Optimal Estimator for a Sample Set with Response Error. Ed Stanek

Transcription

1 Optial Estiator for a Saple Set wit Respose Error Ed Staek Itroductio We develop a optial estiator siilar to te FP estiator wit respose error tat was cosidered i c08ed63doc Te first 6 pages of tis docuet are idetical to c08ed63doc owever, o page 6, we defie a differet estiator tat as ore coefficiets ta te estiator tat was cosidered i c08ed63doc First, we review te otatio ad defiitio of te estiator of te populatio total, ad subsequetly, we develop te variace of te estiator Usig tese expressios, we develop te optial estiator of te fiite populatio total Basic otatio We defie te saple sequece as y ( y y y, were yi is respose for te subject i positio i i sequece, ad i,, We ca express te sequeces i ters of a atrix of costats u, ad y suc tat y u y Values of u we ad are give by u 0, u 0, u 3 0 0, u 4 0 0, u 5 0 ad u 6 0 Te eleets of u ave a value of zero or oe, were uis correspods to te 0 u value i positio i for subject s i sequece u ad u u u Also, we defie a vector u u 3 3 of coefficiets for sequece as β β β β β wit coefficiets βi for te subject i positio i i sequece Godabe s Estiator tat Depeds o Saple Sequeces, were for exaple ( We tere is o respose error, usig tese expressios, we defie a liear estiate for sequece as e yu β Let S ( I I I represet a vector of idicator rado variables, were te realized value of I is oe we saple sequece is selected, ad zero oterwise Uder siple rado saplig witout replaceet, all saple sequeces are equally likely ad ece Ep ( I for all,, Also, defie C08ed64doc /0/008 :38 P

2 ( β vec β β β Wit tese defiitios, as sow o page 3 of c08ed5doc, Godabe s liear estiator for saple sequeces witout respose error is give by E y ( I I u β We express tis equivaletly i ters of v u,, js as a t value of oe if te j sallest label i te saple set is for subject s, ad zero oterwise, ad v v v v v v t v wit eleets v ij tat ave a value of oe if te j sallest v v v label i te saple set is i positio i, ad zero oterwise As a result, ( E y ( I II vβ Tis expressio for te estiator iplies a orderig of sequeces i ters of saple sets, ad perutatios of saple sets for te coefficiets i β Let us defie (, were ( β β β β vec β β β β, ad β i is te coefficiet for te subject i positio i i perutatio of saple set otice tat te coefficiets for,, ad,, are a re-labelig of te coefficiets βi for,, were ( + Tere is a oe-to-oe correspodece betwee β ( β β β ad β ( β β β As a result, β β vec( β β β We express ad tus ( E y ( I II v β β y v, ad ece ( ( vec II β y β I I ys, β ( I I ys ( ( ( ( E β I I ys vec II v β i C08ed64doc /0/008 :38 P

3 Alterative Rado Variables We express te rado variables i tis estiator i a differet aer Sice ( ( I I ys vec I I vec II ys v v, by defiig ys ( v were te vectors are defied by i yvijj yvijj yvijj for i,,, let us defie j j j vec ys v vec( ad ( ( ( ( ( ( ( ( vec vec I I I I I I I I I I I I Usig tis defiitio, were vec II ys vec ( v K ( K I Eleets i K are defied suc tat is a vector wit all eleets equal to zero except t for eleet tat is equal to oe, is a atrix wit all eleets equal to zero t except for diagoal eleet tat is equal to oe, is a vector wit all eleets t equal to zero except for te eleet tat is equal to oe, ad i is a vector will all C08ed64doc /0/008 :38 P 3

4 eleets equal to zero except for i t eleet tat is equal to oe Godabe s estiator based o saple sequeces is give by E β J I vec( + ( vec( β P I D were D Coefficiets for Saple Sets We ca group te perutatio atrix for a saple set, v, wit te coefficiets for te positio i te sequece of a saple set, β, to defie coefficiets usig a differet otatio as We defie ( β v β β β β β wit eleets β j, were β j is te coefficiet for t te subject wit te j sallest label i te saple set for perutatio ad defie β vec β β β β vec β β β otice tat te coefficiets i ( ad ( β are orgaized so tat a coefficiet is idetified wit a ordered label i te saple set, j, ad te perutatio,, of te subjects i te saple set, ad ot te positio i te perutatio, i Tus, for a saple set, β j idetifies te saple subject explicitly, but idetifies te positio of te subject i te perutatio iplicitly troug te defiitio of eleets i v I cotrast, β i idetifies te positio, i, of te coefficiet i perutatio of subjects i te saple set, but idetifies te subject i te positio iplicitly troug te defiitio of te eleets i v Usig tis defiitio of coefficiets, or wile Godabe s liear estiator for saple sequeces as β I v β ad β I v β, C08ed64doc /0/008 :38 P 4

5 E β v vec( ( vec( J I + β v P I D We costrai te coefficiets i Godabe s liear estiator suc tat tey do ot differ by te positio of te saple subjects i a set Tis iplies tat te coefficiets i eac row of β β β ( β β β β β β β β β are costat We represet te coefficiet for row j by β β, ad deote j ( β β β β were β ( β β β Also, let us represet deviatio of te coefficiet fro te average over all positios by te residual, r β β β β r β r β r β r ad also defie j j j, defie ( ( β r r r β β ( β β β P Usig tese defiitios, ad otig tat ( β vec β β β, ( ( r r r β vec β + β β β I β + β r were β r ( r r r vec β β β We defie β vec( β β β ( β vec β β β r r r r ( β vec β β β, r, otig tat ( β P I β Sice j ad ( r β I β+ β Substitutig tis expressio for te coefficiets i Godabe s liear estiator based o saple sequeces, r E β ( I + β vec( v J I r + β ( I + β ( vec( v P I D We sow i c08ed6doc tat te first ter i tis expressio is equal to te expaded FP estiator give by P C I ys vec( I U Fro c08ed60doc (page 33 te C08ed64doc /0/008 :38 P 5

6 coefficiets for te expaded FP are defied by C I β Sice β I v β, C I I v β wic i tur equals C I ( I v + I β I I v β Te FP estiator is give by E P β PY I were Y I UI y, wic we ca express as EP C I ( vec( I K were we defie βp I C Tis is give equivaletly as v ( r E ( P vec v β J J + β I vec v I c08ed63doc, we used tis developet to otivate ad estiator give by EP β J J vec( otice tat I P J ad I P J As a result, tis estiator ca be expressed as EP β J J vec( β I P I P vec (( ( ( We defie a differet estiator tat allows te coefficiets to vary for differet saple sets Tis estiator is give by E β I I vec ( ( r Addig Respose Error We add respose error W tat as expected value Eξ ( W sk 0 ad Eξ ( WskWs k σ s sk if s s ad k k, ad zero oterwise We represet respose by Ysk ys + Wsk We assue te edogeous variaces are kow Te target paraeter is specified as te populatio total T ys μ s C08ed64doc /0/008 :38 P 6

7 We alter te rado variables to accout for te additio of edogeous respose error To do so, we defie W vec Wsk v vec( W W W Usig tis defiitio, we defie vec W vec II W II W II W I I W I I W I I W ( ( ( ( ( ( ( ( Usig tese defiitios, ( vec II ( ys + Wsk v Kvec( + vec( W Te estiator tat we cosider is give by E β ( I I vec( + vec( W Te first step i deteriig te optial coefficiets i tis estiator is developet of te expected value ad variace of vec( + vec( W Tis was doe i c08ed66doc We use te results of tis developet ere Suary of te Expected Value ad Variace of vec( + vec( W We suarize te results to te expected value ad variace of te rado variables give by vec( + vec( W as developed i c08ed66doc Expectatio is take over saple ( sets, E p, perutatios of subjects i saple sets, E, ad edogeous respose error, E ξ Te expected value is give by ( EE p E ξ vec( + vec( W vec( Te variace is give by C08ed64doc /0/008 :38 P 7

8 var p vec( vec( ( ξ + W + ωω Te Expected Value of te Estiator E We cosider a estiator wit coefficiets restricted to deped o saple sets tat is give by E β ( I I vec( + vec( W We deterie te expected value of tis estiator ext Te expected value is give by ( ( EE p E( E EE p E ξ ξ β ( I I vec( + vec( W β I I vec ow ( ( C08ed64doc /0/008 :38 P 8

9 ( I I vec( I I I We express ( or ( I I vec( I I I I I C08ed64doc /0/008 :38 P 9

10 Recall tat vec ys v vec( vec ( I, we use te expasio for ( ( vec( ( I like te rigt ad side of tis expasio, ( ( ABC C A B Expressig I I vec y v s vec y v s vec( y v ( I y vec( v Usig tis result, I ( I y vec( v I order to evaluate As a result, I I vec ( ( ( I y vec( v ( I y vec( v ad ece ow ( I y vec ( v ( ( ( ( ( ( EpE Eξ E EpE E ξ β I I vec + vec W ( I y vec( v ( I y vec( v β ( I y vec( v C08ed64doc /0/008 :38 P 0

11 otice tat te su vec ( v ( v I vec( v I vec ( v cosists of te su of (!! perutatio atrices Over te perutatio atrices, eac eleet i te atrix will ave a value of oe ( As a result, ad ece v J Usig tis result, vec v J I ( ( vec( vec ( J ( I y vec ( J ( I y vec ( J ( EE p Eξ ( E β ( vec( I y J vec ( y J vec ( y J β vec ( y J I order for E to be a ubiased estiator of T, te expected value ust be equal to Let us express vec( y J ( J vec ( y J y so tat,! ties T y C08ed64doc /0/008 :38 P

12 J y ( EE p E( E J y ξ β Jy J J β y J Te estiator will be ubiased we J ( EE p Eξ ( E T J β y J is zero Recall tat β vec( β β β so tat β ( β β β As a result, ( ( EE p Eξ E T β J y β were we represet te coefficiet for row j by ow ( β β β β j β otice tat te subscript j correspods to te positio for te j t j largest subject label i te saple set, ad ece correspods to a particular subject i te saple set We exaie te ter β J i ore detail Expadig ters, β J β ( ( β were β β βj is te average coefficiet for saple set Te ter j j j j j j j C08ed64doc /0/008 :38 P

13 is a vector wit eleets equal to oe for colus correspodig to te subject labels, s, icluded i te saple, ad values of zero for oter colus Tis is a vector of iclusio idicators for te subjects i te saple set Let us defie s to be a iclusio variable for subject s i saple set j Te (, ad we ca express js β J β( β β β Te ubiased coditio is te give by ( EE p Eξ ( E T β β β y, β s ys ad te ubiased costrait correspods to β s ys 0 It is valuable to briefly discuss tis costrait First, te restrictios of te coefficiets will deped upo weter oe or ore of te values of y s are zero For exaple, if y s 0, tere is o eed to ipose ay restrictio o β s For tis reaso, we could cosider te costraits to be specified for all subjects, s, were ys 0, ad specify te as β s ys 0, s were represets te uber of subjects wit o-zero values i te populatio otice also tat if y s 0, a liear estiator tat icludes a coefficiet for y s ca ot be optiized Tis is because te value of te coefficiet ca be set arbitrarily witout a cage occurrig i te estiator Tis proble was oted for te siplest exaple of Godabe s liear estiator We all values of y s are ot equal to zero, te ubiased costrait is as idicated I order for te costrait to old for all possible values of y s, we ca set β s 0 for all s,, Siultaeously, we express tis costrait as J β 0 Tis costrait ivolves te average coefficiet for saple sets tat iclude subject s, ad ot te idividual coefficiets for subject s Tis is siilar to te ubiased costrait used for rado effects i a ixed odel C08ed64doc /0/008 :38 P 3

14 Aside o Relatiosip wit Sequece Rado Variables Tere is a relatiosip betwee te iclusio idicator for subject s i saple set ad idicator rado variables for saple sequeces defied o page 4 of c08ed5doc We discuss D vec D D D, tis relatiosip briefly ere I c08ed5doc, we defie s ( s s s ( D D D D is is is as a vector of idicator rado variables, D is lis, for sequeces tat iclude subject s i positio i, were l,, A ew idex was created for tese rado variables tat idexed te saple sequeces tat icluded subject s Recall tat we referrig to saple sequeces, a rado variable for selectio of sequece is I u, or alteratively, for te correspodig saple set,, ad perutatio of te subjects i te saple set,, by ( II v otice tat tese two rado variables are coected by te relatiosip tat v u We cosider te coectio betwee te realizatios of tese rado variables, ad te realizatio of rado variables defied by D lis A particular saple sequece will result i values of v u wic we idicate (we ad 3 by vjj v jj j j u u v v v jj v jj u u v v j j 3 3 u u 3 3 v jj3 v j j3 j j Coparig te expressios for eleets i tis atrix wit expressios for eleets i te j j vector j illustrates tat oly te presece of a subject i te saple set is idicated, ot j j3 j i additio te positio of te subject i te saple sequece Te su of te realized values of te idicator rado variables Dlis wic we represet by dlis is siilar to te variable j js i Bot variables refer to subject s Te first variable, i i d lis, as a value of oe we C08ed64doc /0/008 :38 P 4

15 subject s is i sequece l Te secod variable, saple set j js, as a value of oe we subject s is i Suary of Ubiased Costraits Te estiator E will be a ubiased estiator of te total we te expressio give by ( ( EE p Eξ E T β J y is equal to zero We all values of ys 0 for s,,, te ubiased costrait is give by β s ys 0 were β βj is te average coefficiet for saple set, ad s js is a iclusio j j variable for subject s i saple set Tis costrait ca be expressed equivaletly as J β 0 Te costrait ca be odified we soe of te values of y s are equal to zero Te Variace of te Estiator E We evaluate te variace of te estiator wit coefficiets restricted to deped o saple sets give by E β ( I I vec( + vec( W We use te expressio for var pξ vec( vec( ( + ω + ωω C08ed64doc /0/008 :38 P 5

16 to siplify te expressio for varp ξ ( β ( I I vec ( vec ( + ω ( I I β β ( I I ( + ωω ( I I β β ( I I ( I I β First, we siplify te expressio for ( I I ( + ωω ( I I ( I I ( ( I I + + ( I I ( ω ω ( I I ow ( I I ( ( I I ( ( I I I I ( ( I I I I We first siplify te expressio for ( ( I I ow C08ed64doc /0/008 :38 P 6

17 ( I I I I I ( I ( I ( I We siplify ( I by expadig ters suc tat ( ( ( ( ow we sowed earlier tat ( ( vec ( ( I I I I I I y v Usig tis result, ( ( I ( I ( I ( I ( I y ( vec ( v vec ( v vec ( v otice tat we ca also express vec ( v ( I vec ( v As a result, ( ( vec vec vec ( vec vec vec ( I ( I y vec ( v vec ( v vec ( v ( I y ( I ( v ( I ( v ( I ( v ( I y ( v ( v ( v Usig te expressio for ( I ( I y ( vec( v vec( v vec( v, C08ed64doc /0/008 :38 P 7

18 I I ( ( ( vec( vec ( vec ( I y v v v ( I ( I ( I y ( vec( v vec ( v vec ( v Siilarly, ( ( ( vec( vec ( vec ( I I I y v v v vec ( v vec ( v vec ( v Usig tese results, ( I I ( ( I I ( I y ( ( I I I I vec vec ( ( vec( vec( vec( I y v v v vec v or ( v ( v ( I y ( C08ed64doc /0/008 :38 P 8

19 ( I I ( ( I I vec ( v vec ( ( ( vec( vec( vec( v ( I y v v v I y vec ( v ow vec ( v vec ( ( vec ( vec( vec( v v v v vec( vec v ( v vec ( v We cosider te patter i to evaluate te su of te vec of perutatio atrices First, otice tat we ca express tese atrices i a patter Let us represet v v v v v v v v v v v v v ( vi vi were vi correspods to a vector of values represetig te saple subject i positio vi i i a saple sequece ote tat vv vv vv vec ( vec ( v v v v v v v v vv vv vv First, we evaluate ters i te sus for diagoal eleets give by C08ed64doc /0/008 :38 P 9

20 v v Cosider a diagoal eleet give by i v v v v v v i i i i i, i v v v v v v i i i i i, i i v v v v v v i, i i i i, i, ij ij ij v v v Tis su correspods to te uber of perutatios wit saple subject j i positio i Tere are (! suc perutatios ext, cosider off diagoal eleets give by vijvij Tis su correspods to te uber of perutatios wit bot te saple subject j ad te saple subject j i positio i Tis vivi! I Sice!, we express ever occurs, so te su is zero As a result, ( tis as viv i I ext, cosider off-diagoal atrices give by viv i vi v i vi v, i viv v i i v v i i v, i viv i vi, v v, i iv v i iv, i were i i Cosider a diagoal eleet give by v v ij i j Tis su correspods to te uber of perutatios wit saple subject j i bot positio i ad positio i Tis ever occurs so te su is zero ext, cosider off diagoal eleets give by vijv Tis su i j correspods to te uber of perutatios wit bot te saple subject j i positio i ad te saple subject j i positio! suc perutatios As a result, i Tere are ( viv (! ( J I wic is equivalet to i Substitutig tese expressios, we fid tat v v J I ( ( i i C08ed64doc /0/008 :38 P 0

21 I ( J I ( J I ( ( vec( vec J I I J I v ( v ( J I ( J I I 0 0 I + J 0 I J I + J I J ( ( ( ( I suary, + vec ( v ( ( ( ( vec v I J I J I J We substitute tis expressio ito I I I I ( ( ( vec ( v vec ( ( ( vec( vec( vec( v ( I y v v v I y vec ( v ( vec ( vec I y ( v v ( I y ( ( + ( ( I y I J I J I J I y ( (( I y ( I J I( I y ( + (( I y ( ( J I J I y ( ( I y I J I I y I J y y ow ( ( ( C08ed64doc /0/008 :38 P

22 ( ( I y J I J I y J I y J y Substitutig tese expressios, ( I I ( ( I I wile ( ( ( (( I y ( I J I( I y ( + (( I y ( ( J I J I y + ( We cosider furter siplificatio of tis ter otice tat we ca express (( I J yy (( J I y J y (( I J yy (( J I y J y + [ I yy J y y J y J y I y J y] + ( I yy y J y J yy y J y I yi Jy ( ( J y I J y Usig tis result, ( I I ( ( I I I ( ( ( y I J y J y I J y Siplificatio of te Expressio Ivolvig te Respose Error Variaces We ext cosider siplificatio of te expressio for ( I I ( ωω ( I I otice tat we ca expad ( I I ( I I ( I I I, ( I I ( I ad ece express C08ed64doc /0/008 :38 P

23 ( I I ( I ωω ( I ( I I I ω ext First ote fro ( I I ( ω ω ( I I ( I I ( I ( ω ω ( I ( I I We expad ad siplify te expressio for ( c08ed66doc (page tat ω ( v I ( σ e σ e σ e tis expressio, I ω v σ e σ e σ e ( ( ( v ( I ( σe σe σe v ( σ e σ e σ e v ( σ σ σ otice tat sice (, we ca express σ σ σ σs, ( ad ece ( I ω v σ I a siilar aer, ( a result, s σs σs ( ( Usig ω I σ s v As I ω ω I v v v σ s v We substitute tis expressio ito ( I I ( ωω ( I I resultig i ( I I ( ωω ( I I ( I σ s ( v v I ( I v σ v ( I s v σ v s C08ed64doc /0/008 :38 P 3

24 Let us exaie te ter v σ s v for a particular exaple, were ad Suppose 0 0 ad v For tis saple set ad perutatio,, 0 0 σ v σs v 0 σ σ σ σ σ σ σ 3 0 σ 0 0 σ σ 0 If v for te sae saple set, te expressio for v σ s v will siplify to 0 σ 0 v σ s v I tis exaple, tere are oly two possible perutatios As a 0 σ 3 result, for tis saple set, σ 3 0 σ 0 v σ s v + 0 σ 0 σ 3 ( σ + σ3 I Te ter represetig te respose error variace for a subject i te saple set is equal for te two subjects i te saple set Tis is wat we were tryig to avoid For tis reaso, soe alterative ideas are eeded C08ed64doc /0/008 :38 P 4