Linearization Variance Estimators for Survey Data: Some Recent Work

Transcription

1 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa Linaiaion Vaianc Eiao fo Suvy Daa: So Rcn Wo A. Dnai an J.. K. Rao A. Dnai Social Suvy Mho Diviion Saiic Canaa Oawa Canaa J.. K. Rao School of Mahaic an Saiic Calon Univiy Oawa Canaa Abac In uvy aling aylo linaiaion i ofn u o obain vaianc iao of calibaion iao of oal an nonlina fini oulaion aa. I i gnally alicabl o any aling ign bu i can la o ulil vaianc iao ha a ayoically ign unbia un a aling. h choic aong h vaianc iao qui oh coniaion uch a i) aoxia unbian fo h ol vaianc of h iao un an au ol an ii) valiiy un a coniional a aling fawo. Dnai an Rao 4) oo a nw aoach o iving aylo linaiaion vaianc iao ha la icly o a uniqu vaianc iao ha aifi h abov coniaion fo gnal ign. Dnai an Rao ) coni h ca of iing on whn ajun fo col nonon an iuaion fo i nonon a u. Dnai an Rao 3) xn h wo o al wih longiuinal uvy which la o nn obvaion an o ulil wigh on h a ln. hy coni a vaiy of longiuinal aling ign coving anl uvy houhol anl uvy a wll a oaing uvy. Dnai an Rao 5) ui oal vaianc iaion in h conx of fini oulaion au o b gna fo uoulaion ol an analyical infnc on ol aa a of in. If h aling facion i all hn h aling vaianc cau alo h ni vaiaion gna by h ign an ol ano oc. Howv whn h aling facion i no ngligibl h ol vaianc houl b an ino accoun in o o conuc vali infnc on ol aa un boh anoiaion oc. In hi a w giv a bif accoun of h Dnai-Rao ho fo vaianc iaion. W alo n iulaion ul on oal vaianc iaion an o xnion. Kywo: Calibaion ol aa oal vaianc.. Inoucion aylo linaiaion i a oula ho of vaianc iaion fo colx aiic uch a aio an gion iao an logiic gion cofficin iao. I i gnally alicabl o any aling ign ha i unbia vaianc iaion fo lina iao unli a aling ho uch a h jacnif an i i couaionally il han h la ho. Howv i can la o ulil vaianc iao ha a ayoically ign unbia un a aling. h choic aong h vaianc iao hfo qui oh coniaion uch a i) aoxia unbian fo h ol vaianc of h iao un an au ol an ii) valiiy un a coniional a aling fawo. Fo xal in h conx of il ano aling an h aio iao Yˆ R = y / x) X of h oulaion oal Y Royall an Cublan 98) how ha a coonly u linaiaion vaianc iao = n ) o no ac h coniional L MSE of Yˆ givn x unli h jacnif vaianc R iao. H J y an x a h al an X i h nown oulaion oal of an auxiliay vaiabl x i h al vaianc of h iual = y y / x) x an n ) no h al an oulaion i. By linaiing h jacnif vaianc iao w obain a iffn J linaiaion vaianc iao = X / x) L which alo ac h coniional vaianc a wll a h unconiional vaianc wh X = X / i h an of x. A a ul o ay b f ov. Vallian 993) obain fo h oaifi iao an conuc a iulaion uy L o ona ha boh an o goo J coniional oi givn h ia o-aa coun. Sänal Swnon an Wan 989) how ha i boh ayoically ign unbia an ayoically ol unbia in h n of E ) = V Yˆ ) wh E no R ol xcaion an V Yˆ ) i h ol vaianc R of Ŷ un a aio ol : E y ) = β x ; R =... an h y a innn wih ol vaianc V y ) = σ x σ >. hu i a J 96

2 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa goo choic fo ih h ign-ba o h ol-ba civ. Dnai an Rao 4) oo a nw aoach o vaianc iaion ha i hoically juifiabl an a h a i la icly o a -y vaianc iao fo gnal ign. hy ali h ho un h ign ba aoach o a vaiy of obl coving gion calibaion iao of a oal Y an oh iao fin ih xlicily o ilicily a oluion of iaing quaion. hy obain a nw vaianc iao fo a gnal cla of calibaion iao ha inclu gnali aing aio an gnali gion iao. hy alo xn h ho o wo-ha aling an obain a aling vaianc iao ha a full u of h fi ha al aa coa o aiional linaiaion vaianc iao. Dnai an Rao ) xn hi ho o h ca of iing on whn ajun fo col nonon an iuaion fo i nonon ba on ooh funcion of obv valu in aicula aio iuaion a u. Dnai an Rao 3) xn h wo o al wih longiuinal uvy which la o nn obvaion an o ulil wigh on h a ln. hy coni a vaiy of longiuinal aling ign coving anl uvy houhol anl uvy a wll a oaing uvy. Dnai an Rao 5) ui oal vaianc iaion in h conx of fini oulaion au o b gna fo uoulaion ol an analyical infnc on ol aa a of in. If h aling facion a ngligibl h aling vaianc cau alo h ni vaiaion gna by h ign an ol ano oc. Howv whn h aling facion i no ngligibl h ol vaianc houl b an ino accoun in o o conuc vali infnc on ol aa un boh anoiaion oc. In hi a w giv a bif accoun of h Dnai- Rao ) ho fo vaianc iaion. In cion w viw h ho fo oal vaianc iaion. W aly h ho o h aio iao an ovi iulaion ul on h foanc of vaianc iao. In cion 3 w xn h ul o iao of ol aa fin a oluion o wigh iaing quaion. Rul in cion 3 a xn o h ca of ulil wigh ajun in cion 4.. Dnai-Rao Linaiaion Mho W a wih a gnal foulaion of h Dnai an Rao 4) aoach o iving aylo linaiaion vaianc iao. hi foulaion will cov boh fini oulaion o cnu) aa θ an ol aa θ un an au u-oulaion ol. An iao θˆ ba on a obabiliy al awn fo a fini oulaion P of i i u o ia boh θ an θ. Howv vaianc iao aocia wih θ an θ a iffn. In h la ca w ia h oal vaianc V ˆ) θ = E V ˆ) θ + V E ˆ θ ) whil h ign vaianc V θˆ ) i ia in h fo ca wh E an an ol vaianc an V no ol xcaion E an xcaion an ign vaianc civly. L... ) b a g = V no ign g vco of ano u u... u ) b a g wigh an = g vco of conan fo =.... L U ˆ = u b a lina iao an uing an oao noaion l u) no h iao of vaianc of Uˆ wh no uaion ov all ln in P. W wi θˆ a f A ) wh A i g aix wih h colun. h choic of A n on h ano oc involv. Fo xal uo θ = i h oulaion oal an w u a aio y iao a ˆ θ = X y ) / x ) = XRˆ wh = / π a h Hovi-hoon wigh wih a = if ln i in h al a = ohwi an π a h incluion obabilii. In hi ca g = = A =... ) f A ) = X y ) / x ) = XYˆ / Xˆ an u) = u) i h ign-ba vaianc iao of h oal wh U ˆ = u : u) = u u ω ) / ω.) ω π π / π obabilii. = an = a a / π = π a h join incluion Suo on h oh han w a in in h ol aa θ = E y ) un a 97

3 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa uoulaion ol on y. In hi ca g = = f A ) = X ) / x ) of oal vaianc of Uˆ i = y an. Fuh h iao u ) = u cov ) u.) wh cov ) i an iao of oal vaianc of an. In h abov ca of v = y ) w hav cov ) = cov y y ) ω ) + v v. ω = v wh.3) In.3) cov y y ) i an iao of h covaianc of y an Whn h ol covaianc of cov y y ) i an a o. y un h au ol. y an y i o h linaiaion vaianc iao of θ ˆ= f A b ) i ily givn by ˆ) θ = ).4) wh ) i obain fo u) by lacing by wh A i a b = f A ) / b b Ab= A u g h aix of abiay al nub wih colun b = b... b ). In h ca of θ w u ) in g.4). Alicaion o Raio Eiao Fo h aio iao θˆ an h fini oulaion oal θ w hav f A ) = X y b ) / x b ) an b hnc = = X / Xˆ ) y Rˆ x ) = X / Xˆ )..5) h vaianc iaion i hn givn by ). Siilaly fo h ol aa θ = E y ) w hav f A ) = X b ) / x b ) an b ˆ X R x = =..6) ˆ X Subiuing in.6) fo wh ˆ) θ = + + u in.) w g ; ; ; ; = X / Xˆ an = ; cov y y ) ω ) / ω = v = + y = X / Xˆ ) y Rˆ x ). ;.7) o ha h fi coonn coon o h ol whil h con coonn coon o h aling ign. ow coni h cial ca of il ano aling wihou lacn SRS). Fo hi cial ca n X ) = =.8) n Xˆ wh = a / n ). Fuh un h aio ol E y ) = β x Cov y y ) =.9) θ = β X an h ol vaianc of y V y ) = E y β x ) i ia obuly by y ) = of.7). h ol coonn uc o X = n )..) n Xˆ o ha ain vali un icificaion of V y ). ow cobining.8) an.) w g X ˆ) θ =..) n Xˆ I i ining o no ha h g-wigh aa auoaically in θˆ ) an ha h fini oulaion cocion n / i abn in θˆ ). h aoach la o a uniqu choic of vaianc iao ha v h g-faco auoaically. I i cuoay o igno h faco X / Xˆ ) an u 98

4 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa ) cu ˆ) θ =..) n A ix vion which inclu h g-faco in h aling coonn only i.. u givn by.8) i givn by H X ˆ) θ = n) + n )..3) ix n Xˆ V E ˆ) θ V Y ) = E y βx ) i ia by = = / n) n ) which i u in lac of in.7). W conuc a all iulaion uy o xain h foanc of iffn vaianc iao boh unconiionally an coniionally on Xˆ. W fi gna R = fini oulaion { y } ach of i = 393 fo h aio ol y = x + x ε.4) / wih ε a innn obvaion gna fo a ) wh h fix x a h nub of b fo h Hoial oulaion ui in Vallian al ). On il ano al of cifi i n i awn fo ach gna oulaion. Ou aa of in i θ = βx = X an h iula oal MSE of h aio iao ˆ θ = X y / x) i calcula a ˆ) ˆ M θ = R θ θ ) wh = θˆ i h valu of h θˆ fo h iula al an y x) a h al an. W calcula h oal vaianc ia θˆ ) an i coonn an fo ach iula al an hi avag an. Figu giv a lo of h avag of vaianc ia an an h iula MSE fo n = 4 L In h ca of n= =. I i n fo Figu ha ac M θˆ ) vy wll wha h u of la o v uniaion a h al i n inca. o xain h coniional foanc of h vaianc iao un il ano aling givn X ˆ = x w conuc anoh iulaion uy iila o Royall an Cublan 98) fo infnc on θ = E y ) uing ol.4). W gna R = fini oulaion { y } ach of i = 393 fo.4) uing h nub of b a x an fo ach oulaion w hn lc on il ano al of i n =. W aang h al in acning o of x -valu an hn gou h ino gou ach of i uch ha h fi gou G conain al wih h all x -valu h nx gou G conain h nx all x -valu an o on o g G...G. Fo ach of h gou o fo w calcula h avag valu of h aio ia ˆ θ = X y / x) an h xanion ia y an h uling coniional laiv bia CRB) in iaing θ = X ; Figu. I i cla fo Figu ha y i coniionally bia unli θˆ : gaiv CRB -4%) fo G incaing o oiiv CRB +4%) fo G. o ha boh y an θˆ a unconiionally unbia fo θ. h coniional bia of θˆ an y in iaing θ i iila o h coniional bia in iaing θ = Y obv by Royall an Cublan 98). W alo calcula h coniional MSE of θˆ an h aocia CRB of h vaianc ia an ba on h avag valu of an ix cu in ach gou; Figu 3. I i vin fo ix Figu 3 ha CRB of ang fo -8% o % cu aco h gou wha xhibi no uch n an i CRB i l han 5% in abolu valu xc fo G an G. Alo h CRB of i lagly 6 ix ngaiv an blow ha of fo h fi half of h gou an abov fo h con half bu cu ix xhibi no viibl n unli. Figu 4 o cu h coniional covag a CCR) of noal hoy confinc inval ba on an ignoing h coonn ) fo noinal lvl of 95%. A xc h u of la o v uncovag bcau h aling facion /393 i ignifican. On h oh han CCR aocia wih i clo o noinal lvl aco gou whil xhibi a n aco gou wih cu CCR anging fo 9% o 97%. Fuh CCR cu ix 99

5 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa aocia wih i blow ha of fo h fi ix half of h gou an abov fo h con half. 3. Eiaing Equaion Suo h uoulaion ol on h on y i cifi by a gnali lina ol wih an E y ) =µ θ) = h θ) wh i a vco of xlanaoy vaiabl an h.) i a lin funcion. h ol aa of in i θ. Fo xal h choic h a) = a giv lina gion a a ol an h a) = /+ ) la o h logiic gion ol fo binay on y. W fin cnu iaing quaion CEE) a l θ) = l θ) = wih E θ) = l an h oluion o CEE giv h cnu aa θ. W hav l θ) = y µ θ)) fo lina an logiic gion ol. W u a gnal cla of calibaion iao wih wigh w = F x αˆ ) wh h vco aa α i in by olving a of calibaion conain x α ) x = X 3.) F wh X i h nown oal of a q vco of calibaion vaiabl x. Fo xal h choic a F a) = + a giv GREG wigh an F a) = la o aing aio wigh. W u h calibaion wigh o ia h CEE. h calibaion wigh iaing quaion a givn by lˆ θ) = x αˆ) l θ) =. 3.) F h oluion o 3.) giv h calibaion wigh iao θˆ of boh ign unbia fo fo θ i.. E θ an θ. I i aoxialy θ an ign-ol unbia θ ˆ) = θ an E θ ˆ) = θ. W focu E h on oal vaianc iaion aocia wih θ. Dnai an Rao 4) ui h ca of θ un h gnal cla of calibaion wigh whil Dnai an Rao 5) ui h ca of ol aa θ un GREG wigh. I follow fo 3.) ha θˆ i of h fo f A ) wih = l θ)) wh A ) an A i a + ) aix wih f i a vco h colun. Following h ilici iffniaion ho of Dnai an Rao 4) = f A ) / b i Z b Ab= A valua a - ˆ Z = [ Jˆ θˆ)] F x αˆ) B l αˆ) x I ) 3.3) wih ˆ f x α) x x ] - B l α) = [ f x α) x l θˆ) 3.4) J ˆ θ ) = x α ˆ) l θ ) / θ ) 3.5) F I i h iniy aix an f a) = F a) / a. h linaiaion vaianc iao of θˆ i obain fo.) an.3) by lacing u by h +) aix Z v by l θ)) an cov y y ) by an iao of h covaianc aix of l θ) un h au ol. Af ilificaion w g θ ˆ) = + 3.6) wh i h aling ia covaianc aix givn by wih = [ J ˆ θˆ)] - [ ω ) / ω ] F x θˆ) αˆ) F x θ l θ B l α) αˆ) θˆ)[ J ˆ θˆ)] - 3.7) ˆ) = ˆ) ˆ ˆ x. 3.8) h ol ia covaianc aix n on h au ol covaianc ucu. If Cov l θ) l θ)) = fo an V l θ)) = E l θ) l θ)) i ia obuly by l θˆ) l θˆ) hn h ol ia covaianc aix uc o = [ ˆ ˆ)] F ˆ) ˆ) ˆ)[ ˆ θˆ)] - - J θ x α l θ l θ J.3.9) o ha fo h lina gion an logiic gion ol Cov l θ) l θ)) = fo if h y a uncola un h ol noing ha l θ) = y µ θ)). W conuc a all iulaion uy o xain h unconiional ign-ol) foanc of h calibaion wigh iao θˆ of h ol 9

6 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa aa θ un a Poion gion ol an il o-aificaion ajun. In aicula w coa h fficincy of θˆ laiv o θ uing only ign wigh in h wigh iaing quaion 3.): l θ) = l θ) =. W alo xain h unconiional foanc of h vaianc iao ˆ θ) an θ ) in acing h oal vaianc of θˆ an θ civly. o ha ˆ θ) i givn by 3.6) an θ ) i obain fo 3.6) by changing F αˆ ) o an θˆ ) o l θˆ). W alo coni a naïv vaianc iao θˆ) aocia wih θˆ which u w in h n lac of in θ ). W gna R = fini oulaion y } x { ach of i = 393 fo y P λ ) an B ) wih λ = x θ ) + θ an = x δ + δ x ) /{+ x δ + δ x )} wh h fix vaiabl x i h nub of b in hoial fo h hoial oulaion. h choic of δ an = δ =. la o an avag of abou 6 % fo. h gou inicao w gna fo h fi oulaion an hn fix in h iulaion of aining oulaion o ha { } ay b ga a fix xlanaoy vaiabl. Ou aa of in i θ = θ θ ) ). = Fo ach gna oulaion on il ano al of i n = 3 wa awn. o iln o-aificaion ajun w fi gou h oulaion uni ino wo cla wih 7 uni having x < 35 in cla an uni wih x 35 in cla. L u b h cla bhi c inicao fo ln in cla c =. W u GREG wigh ajun wih nown =. x = u u ) an an X = ) wh = 7 W calcula h ia θˆ θ an h vaianc ia ˆ θ) θ ) an ˆ) fo ach θ n gna al an hi an θˆ θ θˆ ) θ ) an θˆ ) an h vaianc of θˆ an θ n no ˆ V θ) an V θ ). W hav h following ul: ) V ˆ θ ) =. 39 an V ˆ θ ) =. 67 coa o V ) =.33 an V θ ) =. 6 ugging ha o-aificaion i no ffciv fo iaing ol aa whn h ol fi h aa wll; in fac i la o ligh inca in vaianc. hi ul i in agn wih h obvaion a by Rao Yung an Hiioglou ). ) ˆ ) =. 3 ˆ θ ) =. 5 coa o V ˆ θ ) =.39 an V ˆ θ ) =. 67 ugging ha h vaianc iao ac h cooning oal vaianc in h ca of calibaion. Siilaly ) =. an θ ) =. 48 coa o V ) =.33 an V θ ) =. 6 howing ha alo ac h oal vaianc wihou calibaion. Finally h naiv vaianc iao of θˆ alo ac h oal vaianc of θˆ : ˆ n ) =. an ˆ n θ ).45 coa o V ˆ θ ) =. 39 an = V ˆ θ ) = Mulil Wigh Ajun In h nc of iing on wighing ajun i ofn u o cona fo col nonon. L no h aial on h inicao vaiabl fo h ln i.. = if h i col nonon an = if h i aial on. A wily-u aoach o aju fo col nonon i o loy a nw of wigh w h wih ln qual o w = F αˆ) 4.) wh h vco aa α whn ico vaiabl x = x x... ) a availabl fo all qx al ln i fin by olving a of ochaic calibaion conain x x x α) = x 4.) wh x α) = [ lb+ ub x x α)] /[+ x x α) ] F x α) = / x α) wih low an u boun lb ub) gnally o ). o ha 4.) can b win a x x α) ) =. Suo an aiional vco of calibaion vaiabl =... ) wih now oal = q... q ) i 9

7 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa availabl in aiion o h vco x. h vco x i au o b la o h on obabiliy of ln whil h vco i au o b la o h vaiabl of in. In hi ca h final wigh a of h fo w = w G βˆ) 4.3) wh h aa β i in by olving a of conan calibaion conain w G β) =. 4.4) Af ajun fo col nonon an u of auxiliay infoaion h iao θˆ of h ol aa θ un h ol E y ) = µ θ) i obain a h oluion o L ˆ l θ) = F x αˆ) G βˆ) l θ) =. 4.5) = ) wh 3 = = an = l ). hn θˆ can b win a θ 3 θ ˆ= f A ) wh A i a + ) aix wih h colun. Following h ilici iffniaion ho of Dnai an Rao 4) = A ) / b i valua a: f b Ab= A wih = ) 4.6) 3 ˆ = [ Jˆ θˆ)] B αˆ) x x ˆ) α F x α G β Bˆ l β Bˆ ˆ) ˆ) ˆ) αˆ x ) ˆ ˆ)] ) = [ J θ + = [ Jˆ θˆ)] F x αˆ) G βˆ ) I 3 wh B l β) = ) w g β) w g β) l θˆ ) ˆ ˆ ˆ α) = [ Q α)] [ x α) / α] x / B ˆ Q α) = x [ x α) / α ] an ˆ = G βˆ)[ l θˆ) B l βˆ) ]. x h linaiaion vaianc iao of θˆ i givn by θ ˆ) = cov ) 4.7) α) wh cov ) = ˆ ξ ˆ ξ / ˆ ξ cov ˆ ˆ ˆ ) / ˆ + ξ ξ ξ ξ + ω ) / ω v v ˆ ξ = Eˆ ) ˆ ξ = Eˆ ) l θˆ) l θˆ)) 4.8) l θˆ) l θˆ) l θˆ) l θˆ) v = l θ)) an E i h on xcaion. Subiuing 4.8) in 4.7) w g θˆ) = wh + + [ ˆ ξ ˆ ξ / ˆ ξ ] ; + [ ˆ ξ ˆ ξ ˆ ξ ) / ˆ ξ ] + [ ω ) / ω ] cov l θˆ) l θˆ)) ; = [ Jˆ θˆ)] F x αˆ) G βˆ ) I ; ; ; ; J θ F x α Bˆ = [ ˆ ˆ)] ˆ) αˆ x ) ; ) = [ Jˆ θˆ)] an ˆ ; 4.9) ˆ F x αˆ) B αˆ) x [ x ˆ)] ) α ; o ha h fi coonn coon o h ol h con coonn coon o h on an h hi coonn coon o h aling ign. W gna R = oulaion ach of i = uing Hann al. 983) ol wh 3 / E y x) =.4+. 5x an Va y x) =.65x wih boh x an y having gaa iibuion. h valu of x w gna fo h fi oulaion only an ainain fix uing h iulaion. W u x α=.4 x which la o an avag on a of abou 64% wh x = x ). h iaing quaion fo ln i / l θ) = c y x θ ) wih c = x 3 x / ) an h vco aa i θ =.4.5) θ ). Fo ach oulaion on il ano al of i n = i awn. o iln h ajun w u in boh ajun h a vco of 9

8 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa auxiliay vaiabl = x. Fo h col nonon ajun w u 4.) wih lb ub) = ) ; an fo h con wigh ajun w u wih G = X ) in cobinaion l u ) + u l) x A β) β) = wh u ) + l) x A β) A= u l) /[ u ) l)] β = β β ) l =. 6 an u =.4. h iao of ol aa i givn by θ ˆ - = [ w c x ] w c y. Only 7 ca convg. h nonconvgnc i ainly u o nonon ajun. Fo h 7 ca h bia in h iaion of θ i ngligibl: θ =. 4 v. θ.448 ˆ = an θ =. 5 v ˆ θ = wh ˆ θ an ˆ θ no h avag valu of ˆ θ an ˆ θ. W hav h following ul on h avag valu V θˆ) : an coa o iula V ˆ θ ) =.5 ˆ ) =. ˆ ) =. ˆ ) =.4 ˆ ) =. 9 V ˆ θ ) =.7 ˆ θ ) =. 9 ˆ θ ) =. 9 ˆ θ ) =. ˆ θ ) =. 3 V ˆ θ ˆ θ ) =.3 ˆ ˆ θ θ ) =. 3 ˆ ˆ θ θ ) =.9 ˆ ˆ θ θ ) =. 8 ˆ ˆ θ θ ) =.. h cn covag of 95% noal hoy confinc inval aocia wih a 94.4 fo θ an 96.3 fo θ. Rfnc Dnai A. an Rao J.. K. ) Linaiaion Vaianc Eiao fo Suvy Daa wih iing on Pocing of h Scion Suvy Rach Mho Aican Saiical Aociaion Dnai A. an Rao J.. K. 3) Linaiaion Vaianc Eiao fo Longiuinal Suvy Daa Fal Coi on Saiical Mhoology Rach Confnc Dnai A. an Rao J.. K. 4) Linaiaion Vaianc Eiao fo Suvy Daa wih icuion) Suvy Mhoology Dnai A. an Rao J.. K. 5) Linaiaion Vaianc Eiao fo Mol Paa fo Colx Suvy Daa Pocing of Saiic Canaa Syoiu. Hann M. H. Maow W. G. an ing B. J. 983) An Evaluaion of Mol-Dnn an Pobabiliy Saling Infnc in Sal Suvy. Jounal of h Aican Saiical Aociaion Rao J..K. Yung W. an Hiioglou M. ) Eiaing Equaion fo h Analyi of Suvy Daa uing Poaificaion Infoaion Sanyhā: h Inian Jounal of Saiic Royall R. M. an Cublan W. G. 98) An Eiical Suy of h Raio Eiao an Eiao of i Vaianc Jounal of h Aican Saiical Aociaion Sänal C.-E. Swnon B. an Wan J. H. 989) h Wigh Riual chniqu fo Eiaing h Vaianc of h Gnal Rgion Eiao of h Fini Poulaion oal Bioia Vallian R. 993) Poaificaion an Coniional Vaianc Eiaion Jounal of h Aican Saiical Aociaion Vallian R. Dofan A. H. an Royall R. M. ) Fini Poulaion Saling an Infnc: A Picion Aoach Wily. 93

9 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa 5 5 Valu Sal i Siula MSE va.. Saling Coonn Figu : Avag of vaianc ia fo lc al i coa o ia MSE of h aio iao. = va.. = Saling coonn % Rlaiv Bia 5% % 5% % -5% 38 -% -5% Gou xanion aio Figu : Coniional laiv bia of h xanion an aio iao 94

10 Pa n a h ICES-III Jun 8-7 Monal Qubc Canaa Rlaiv bia 3% % % % -% -% -3% -4% Gou cuoay ix Figu 3: Coniional laiv bia of vaianc iao an cu ix 98% 96% 94% Covag 9% 9% 88% 86% 84% 8% Gou cuoay ix aling Figu 4: Coniional covag a of noal hoy confinc inval ba on fo noinal lvl of 95% cu ix an 95