Estimation of a proportion under a certain two-stage sampling design

Transcription

1 Etmaton of a roorton under a certan two-tage amng degn Danutė Kraavcatė nttute of athematc and nformatc Lthuana Stattc Lthuana Lthuana e-ma: raav@tmt Abtract The am of th aer to demontrate wth exame that the degn-baed etmator for the roorton of the frt-tage amng eement havng aocated at eat one econd-tage eement wth the attrbute of nteret ung the two-tage amng degn baed The tuaton encountered n the Adut Educaton Survey (AES) when etmatng the hare of ndvdua n non-forma educaton nvoved n job-reated earnng actvte ntroducton A new robem reated to the etmaton of a roorton ha aren n the Adut Educaton Survey (herenafter referred to a the AES ) The arameter of nteret the hare of ndvdua n non-forma educaton nvoved n job-reated earnng actvte n the aer the robem decrbed n the genera framewor and t hown by the exame that the degn-baed etmator for th arameter baed The drecton for further reearch drawn ouaton and arameter Let u denote by U { u u } ubunt of e cont of u the ouaton of the unt to each of whch a cuter of aocated Thu the ouaton of a ubunt U + + eement Suoe ome of the ubunt have an attrbute of nteret and ome of them do not have t Let u ntroduce a tudy varabe n ouaton U wth vaue f there at eat one ubunt among ubunt aocated wth unt u and otherwe Then the number of unt n the ouaton havng aocated at eat one ubunt wth the attrbute equa to the tota of varabe :

2 t () The hare (roorton) of the unt n U havng aocated at eat one ubunt wth the attrbute equa to the mean of varabe : µ Let u conder the etmaton of t / arameter t and µ from the urvey data Same and etmator The ame degn of ubunt conttutng ouaton U can be decrbed by a -tage amng degn wth ome robabtc ame me random ame them f ther number e than of unt n U at the frt tage and a of m ubunt n the cuter aocated wth unt m ) at the econd tage: U U u (or a of At the econd tage ame wthout oong the generaty et u conder Let u denote by the m for d e m can be any otve number but for mcty and m for for the frt tage amng degn weght wth ( ) : The degn-baed etmator uggeted for the number of unt havng aocated at eat one ubunt wth the attrbute t : d () where ẑ the degn-baed etmator of : f at eat one ubunt wth the attrbute beong to and otherwe For the hare of unt havng aocated at eat one ubunt wth the attrbute the uggeted degn-baed etmator µ () t / Th etmator uuay ued n the ot AES of tattca offce

3 Hyothe: etmator () and () are baed eg Et t E µ µ the exectaton taen here wth reect to the two-tage amng degn The foowng exame confrm the hyothe 4 Exame Let u tudy a ma ouaton { u u u } U contng of unt Unt u aocated wth one ubunt wthout an attrbute denoted by nonattr; unt u aocated wth one ubunt wth the attrbute denoted by attrb; unt u aocated wth two ubunt: one wth an attrbute (attrb) and one wthout an attrbute (nonattr) For th ouaton the number of unt wth the attrbute and ther hare equa to t µ / Let u draw the frt-tage me random ame of n eement from ouaton U The obe ame accordng to th amng degn and ther amng degn robabte are: u ) u ) u ) ( u ( u ( u ( ) ( ) ( ) ; Let u mfy the ame degn tang for the ame of ubunt m for and m for The econd tage amng degn robabte are a foow: Let u etmate t n thee ame: ( nonattr u ) ( attrb u ) ( nonattr u ) ( attrb u ) nonattr u ) ( attrb u ) ( () ( u u) : t ( + ) (+ ) µ ( ) n For the eement u we etmate f the unt wth the attrbute eected for the econd-tage ame and otherwe () ( u u) : f { nonattr} then () t ( + ) (+ ) n µ

4 f { attrb} then u ) : f { nonattr} ( u then f { attrb} ( + ) (+ ) n () t ( + ) (+ ) n (4) t () µ (4) µ (5) then (5) t ( + ) (+ ) n Let u cacuate the exectaton of t wth reect to the amng degn: () ( ) ( ) () () ( ) + t ( nonattr u ) t ( attrb u ) E t t + + (4) (5) ( t ( nonattr u ) + t ( attrb u )) ( ) µ t t mean that etmator t baed Conequenty and etmator Et E µ µ µ of the roorton of the unt wth the attrbute ao baed t cear by ntuton that etmator () underetmatng the true number of the unt wth the attrbute becaue there are obe cae when the amed unt baed on the amed ubunt cafed a wthout an attrbute when n reaty t a non-amed ubunt wth an attrbute aocated wth t but there are no obe cae when the amed unt cafed a beng aocated to the ubunt wth an attrbute when n reaty t not o 5 obe drecton for the foowng reearch The other nd of etmator for the roorton of the frt-tage amng eement under the two-tage amng degn a foow Some auxary aumton about the ouaton of econdary eement have to be tated Let u uoe the number the number of ubunt wth attrbute defne robabte of ubunt aocated wth unt random u fxed and nown but Let u ( ) ( ) ( )

5 The number of the amed ubunt wth attrbute We have the foowng reatonh: Y ao random Y mn( ) _ f _ Y > > > _ f _ Y Let u nvetgate a random event > Let u denote the random varabe J _ f _f > Varabe J obtan wth the robabty ( J ) ( > ) ( Y > ) + ( > Y ) ( Y ) () Let u cacuate th robabty f then Y concde wth ( Y ) > and and ( > ) ( Y ) ( ) Y f > then ( Y ) ( Y ) (4) > ( Y ) ( Y ) ( ) ( ) ( Y ) ( ) ( ) Y ( Y ) Y ( Y ) ( Y ) ( ) CC ( Y ) C Hence from (4) we get

6 ( Y ) > ( ) ( ) (5) nertng (5) nto () we can cacuate Let u ntroduce a new etmator for the number of unt aocated wth the ubunt wth attrbute: rooton Suoe robabte ( ) and nown Then t d EJ (6) > are fxed the exectaton of etmator t (4) under the amng degn t varance Var( t ) + ( ) etmator of varance Var ( t ) + n ractce robabte ( ) E t unbaed are not nown and they have to be etmated Then the etmator of the tota and t varance become more comcated The exame of thee aroxmate robabte of the Lthuanan AES for the hare of job-reated earnng actvte n non-forma educaton hown n the fgure beow () 8 6 4