[經(jīng)濟(jì)學(xué)]sling抽樣技術(shù)統(tǒng)計(jì)學(xué)專業(yè)課課件-資料下載頁(yè)
2025-02-19 18:30本頁(yè)面
【正文】 is used to adjust estimates from the sample so that they reflect demographical totals. ?Population:4000 ?Sample: 400 2700 females 1300 males 240 females 160 males to bee a teacher ?Simple estimation: ?Ratio estimation: 240 females 160 males 84 females 40 males 124 4 0 0 0 1 2 4 0400??8 4 4 02 7 0 0 1 3 0 0 1 2 7 02 4 0 1 6 0? ? ? ??Ratio estimation is used to adjust nonresponse. ? Bias and Mean Squared Error of Ratio Estimators Ratio estimation is biased, unlike SRS estimation, but with a reduced variance as a pensator for the presence of bias. Since: we get: ? ?? ?1? ?y xxy r y x y y yxxt ttt t t t t ttt?? ?? ? ? ? ? ?????? ??? ? ?( ) ( ) ? yy r y y x x yxtE t t E t E t t tt??? ? ? ? ?????? ?? ? ?? ? ?C o v ( , ) C o v ( , )x x x x xE B t t B t t B t??? ? ? ? ? ? ? ???? Then we have: ? Consequently, as shown by Hartley and Ross(1954). ? ?() ( , )? ?B i a s ( ) ( ) y r yxUE t t C o v B xB E B Btx? ?? ? ? ?? ?? ? ( , ) V ( ) V ( )| B ia s ( ) | ( , ) 1?? ?V ( )V ( ) V ( )U UCo rr B x B xB Co v B xx BB x B??? ? V ( )( , ) V ( ) V ( )C V ( ) 1?V ( )UUxC orr B x B x xxx B? ? ?Another way to show this is: The bias of is small if: 1, the sample size n is large。 2, the sampling fraction n/N is large。 3, is large 4, is small。 5, the correlation R is close to 1. 222V ( ) ( , )?( ) 1 x x yUUB S R S Sn B x C o v x yE B BN n x x? ???? ? ? ??????BUxxSThe proof for it is: ()? UUx x xy y B x y B xB B Bx x x x????? ? ? ? ? ?? ? () UUUy B x x xy B xx x x??????? ? ? ? ()? UUy B x x xE B B Exx????? ? ??? ???? ?21 () UUE y B x x xx? ? ? ?????? ?S i n c e : 0UUE y B x y B x? ? ? ? ? ?21 ()U U UUE y y y B x x xx? ? ? ? ? ?????? ? ? ?21 ( ) ( )U U U UUE y y x x y B x x xx? ? ? ? ? ? ?????? ? ? ?222211 1y x x x x yUUnR S S B S B S R S Sx N n x??? ? ? ? ? ?????? ? ? ?21 ( ) ( )U U U UUE y y x x B x B x x xx? ? ? ? ? ? ?????? ? () 1 amp。 ( ) 1 ( )UUyxE y y x x nR V x V xS S N n?? ???? ????? ? ??? ??????In the works of HartleyRoss, an unbiased estimator for the parameter B is given after the following is proved: The unbiased estimator is: ? ?( 1 )( 1 )UNnE r y x r BN n x???? ? ? ??????? ?1( 1 ) 1? . w h e r e( 1 )niHRiUiyNnB r y r x rN n x n x??? ? ? ? ?? ?The approximated MSE of is: The approximated MSE of will be small if: 1, the sample size n is large。 2, the sampling fraction n/N is large。 3, the deviation about the line is large。 4, is large。 5, the correlation R is close to +1. 2 2 2222221?( ) ( ) 1 y x y xUUS B R S S B SnE B B E y B xx N n x????? ? ? ? ?????y Bx?Ux?B?BThe proof for it is: And: ? ?? UUyyyE B B E B Ex x x????? ? ? ? ?????????? ?2? ? ? ?( ) ( ) ( )M S E B V B E B B V B? ? ? ?? ? ? ?? ?2222( ) 011()d y B xUUi i iUUd y B xE d y B xE d E d E dxx????? ? ??? ? ? ? ? ? ? ? ? ? ?? ? ? ?f o r l a r g e e n o u g h 0UUUUE y yyynExx??? ?? ? ?????? ? ? ?22221?Uy B xE B B E E y B xxx???? ? ? ????? And: ? ?? ? 22211 1 111NiidiUUd E dnSx x N n N????? ? ? ? ? ?????? ?? ? ? ?2211 11NNi i i U U iiiy B x y y y B xNN??? ? ? ??????? ? ? ? 21 1Ni U i Uiy y B x xN?? ? ????????? ? 2211111NiiiUy B xnx N n N????? ? ? ? ?????? ? Therefore we get: ? ? ? ? ? ? ? ?22 211 21Ni U i U i U i Uiy y B y y x x B x xN???? ? ? ? ? ? ? ???? ?2 2 2 2 2 222y x y x y x y xS B S B S S B R S S B S? ? ? ? ? ?2 2 2222221?( ) ( ) 1 y x y xUUS B R S S B SnE B B E y B xx N n x????? ? ? ? ?????In practice, B is unknown, then let It follows from above, we will have: 2222?()? ?( ) 1 1( 1 )iie i SUUy B xsnnVBN n x N n n x??? ? ? ?? ? ? ?? ? ? ? ?? ? ? ??22? ? ??[ ] [ ] 1 ey r xsnV t V t B NNn??? ? ?????2? ? ??[ ] [ ] 1 erUsnV y V x BNn??? ? ??????i i ie y B x??The CIs can be constructed as: Let’s just study two examples in the textbook. ? ?1 . 9 6 S E ( )BB?? ?1 . 9 6 S E ( )rryy?? ?1 . 9 6 S E ( )y r y rtt? Accuracy of the MSE Approximation For () to be a good approximation of MSE, we want a large sample size n (30), and Advantage of ratio estimation C V ( ) 0 . 1 , C V ( ) 0 . 1xy??2M S E ( ) ( ) 1 ySny V y Nn??? ? ?????2 2 22?M S E ( ) 1 y x y xrS B R S S B SnyNn??????????Then, So to the accuracy of the approximation, holds, if and only if: 222?M S E ( ) M S E ( ) 1 x y xrB R S S B SnyyNn????? ? ??????M S E ( ) M S E ( )ryy?? ?21 y x xR S B S B SnNn??????????C V ( )2 2 C V ( )xyBS xRSy?? Regression Estimation ? Ratio estimation works best if the data are well fit by a straight line through the origin. But sometimes, data appear to be evenly scattered about a straight line that does not go through the origin, that is, the data look as though the usual straightline regression model would provide a good fit. 01BByx??0 1 1? ? ?? ()r e g U Uy B B x y B x x? ? ? ? ? where: He
教學(xué)課件相關(guān)推薦
鄂ICP備17016276號(hào)-1