceedgs of te Bueu of te Cesus Aul Resec cofeece, F, RE (99) We e feeces fom multple mputto vld? Poceedgs of te Secto o Suve Resec Metods, Amec Sttstcl Assocto, 7 3 F, RE (996) Altetve pdgms fo te lss of mputed suve dt Joul of te Amec Sttstcl Assocto, 9, Fulle, WA (998) Replcto vce estmto fo two pse smples Sttstc Sc, 8, Hse, M, Huwtz, W d Mdows, WG (953) Smple Suve Metods d Teo, Vol, ew Yok: Jo Wle & Sos, c Hse, M, d Teppg, BJ (985) Estmto fo Vce AEP Upublsed memodum, Westt, Wsgto, DC Ro, JK (996) O vce estmto wt mputed suve dt Joul of te Amec Sttstcl Assocto, 9, Ro, JK, d So, J (99) Jckkfe vce estmto wt suve dt ude ot deck mputto Bometk, 79, 8 8 so tt V { µ } (5) stsfes Sttstcs Cd, Ctlogue o 00

9 Suve Metodolog, Jue Ro, JK, d Stte, RR (995) Vce estmto ude twopse smplg wt pplcto to mputto fo mssg dt Bometk, 8, Rub, DB (987) Multple mputto fo oespose Suves ew Yok: Jo Wle & Sos, c Rub, DB (996) Multple mputto fte 8+ es Joul of te Amec Sttstcl Assocto, 9, Rub, DB, d Sceke, (986) Multple mputto fo tevl estmto fom smple dom smples wt goble oespose Joul of te Amec Sttstcl Assocto, 8, Sädl, C E (99) Metods fo estmtg te pecso we mputto s bee used Suve Metodolog, 8, 4 5 Sädl, C E, d Swesso, B (987) A geel vew of estmto fo two pses of selecto wt pplctos to twopse smplg d oespose tetol Sttstcl Revew, 55, Scfe, J (997) Alss of complete Multvte Dt Cpm & Hll So, J, Ce, Y d Ce, Y (998) Blced epeted eplcto fo sttfed multstge suve dt ude mputto Joul of te Amec Sttstcl Assocto, 93, So, J, d Steel, P (999) Vce estmto fo suve dt wt composte mputto d oeglgble smplg fcto Joul of te Amec Sttstcl Assocto, 94, Sttstcs Cd, Ctlogue o 00