Outlier Robust Imputation of Survey Data

Transcription

1 ASA Secon on Surve Reearch Meho Ouler Robu Impuaon of Surve Daa Ramon L. Chamber Deparmen of Socal Sac Unver of Souhampon, Hghfel Souhampon, SO7 BJ, Une Kngom Ruln Ren ORC Macro Inernaonal Inc. 785, Belvlle Drve Calveron, MD 75, USA Abrac: Ouler robu meho of urve emaon, e.g. rmmng, norzaon, are ell non (Chamber an Koc, 993. Hoever, uch meho o no are he mporan praccal problem of creang an ouler free aa e for general an publc ue. In parcular, ha requre n h uaon a aa e from hch he ouler robu urve emae can be recovere b he applcaon of anar meho of urve emaon. In h paper e ecrbe an mpuaon proceure for oulng urve value, calle revere calbraon, ha acheve h am. Th meho can alo be ue o correc gro error n urve aa, a ell a o mpue mng value. The paper conclue h an evaluaon of he meho bae on a realc urve aa e. Ke or: Aular nformaon; Fne populaon; Sample urve; Ouler; Gro error; Mng aa; Impuaon; Robu emaon; Wnorzaon; Calbraon.. Inroucon Oulng aa value are frequenl encounere n ample urve, parcularl urve meaurng economc an fnancal phenomena. Chamber (986 clafe hee value no o group. The fr are repreenave ouler value. Thee are correcl meaure ample value ha are oulng relave o he re of he ample aa an for hch here no reaon o beleve ha mlar value o no e n he non-ample par of he urve populaon. The econ group con of non-repreenave ouler value. Thee are gro error n he ample aa, caue b efcence n urve proceng (e.g. mcong. Such error have nohng o o h he value n he nonample par of he urve populaon. Eher pe of ouler can have a ubanal mpac on he evenual urve emae f gnore. Tpcall, nonrepreenave ouler are eece an correce urng he urve eng proce, hle repreenave ouler are hanle n he urve emaon proce, generall b he ue of ouler robu or rean emaon proceure. Degn-bae approache o ealng h ouler n urve emaon are ecrbe b Kh (965, Searl (966 an Hroglou an Srnah (98. Chamber (98, 986 evelope moelbae ouler robu emaon echnque for ample urve. Recen or n h area ecrbe n Chamber an Koc (993, Lee (99, 995, Hullger (995, Welh an Ronche (998 an Duchene (999. The reearch ecrbe n h paper ha been carre ou hn he Eure Proec (, hch ame a he evelopmen an evaluaon of ne meho for eng an mpuaon, an n parcular he evelopmen of mpuaon meho ha can be ue h ouler n urve aa. Afer carrng ou urve emaon, he acan ofen ha o elver a aa e for general an publc ue. I har o magne ha a noneper uer of h aa e ll emplo he ame ophcae robu echnque ha he acan ha apple o hoe par of he aa e conanng ouler. Conequenl he urve acan mu elver a clean aa e, h ouler value approprael mofe, uch ha he aa e uable for general ue h anar acal ofare. Ieall, h here one can recover he reul obane from he robu emaon meho ung h anar ofare. Th can be acheve b ung an ouler mpuaon proceure ha e call revere calbraon. In h paper e ecrbe h meho an compare h more anar mpuaon meho ha are pcall ue for mpuaon of mng aa. The rucure of h paper a follo: n he ne econ e ecrbe he revere calbraon approach o ouler mpuaon. In econ 3 an 4, e ecrbe ho he clacal mpuaon meho for mng value uch a he regreon mpuaon an he neare neghbor mpuaon can be ue o ouler mpuaon, an ho he can be mprove o aap o h uaon. In econ 5, e preen ome numercal reul o evaluae he mpuaon meho b ung a realc urve aa e. Th aa e ha been creae hn he Eure Proec an bae on he Annual Bune Inqur (hereafer abbrevae a he ABI urve carre ou b he UK Offce for Naonal Sac (hereafer abbrevae a he ONS.. Ouler mpuaon b revere calbraon Impuaon meho have raonall been ue for mng aa. The bac ea n h cae ha, b fllng n he mng value n a aa e, anar meho of nference, hch pcall aume complee aa, are applcable. In h econ e ae h ea an appl o anoher common urve aa problem. Th he preence of ouler n hee aa. A noe n he prevou econ, uch ouler can be repreenave or nonrepreenave. Once he ouler n he urve aa have been enfe an clafe n h a, e 3336

2 ASA Secon on Surve Reearch Meho can rea hem approprael. Non-repreenave ouler are ver mlar n concep o mng aa. B efnon hee value are, for one reaon or anoher, rong. Conequenl, he nee o be change bac o her correc value. Th can be one b re-nerve of he urve reponen ha prove hee value, n he ame a ha one can carr ou follo-up nerve of urve nonreponen. Alernavel hee value can be replace b mpue value erve from he nonouler or nler n he urve aa e, mlar o he a mpue value bae on reponen aa are ue o replace mng aa. Noe ha h approach mae he aumpon ha, cononal on non (an correc value for covarae, he error creaon proce leang o non-repreenave ouler nepenen of he proce unerpnnng generaon of he rue value for hee ouler. Repreenave ouler, on he oher han, are more ffcul o hanle. B efnon, here nohng o be gane b re-nerve of he reponen ha prove hem (beon he nolege ha hee value are n fac correc. Impuaon of hee value bae on relaonhp n he nler aa value alo napproprae, nce hee ouler value clearl o no have he ame relaonhp. Moern ouler rean meho of emaon allo for h fference, bu conrol he mpac of he correponng ouler conrbuon o he overall urve emae. Wha requre n h cae a meho of ouler mpuaon ha mmc h behavour.. Revere calbraon mpuaon A bac aumpon ha all repreenave ample ouler are enfable. To mnme noaon, e nall aume ha applcaon of urve eng an follo-up proceure mple ha here are no mng value or nonrepreenave ouler n he ample aa. Tha, all ouler n hee aa are repreenave. Le enoe he ample of n un an le {, } enoe a arge e of emaon egh ha e h o appl o all he ample value, ouler a ell a nler, n orer o emae he populaon oal of nere. Ofen hee egh ll be he nvere of ncluon probable or regreon (e.g. GREG or BLUP egh. Ther man characerc ha he are non for each ample un an are fe. Our problem hen one of mpung ample aa value uch ha hen hee mpue value are mulple b he {, } an umme over he ample, he hen lea o an accepable emae of he populaon oal. B accepable e mean here ha h emae equal one ha e oban hen e appl an approprae ouler rean echnque o he ample aa. For eample, uppoe ha : uch an emae, here he {, } are ouler rean egh. Then h conon afe hen : here he {, } enoe he mpue ample value. Le be he ub-ample of ze n conng of he repreenave ample ouler an le be he ub-ample of ze n n n ha con of he ample nler. A naural rercon, n hch cae he problem can be re-epree a one of efnng a e of mpue value, } ha afe : { ( A naural a of choong he, o ha he reman a cloe a poble o he rue value, ubec o he conran (. In urn, h requre ha e pecf a ance meaure (, beeen he mpue value an he rue value ha mu be hen mnmze ubec o h conran. I ea o ee ha h equvalen o a calbraon problem here he urve varable pla he role of ample egh an he ample egh varable pla he role of he urve varable. I ell non (Devlle an Särnal, 99 ha : F ( λ ( here F ( a calbraon funcon ha afe F (, F ( > an λ a conan eermne b F ( λ. Suppoe ha >. A mple ance meaure : (, ( / q (3 here q >, are conan ha can be choen b he acan. Ung h ance meaure, e have F ( + q (Devlle an Särnal, 99. From ( follo : + q (4 q The econ erm on he rgh-han-e of (4 negave f he ouler are manl bg ouler,.e. ae value much larger han he value aocae h he nler n he ample. Conequenl he oberve rue value aocae h a repreenave ouler ecreae. In conra h erm pove f he ouler are manl mall ouler,.e. ae value much maller han he 3337

3 ASA Secon on Surve Reearch Meho value aocae h he nler n he ample. In h cae he rue value aocae h a repreenave ample ouler ncreae. Th conen h he general ea of ouler mofcaon or norzaon. A poenal avanage of revere calbraon mpuaon ha a calbraon program CALMAR (Sauor, 993 avalable, conanng everal fferen ance funcon (,. Sanar choce of q are q or q. In he laer cae (4 mplfe o a rao-pe mpuaon :. (5 Noe ha neher (4 nor (5 guaranee ha he mpue value af eng rule. Epecall for (4 hch ma prouce negave mpue value. To preven negave value, e can ue one of he alernave ance meaure propoe n Devlle an Särnal (99 or ue he ance meaure (3 h q, hch lea o rao-pe mpuaon (5. Alernavel, e can negrae he eng rule no he calbraon proceure.. The general cae The revere calbraon meho ecrbe above rea all ouler mlarl. In parcular he are all eher ecreae or ncreae n value. Th enble f hee value are all of one pe,.e. all bg or all mall. Hoever, n pracce ouler relave o a regreon moel for en o be a m of hee o pe, an hee o fferen pe of ouler nee o be reae fferenl n mpuaon (he mall ouler nee o be ncreae an he bg ouler nee o be ecreae. Furhermore, here are pcall alo mng value for n he ample aa, an hee nee o be mpue a he ame me a hee ouler are mpue. Suppoe ha a ample ubec o boh ouler an mng value. Le be he ubample of nler an reponen, an le be he ub-ample conng of ouler an mng value. Suppoe alo ha a relable populaon oal emae obane b ome ouler rean proceure ha ae non-repone no accoun. Le be an emae of he populaon oal of he nler an reponen. Then an emae of he populaon oal of he ouler an non-reponen can be obane a. Wha e mean b a populaon here open o nerpreaon. In fac, e have four populaon (or, o be more prece, oman. Thee are he reponen nler populaon, he nonreponen populaon, he reponen mall ouler populaon an he reponen bg ouler populaon. We aume ha our overall arge populaon emae can be broen on no four componen ha effecvel repreen our be emae for he oal of each of hee oman. Smlarl e aume ha he ample un can be ve among hee four oman. The revere calbraon proce hen raghforar. We au he oberve ample value n each of he o ouler oman o ha hen mulple b her arge egh he recover her correponng componen of he overall emae. Fnall, e mpue ample value for he mng cae n orer o recover he la componen of he emae. To be more prece, le enoe he reponng ample un correponng o large ( ouler, he reponng ample un correponng o mall ouler, an m ( he nonreponng ample un. The correponng ecompoon of he emae populaon oal ( ( + ( m + + +, h. The revere calbrae mpue value are hen gven b : + ( ( q ( q + q q + ( m ( m q ( m q,,, ( ( m ( here he value repreen nal (uncalbrae mpue value for he mng aa cae. An obvou choce for he fe value for h cae generae b he oberve ample nler, hch correpon o aumng ha all nonreponen are nler. Oberve ha hee mpue value lea o rao pe mpuaon hen q : ( ( ( m ( m ( ( m ( For meho of ecompong he overall robu oal emaon no oman componen, he reaer referre o Ren an Chamber (. 3338

4 ASA Secon on Surve Reearch Meho 3. Impuaon b regreon I clear ha he meho for mng aa mpuaon can be ue o mpue ouler value, b reang he ouler value a mng (e.g. he regreon mpuaon. Suppoe ha a covarae an he urve varable are lne b a lnear moel : β + ε, U here { ε } are he regreon reual, E ( ε, ( Var σ v ( ε ; v ( > a non funcon ; β he unnon regreon coeffcen. Le be a ample conanng ouler value n a ub-ample, +. B reang he ouler value a mng, an emaon of β bae on he non-ouler obervaon gven b : v ( β (3 v ( B reang he ouler value a mng, he clacal regreon mpuaon of he ouler value are her moel bae precon : β, (4 Th mpuaon rea he ouler value a mng an herefore ae no accoun of he fac ha he ouler value are rue an correcl oberve value. In fac, he regreon mpuaon for an ouler value cononal epecaon uner he aumpon ha an nler value an follo he ame moel a for all he nler : (, β E, Th logcall rong nce e no ha he ouler value o no follo he ame moel a for he nler. If e loo a he populaon oal emaon bae on he complee aa e afer mpuaon hch naurall he regreon emaon : lr U + β + β ( ( (5 here he non populaon oal of he covarae, {, } are he amplng egh. Th equvalen o u hro ou he ouler n he populaon oal emaon. When he ouler value are manl eremel large value, h emaon ll uuall uner emae he populaon oal. Hoever, h meho of mpuaon can be mprove b ang he ouler value no accoun n he mpuaon. A mple aapaon o a a correcon erm n he regreon mpuaon (4 : β + δ, (6 here δ a correcon erm hch can be fe or ranom. A fe correcon con o a o he clacal mpuaon of (4 a fe pove quan z α / σ v( hen relae o an ouler value hch locae largel above he regreon lne, an a fe negave quan z α / σ v( hen relae o an ouler value hch locae largel uner he regreon lne, a hon n fgure. The quan δ can be epree b : ( β z α / σ v( δ gn, (7 here z α / he upper α / crcal value of a N (, varable ; σ an emaor of he reual varance bae on he nler : σ ( e e e β h e an e,. v( Th mofcaon ha an nuve ene a hon n fgure : nea of pullng on or puhng up an ouler value ono he regreon lne, e pull on or puh up ll he borer of he confence regon of he regreon. Th mean ha e mof an ouler value a mnmum a poble ll become an accepable nler value. Fgure. Ouler mpuaon b regreon Yello o repreen clacal regreon mpuaon; Green o repreen mofe regreon mpuaon. A ranom correcon con o a o he clacal mpuaon of (4 a ranom pove quan z σ v( hen relae o an ouler value hch locae largel above he regreon lne, an a ranom negave quan z σ v( hen relae o an ouler value hch 3339

5 ASA Secon on Surve Reearch Meho locae largel uner he regreon lne. Tha, b a quan : ( β z σ v( δ gn, (8 here { z, } a ample of obervaon from N (,. I ea o ee ha he epece value of z E ( z / π. B h fac, he ranom correcon mofe on average le he clacal regreon mpuaon han he fe correcon nce, for eample, for α, 5, z α /.96 > /π. If e loo a he populaon oal emaor bae on he complee aa e afer mpuaon : lr + β + β ( ( + δ (9 Compare h epreon (5, he era erm δ n he above epreon he compenae conrbuon of he ouler value o he emaon of he populaon oal. When he ouler value are manl eremel large value, h erm pove ; n conrar, h erm negave. I can be vee a a ba correcon erm for correcng ba caue b clacal regreon mpuaon of ouler value. See able an 3 for numercal reul here e oberve a pove correcon erm. 4. Impuaon b he neare neghbor For a gven ouler value,, aume nong all he oberve value of an aular varable, a for he clacal regreon mpuaon, he clacal neare neghbor mpuaon of he ouler value rea he ouler value a mng b earchng neare neghbor : Arg Mn { (, } hen gvng an mpue value, here a ance meaure. For eample, a uuall ue ance meaure (,. A n he clacal regreon mpuaon, he ouler value elf no aen no accoun n he mpuaon, or n he earchng of neare neghbor. Hoever, he neare neghbor mpuaon ma be preferable o he clacal regreon mpuaon nce loo le he mofe regreon mpuaon hen he ample ze large an ha he oberve value are ene n he ene ha, hen e have : β + ε β + ε ( here ε he regreon reual an can be een a a correcon erm correponng o δ n he mofe regreon mpuaon (6, bu ma no ala rec he correcon o he goo recon. The numercal reul hon n able prove ha he populaon oal emaon afer neare neghbor mpuaon ver cloe o ha obane afer mofe regreon mpuaon To preven he unceran correcon, he ouler value elf houl be aen no accoun n he earchng of neare neghbor. A mple aapaon of he clacal neare neghbor con o ue a ance meaure hch meaure he, an ance beeen he o aa pon ( (, [(, ( ], ha, o ue a ance meaure,,. The neare neghbor of f mnmze he ance { [ (,, (, ]} Arg Mn ( The mpue value for hen. A general ance meaure : [(,, ( ], ( + ( ( ( Hoever, epreon ( rea he ouler value an he covarae value equall n he earchng of he neare neghbor. The ouler value coul omnae he earchng recon hen an eremel large value. Th nconvenence coul be mprove b mofng he ance meaure a a eghe meaure, e call eghe neare neghbor : [(,, (, ] ( + ( α ( ( α (3 here α a eghng facor choen b acan hch reflec he level of he mporance ha he acan pu on he ouler value. I can be a unque egh for all of he ouler value or a pecfc value for each of he ouler value. For eample, n he laer cae, e can ue α /( +. One can recover he clacal neare neghbor hen α, an he mofe neare neghbor ( hen α. 5. The neare neghbor mpuaon of ala f he neare neghbor of. Fgure belo llurae ho he mofe neare neghbor earche. 334

6 ASA Secon on Surve Reearch Meho prncple ha chooe he one hch ll pa he eng. When all of he compeor pa he eng, he choce of he neare neghbor ha no mporance. One can chooe he neare neghbor ranoml among all of he compeor. 5. Numercal evaluaon Fgure. Ouler mpuaon b neare neghbor Yello o repreen clacal neare neghbor mpuaon; Blue o repreen clacal neare neghbor; Green o repreen he mofe neare neghbor mpuaon. From able 3 e ee ha he mpue value b mofe neare neghbor have beer correlaon h he rue ouler value han he regreon mpuaon here poor correlaon ere oberve. The correlaon beeen he mpue value an he orgnal ouler value one of he evaluaon creron for ouler mpuaon (Chamber,. We are loong for mpue value hch are accepable an reflec he ruh n mamum. In h ene, he neare neghbor mpuaon preferable o he regreon mpuaon. Hoever, ha he ame rabac a for he regreon mpuaon nce ma prouce nval mpuaon, ha, he mpue value o no pa he eng proceure. Alo, he eghng facor α pla an mporan role n he earchng of he neare neghbor, epecall hen he ouler are eremel large value. So far, no erou heor can rec he choce of h facor. In he pracce, he mpuaon proceure a recurve proceure hch ue epermenal value of α an combne he eng rule h mpuaon. Impuaon proceure op f all mpue value pa he eng. The earchng for he neare neghbor of an ouler value ma no necearl be rerce o nler value nce an ouler value oulng aocae h, bu ma no be oulng aocae h, an va vera. Bu appear ha he mpuaon proce converge (all mpue value pa he eng more qucl hen he earchng of neare neghbor lme n he nler. The mo unfavorable cae for an unlme earchng ha o ouler are neare neghbor one for anoher, an are ll oulng h echange value. In h cae, he mpuaon proceure oe no converge. Remar : I clear ha he neare neghbor of an ouler value ma no be unque. When h occur, e mu chooe one among all he compeor. The We evaluae he mpuaon meho ue n he prevou econ ung he 997 ecor one ABI aa, a prepare for he Eure Proec (. In parcular e focu on one aular varable ( urnreg correponng o he reger value of emae urnover for a bune an o anal varable (. Thee are oal urnover (urnover, an oal purchae (puro. Snce urnreg a reger varable e no overall oal a ell a raum oal. The raa hemelve correpon o ze raa efne n erm of he reger meaure of he number of emploee an he urnreg value for he bune. Sample egh (egh are alo avalable. The aae ha 699 cae an ha man repreenave ouler. Table gve he number of ouler an he non robu an robu emae of he populaon oal. Ouler ere eece ung an acro-raum forar earch proceure (Henge an Chamber, bae on a lnear regreon moel n he log cale of he aa. The robu emae of he populaon oal ere obane b ung Chamber (986 moel bae robu emaor, h he regreon coeffcen emae b ung onl he nler. The non-robu emae are he moel bae clacal regreon emae h ouler beng gnore, hch can be epree a a eghe um : ŷ reg here {, } are he regreon egh hch are ue a he arge egh for populaon oal emaon afer mpuaon : + ( v ( / v ( The calbraon egh q ue n he revere calbraon mpuaon n h econ choen a q, hch lea o a rao pe mpuaon a gven n (. Table. Number of ouler, non-robu an robu emae of he populaon oal Number of ouler Non-robu regreon emae Robu regreon emae Turnover 6 69,545,47 5,938,74 Puro 9,575,8 8,73,48 334

7 ASA Secon on Surve Reearch Meho Table gve he populaon oal emae afer mpuaon b meho of mpuaon. Non robu an robu emae before mpuaon are alo gven n he ame able for comparon. We can ee ha he revere calbraon mpuaon (bg an mall ouler ere mpue eparael recovere eacl he robu emae. Whle he mofe regreon an mofe neare neghbor mpuaon prouce lghl hgher emae compare o he robu emae. The clacal neare neghbor mpuaon prouce emae ver cloe o he mofe regreon mpuaon, a pone ou n econ 4. Table. Populaon oal emaon before an afer mpuaon b mpuaon meho Non-robu emaon before mpuaon Robu emaon before mpuaon Revere calbraon Clacal emaon afer mpuaon Turnover Puro 69,545,47 9,575,8 5,938,77 8,73,48 5,938,77 8,73,48 Regreon 5,5,6 8,483,77 Neare neghbor Mofe regreon Mofe neare neghbor 53,479,4 8,35,764 53,5,96 8,98,3 54,73,45 8,553,68 The fference beeen he non robu emae an he robu emae, beeen he clacal emae before an afer mpuaon, are no ubanal nce he ouler mpac on he populaon oal emaon n h aa e no ramac. In ome cae, he mpac coul be faal nce a fe ouler aocae h her egh repreen a large percen n he eghe oal. On he oher han, he numercal reul preene n h econ ue a relavel clean veron of he ABI aa e ha free of gro error an mng value. A more eale evaluaon of he revere calbraon mpuaon can be foun n Ren an Chamber ( here a ranng aa veron of he ABI aa a ue hch conan ouler, gro error an mng value. The faal mpac on he populaon oal emaon can be clearl een n ha paper f he are no reae properl. Though he fference beeen he clacal emae before an afer mpuaon are no ver mporan a hon n able, hoever, he fference beeen he average value on he ouler aa pon before an afer mpuaon are gnfcan, a hon n able 3. The fference can alo be een n fgure 4, here he horzonal a repreen he ouler value before mpuaon, he vercal a repreen her correponng mpue value. The plo are n log cale of he aa for ea eeng. In column ( of able 3, e preen he reul for revere calbraon mpuaon here bg an mall ouler are mpue eparael. The number n he brace repreen he coeffcen of correlaon beeen he rue ouler value an her mpuaon. I hon ha he revere calbraon mpuaon acheve perfec correlaon nce mpl a rao pe mpuaon. Regreon mpuaon an neare neghbor mpuaon are preene n column ( an (3. The fr ro n each cell repreen he clacal regreon mpuaon an he clacal neare neghbor mpuaon, repecvel. The econ ro repreen he mofe regreon an he mofe neare neghbor mpuaon. The mofe neare neghbor mpuaon he eghe neare neghbor; a eghe ance meaure ue h a unque eghng facor for all of he ouler. The numercal reul of our mulaon u ho ha h facor a ver enve facor o he meho. Table 3. Average value on he ouler aa pon before an afer mpuaon an he correlaon beeen hem True value Turnover (. Puro (. ( ( (3 4 (.7 73 (.4 73 (.3 9 (.5 7 ( ( (.9 87 (.65 ( Revere calbraon mpuaon ; ( Regreon mpuaon ; (3 Neare neghbor mpuaon. In fgure 3 e preen ome caer plo of urnover agan urnreg n log cale of he aa, before an afer mpuaon. In fgure 4, e preen ome caer plo of he mpue value agan he rue ouler value n log cale of he aa for he varable urnover. All mpuaon are one roun mpuaon, ha, no eng proceure apple o he mpue value. From able 3 an fgure 4 can be een ha he revere calbraon mpuaon acheve he be lnear relaonhp beeen he mpue value an he rue ouler value, ne he eghe neare neghbor mpuaon. A concluon, he revere calbraon mpuaon can be a compeve alernave o he convenonal mpuaon meho, epecall for mpuaon of ouler value. I ha he avanage of recoverng he ouler robu emae b applng he clacal emaor o he mpue an complee aa e. I prouce mpue value beer correlae o he rue ouler value. 334

8 ASA Secon on Surve Reearch Meho Fgure 3. Scaer plo of urnover ( agan urnreg ( n log cale, before mpuaon an afer mpuaon. (re colore pon are ouler aa pn Before mpuaon Revere calbraon Regreon Neare neghbor Mofe regreon Mofe neare neghbor Wh bg an mall ouler mpue eparael. Fgure 4. Scaer plo of mpue value ( agan oberve ouler value ( for urnover, n log cale Revere calbraon Neare neghbor Mofe regreon Regreon Revere calbraon Mofe neare neghbor Wh bg an mall ouler mpue eparael. Reference Chamber, R. L. (98. Robu Fne Populaon Emaon. PhD. The. The John Hopn Unver, Balmore. Chamber, R. L. (986. Ouler robu fne populaon emaon. Journal of he Amercan Sacal Aocaon, 8, Chamber, R. L. an Koc, P. N. (993. Ouler robu ample urve nference. Inve Paper, Proceeng of he 49h Seon of he Inernaonal Sacal Inue, Frenze. Chamber, R. L. (. Evaluaon crera for acal eng an mpuaon. Eure Proec Repor. Devlle, J. C. an Särnal, C. E. (99. Calbraon emaor n urve amplng. Journal of he Amercan Sacal Aocaon, 87, Duchene, P. (999. Robu calbraon emaor. Surve Mehoolog, 5, Eure Proec (. Eure Proec ocumen. ONS. Henge, A. an Chamber, R., L.. (. Robu mulvarae ouler eecon va he forar earch. Eure Proec Repor. Hroglou, M. H. an Srnah, K. P. (98. Some emaor of he populaon oal from mple ranom ample conanng large un. Journal of he Amercan Sacal Aocaon, 76, Hullger, B. (995. Ouler robu Horvz- Thompon emaon. Surve Mehoolog,, Kh, L. (965. Surve Samplng. John Wle & Son, Ne Yor. Lee, H. (99. Moel-bae emaor ha are robu o ouler. Proceeng of he 99 Annual Reearch Conference. U.S. Bureau of he Cenu. Lee, H. (995. Ouler n bune urve. In Bune Surve Meho, (E. B.G. Bo, D. A. Bner, B. N. Chnnappa, A. Chranon, M. J. College an P. S. Ko, John Wle & Son, Ne Yor. Ren, R. an Chamber, R. L. (. Ouler Robu Impuaon b Revere Calbraon. Eure Proec Repor. Roall, R. M. (97. On fne populaon amplng uner ceran lnear regreon moel. Bomera, 57,

9 ASA Secon on Surve Reearch Meho Sauor, O. (993. La macro CALMAR: Rereemen un échanllon par calage ur marge. Techncal Repor F93: INSEE. Searl, D. T. (966. An emaor hch reuce large rue obervaon. Journal of he Amercan Sacal Aocaon, 6, - 4. Welh, A. H. an Ronche, E. (998. Bacalbrae emaon from ample urve conanng ouler. Journal of he Roal Sacal Soce, B, 6,