Chapter 7 Varying Probability Sampling

Transcription

1 Chapte 7 Vayg Pobablty Samplg The smple adom samplg scheme povdes a adom sample whee evey ut the populato has equal pobablty of selecto. Ude ceta ccumstaces, moe effcet estmatos ae obtaed by assgg uequal pobabltes of selecto to the uts the populato. Ths type of samplg s kow as vayg pobablty samplg scheme. If Y s the vaable ude study ad X s a auxlay vaable elated to Y, the the most commoly used vayg pobablty scheme, the uts ae selected wth pobablty popotoal to the value of X, called as se. Ths s temed as pobablty popotoal to a gve measue of se (pps) samplg. If the samplg uts vay cosdeably se, the SRS does ot takes to accout the possble mpotace of the lage uts the populato. A lage ut,.e., a ut wth lage value of Y cotbutes moe to the populato total tha the uts wth smalle values, so t s atual to expect that a selecto scheme whch assgs moe pobablty of cluso a sample to the lage uts tha to the smalle uts would povde moe effcet estmatos tha the estmatos whch povde equal pobablty to all the uts. Ths s accomplshed though pps samplg. ote that the se cosdeed s the value of auxlay vaable X ad ot the value of study vaable Y. Fo example a agcultue suvey, the yeld depeds o the aea ude cultvato. So bgge aeas ae lkely to have lage populato ad they wll cotbute moe towads the populato total, so the value of the aea ca be cosdeed as the se of auxlay vaable. Also, the cultvated aea fo a pevous peod ca also be take as the se whle estmatg the yeld of cop. Smlaly, a dustal suvey, the umbe of wokes a factoy ca be cosdeed as the measue of se whe studyg the dustal output fom the espectve factoy. Dffeece betwee the methods of SRS ad vayg pobablty scheme: I SRS, the pobablty of dawg a specfed ut at ay gve daw s the same. I vayg pobablty scheme, the pobablty of dawg a specfed ut dffes fom daw to daw. It appeas pps samplg that such pocedue would gve based estmatos as the lage uts ae oveepeseted ad the smalle uts ae ude-epeseted the sample. Ths wll happe case of sample mea as a estmato of populato mea whee all the uts ae gve equal weght. Istead of gvg equal weghts to all the uts, f the sample obsevatos ae sutably weghted at the estmato

2 stage by takg the pobabltes of selecto to accout, the t s possble to obta ubased estmatos. I pps samplg, thee ae two possbltes to daw the sample,.e., wth eplacemet ad wthout eplacemet. Selecto of uts wth eplacemet: The pobablty of selecto of a ut wll ot chage ad the pobablty of selectg a specfed ut s same at ay stage. Thee s o edstbuto of the pobabltes afte a daw. Selecto of uts wthout eplacemet: The pobablty of selecto of a ut wll chage at ay stage ad the pobabltes ae edstbuted afte each daw. PPS wthout eplacemet (WOR) s moe complex tha PPS wth eplacemet (WR). We cosde both the cases sepaately. PPS samplg wth eplacemet (WR): Fst we dscuss the two methods to daw a sample wth PPS ad WR.. Cumulatve total method: The pocedue of selecto a smple adom sample of se cossts of - assocatg the atual umbes fom to uts the populato ad - the selectg those uts whose seal umbes coespod to a set of umbes whee each umbe s less tha o equal to whch s daw fom a adom umbe table. I selecto of a sample wth vayg pobabltes, the pocedue s to assocate wth each ut a set of cosecutve atual umbes, the se of the set beg popotoal to the desed pobablty. If X, X,..., X ae the postve teges popotoal to the pobabltes assged to the uts the populato, the a possble way to assocate the cumulatve totals of the uts. The the uts ae selected based o the values of cumulatve totals. Ths s llustated the followg table:

3 Uts Se Cumulatve X X X X X X T X T X X T T T X X X Select a adom umbe R betwee ad T by usg adom umbe table. If T R T, the th ut s selected wth pobablty X, =,,,. T Repeat the pocedue tmes to get a sample of se. I ths case, the pobablty of selecto of th ut s T T X P T T P X. ote that T s the populato total whch emas costat. Dawback : Ths pocedue volves wtg dow the successve cumulatve totals. Ths s tme cosumg ad tedous f the umbe of uts the populato s lage. Ths poblem s ovecome the Lah s method. Lah s method: Let M Max X,.e., maxmum of the ses of uts the populato o some coveet,,..., umbe geate tha M. The samplg pocedue has followg steps:. Select a pa of adom umbe (, ) such that, M.. If X, the th ut s selected othewse eected ad aothe pa of adom umbe s chose. 3. To get a sample of se, ths pocedue s epeated tll uts ae selected. ow we see how ths method esues that the pobabltes of selecto of uts ae vayg ad ae popotoal to se. 3

4 Pobablty of selecto of th ut at a tal depeds o two possble outcomes ethe t s selected at the fst daw o t s selected the subsequet daws peceded by effectve daws. Such pobablty s gve by P( ) P( M ) X M. * P, say. X Pobablty that o ut s selected at a tal M X M X Q, say. M Pobablty that ut s selected at a gve daw (all othe pevous daws esult the o selecto of ut ) P QP Q P... * * * * P Q X / M X X X / M X X total X. Thus the pobablty of selecto of ut s popotoal to the se sample. X. So ths method geeates a pps Advatage:. It does ot eque wtg dow all cumulatve totals fo each ut.. Ses of all the uts eed ot be kow befoe had. We eed oly some umbe geate tha the maxmum se ad the ses of those uts whch ae selected by the choce of the fst set of adom umbes to fo dawg sample ude ths scheme. Dsadvatage: It esults the wastage of tme ad effots f uts get eected. X The pobablty of eecto. M The expected umbes of daws equed to daw oe ut Ths umbe s lage f M s much lage tha X. M. X 4

5 Example: Cosde the followg data set of 0 umbe of wokes the factoy ad ts output. We llustate the selecto of uts usg the cumulatve total method. Factoy o. umbe of wokes Idustal poducto Cumulatve total of ses (X) ( thousads) ( metc tos) (Y) 30 T 5 60 T T T T T T T T T Selecto of sample usg cumulatve total method:.fst daw: - Daw a adom umbe betwee ad Suppose t s 3 3 -T 4 T 5 - Ut Y s selected ad Y5 8 etes the sample.. Secod daw: - Daw a adom umbe betwee ad 64 - Suppose t s 38 - T7 38 T8 - Ut 8 s selected ad Y8 7 etes the sample - ad so o. - Ths pocedue s epeated tll the sample of equed se s obtaed. 5

6 Selecto of sample usg Lah s Method I ths case M Max X 4,,...,0 So we eed to select a pa of adom umbe (, ) such that 0, 4. Followg table shows the sample obtaed by Lah s scheme: Radom o Radom o Obsevato Selecto of ut X3 0 tal accepted ( y 3) X6 tal eected X4 4 tal eected 9 9 X 5 tal eected 9 X9 tal accepted ( y 9) ad so o. Hee ( y3, y 9) ae selected to the sample. Vayg pobablty scheme wth eplacemet: Estmato of populato mea Let Y : value of study vaable fo the th ut of the populato, =,,,. X : kow value of auxlay vaable (se) fo the th ut of the populato. P : pobablty of selecto of th ut the populato at ay gve daw ad s popotoal to se X. Cosde the vayg pobablty scheme ad wth eplacemet fo a sample of se. Let value of th obsevato o study vaable the sample ad Defe the y,,,...,, p y be the p be ts tal pobablty of selecto. 6

7 s a ubased estmato of populato mea Y, vaace of s Y whee P Y P ad a ubased estmate of vaace of s Poof: s ( ). ote that ca take ay oe of the values out of Z, Z,..., Z wth coespodg tal pobabltes P, P,..., P, espectvely. So E ( ) ZP Y P P Y. Thus E ( ) E ( ) Y Y. So s a ubased estmato of populato mea Y. The vaace of s Va( ) Va ow Va Va( ) E E( ) E Y s ' ( ) ( ae depedet WR case). Z Y P Y Y P P (say). 7

8 Thus Va( ). To show that s s a ubased estmato of vaace of, cosde ( ) E( s) E( ) o ( ) Es E E ( ) E ( ) Va( ) E( ) Va( ) E( ) Y Y Y usg Va( ) Y P P ( ) s E Va( ) s y Va( ). ( ) p ote: If P, the y, Y Va( ) Y. whch s the same as the case of SRSWR. y 8

9 Estmato of populato total: A estmate of populato total s ˆ y Ytot. p. Takg expectato, we get ˆ Y Y Y EY ( tot ) P P... P P P P Y Ytot Y. tot Thus Y ˆtot s a ubased estmato of populato total. Its vaace s Va Yˆ ( tot ) Va( ) Y Y P P Y Y P Y P tot Y tot P. A estmate of the vaace ˆ s Va( Ytot ). Vayg pobablty scheme wthout eplacemet I vayg pobablty scheme wthout eplacemet, whe the tal pobabltes of selecto ae uequal, the the pobablty of dawg a specfed ut of the populato at a gve daw chages wth the daw. Geeally, the samplg WOR povdes a moe effcet estmato tha samplg WR. The estmatos fo populato mea ad vaace ae moe complcated. So ths scheme s ot commoly used pactce, especally lage scale sample suveys wth small samplg factos. 9

10 Let U : th ut, P : Pobablty of selecto of U at the fst daw,,,..., P P Pobablty of selectg U at the : ( ) P () P. th daw Cosde P() Pobablty of selecto of U at d daw. Such a evet ca occu the followg possble ways: U s selected at d daw whe st d - U s selected at daw ad U s selected at daw st d - U s selected at daw ad U s selected at daw st d - U s selected at daw ad U s selected at daw st d - s selected at daw ad U s selected at daw - U st d - U s selected at daw ad U s selected at daw So P () ca be expessed as P P P P P P P P... P P... P () P P P P P P P ( ) P P P P P P P ( ) P P P P P P P P P P P P P P P P fo all uless () () P. 0

11 y P () wll, geeal, be dffeet fo each =,,,. So E wll chage wth successve daws. p y Ths makes the vayg pobablty scheme WOR moe complex. Oly wll povde a ubased p y estmato of Y. I geeal, ( ) wll ot povde a ubased estmato of Y. p Odeed estmates To ovecome the dffculty of chagg expectato wth each daw, assocate a ew vaate wth each daw such that ts expectato s equal to the populato value of the vaate ude study. Such estmatos take to accout the ode of the daw. They ae called the odeed estmates. The ode of the value obtaed at pevous daw wll affect the ubasedess of populato mea. We cosde the odeed estmatos poposed by Des Ra, fst fo the case of two daws ad the geeale the esult. Des Ra odeed estmato Case : Case of two daws: Let y ad y deote the values of uts U() ad U () daw at the fst ad secod daws espectvely. ote that ay oe out of the uts ca be the fst ut o secod ut, so we use the otatos U() ad U() stead of U ad U. Also ote that y ad yae ot the values of the fst two uts the populato. Futhe, let p ad p deote the tal pobabltes of selecto of U () ad U (), espectvely. Cosde the estmatos y p y y p /( p) y. y p ote that p ( p ) p s the pobablty PU ( () U () ).

12 Estmato of Populato Mea: Fst we show that s a ubased estmato of Y. E ( ) Y. ote that Cosde P. y y Y Y Y p p P P P ( ) E ote that ca take ay oe of out of the values,,..., E Y Y Y P P... P P P P Y E ( ) Ey y ( p ) p ( P ) E( y) EE y U() ( Usg E( Y) EX[ EY( Y X)]. p whee E s the codtoal expectato afte fxg the ut U () selected the fst daw. y Sce p ca take ay oe of the ( ) values (except the value selected the fst daw) Y P wth pobablty P, P so ( P) y * Y P E y U() ( P) E U() ( P). p p P P. whee the summato s take ove all the values of Y except the value y whch s selected at the fst daw. So ( P ) * E y U() Y Ytot y. p Substtutg t E ( ), we have E ( ) Ey ( ) EY ( tot y) E( y) E( Ytot y) Ytot EY ( tot ) Y.

13 Thus E ( ) Y Y Y. E ( ) E ( ) Vaace: The vaace of fo the case of two daws s gve as Va( ) P P Y P Y 4 Y Y tot tot P P Poof: Befoe statg the poof, we ote the followg popety ab a b b whch s used the poof. The vaace of s Va E E ( ) ( ) ( ) y y( p) E y Y p p y ( p ) E y ( p ) Y atue of atue of vaable vaable depeds depeds oly o st upo ad st daw d daw 4 p p Y ( ) ( P) Y P PP = Y 4 P P P Y ( ) PP Y ( P) PP P ( P ) PP = YY Y 4 P P P P PP P Y ( P) P Y ( P) P = ( ). YY P Y 4 P P P P 3

14 Usg the popety ab a b b, we ca wte Y ( P) Y Y Va( ) ( ) ( )( )] P P P P Y P Y Y Y 4 P( P) P P Y Y Y (P ) ( P P P) Y( P)( Y Y) Y 4 P P P Y 4 Y Y P Y P Y P P P P PY Y Y Y P YP Y Y ] Y Y 4 Y P Y Ytot YtotYP Y P P Y P Y tot Ytot Y Y 4 P 4 Y tot YtotYP 4 Y Y P P Ytot Y YtotYP Ytot Ytot P ( 4 ) 4 P P Y tot Y P Ytot ( Y YtotYP Ytot Ytot P Ytot ) P 4 Y P P Y Y Y YP P Y 4 tot tot tot P P Y Y P Ytot P Ytot P 4 P Y Y Y P Y P Ytot P Ytot P P P 4 4 vaace of WR educto of vaace case fo WR wth vayg pobablty ` 4

15 Estmato of Va( ) Va( ) E( ) ( E( )) E ( ) Y Sce E ( ) EE ( u) EY YE( ) Y. Cosde E E ( ) E ( ) E ( ) Y Va( ) Va( ) s a ubased estmato of Va( ) Alteatve fom Va( ) ( ) 4 y y y p 4 p p y y( p) ( p ) 4 p p ( p ) y y 4 p p. Case : Geeal Case Let () () ( ) ( ) ( U, U,..., U,..., U ) be the uts selected the ode whch they ae daw daws whee U ( ) deotes that the th ut s daw at the th daw. Let ( y, y,.., y,..., y ) ad ( p, p,..., p,..., p ) be the values of study vaable ad coespodg tal pobabltes of selecto, espectvely. Futhe, let P(), P(),..., P( ),..., P ( ) be the tal pobabltes of U, U,..., U,..., U, espectvely. () () ( ) ( ) 5

16 Futhe, let y p y y y... y ( p... p ) fo,3,...,. p Cosde as a estmato of populato mea Y. We aleady have show case that E( ) Y. ow we cosde E ( ),,3,...,. We ca wte E ( ) EE U (), U(),..., U ( ) whee E s the codtoal expectato afte fxg the uts U(), U(),..., U( ) daw the fst ( - ) daws. Cosde y y E ( p... p ) E E ( p... p ) U, U,..., U p p () () ( ) y E ( P P... P ) E U, U,..., U. () () ( ) p () () ( ) y Y Sce codtoally ca take ay oe of the ( - -) values,,,..., wth pobabltes p P P, so P P... P () () ( ) y Y P (... ) (... ) * E p p E P P P. p () () ( ) P ( P P... P ) () () ( ) E * Y whee * deotes that the summato s take ove all the values of y except the y values selected the fst ( -) daws lke as ( (), (),..., ( )),.e., except the values y, y,..., y whch ae selected the fst ( -) daws. 6

17 Thus ow we ca expess y E ( ) EE y y... y ( p... p ) p * E Y() Y()... Y( ) Y E Y() Y()... Y( ) Y ( (), (),..., ( )) E Y Y... Y Y Y Y... Y E Ytot Ytot () () ( ) tot () () ( ) Y fo all,,...,. The E E Y Y. Thus s a ubased estmato of populato mea Y. The expesso fo vaace of geeal case s complex but ts estmate s smple. Estmate of vaace: Va( ) E( ) Y. Cosde fo s, E ( ) EE ( U, U,..., U ) s s s E Y YE( ) Y because fo s, wll ot cotbute ad smlaly fo, s s wll ot cotbute the expectato. 7

18 Futhe, fo s, Cosde Substtutg E ( ) EE ( U, U,..., U ) s s E sy YE( ) Y. s E s E( s) ( ) ( s) s ( ) ( s) s ( ) Y ( ) Y. Y Va( ), we get Va( ) = E( ) Y = E ( ) E E ( s) ( ) ( s) s Va( ) s. ( ) ( s) s s ( s) s s, ( s) s Usg The expesso of Va ( ) ca be futhe smplfed as Va( ) ( ) ( ) ( ). ( ) 8

19 Uodeed estmato: I odeed estmato, the ode whch the uts ae daw s cosdeed. Coespodg to ay odeed estmato, thee exst a uodeed estmato whch does ot deped o the ode whch the uts ae daw ad has smalle vaace tha the odeed estmato. I case of samplg WOR fom a populato of se, thee ae uodeed sample(s) of se. Coespodg to ay uodeed sample(s) of se uts, thee ae! odeed samples. Fo example, fo f the uts ae u ad u, the - thee ae! odeed samples - ( u, u) ad ( u, u ) - thee s oe uodeed sample ( u, u ). Moeove, Pobablty of uodeed Pobablty of odeed Pobablty of odeed sample ( u, u ) sample ( u, u ) sample ( u, u ) Fo 3, thee ae thee uts u, u, u 3 ad -thee ae followg 3! = 6 odeed samples: ( u, u, u3),( u, u3, u),( u, u, u3),( u, u3, u),( u3, u, u),( u3, u, u ) - thee s oe uodeed sample ( u, u, u 3). Moeove, Pobablty of uodeed sample = Sum of pobablty of odeed sample,.e. Pu (, u, u) Pu (, u, u) Pu (, u, u) Pu (, u, u) Pu (, u, u) Pu (, u, u), Let s, s,,..,,,,...,!( M) be a estmato of populato paamete based o odeed sample s. Cosde a scheme of selecto whch the pobablty of selectg the odeed sample ( s ) s p s. The pobablty of gettg the uodeed sample(s) s the sum of the pobabltes,.e., p s M p s. Fo a populato of se wth uts deoted as,,,, the samples of se ae tuples. I the th daw, the sample space wll cosst of ( )...( ) uodeed sample pots. 9

20 the p so P selecto of ay odeed sample ( )...( )! selecto of ay psu Pselecto of ay uodeed sample! P ( )...( ) odeed sample p s M(!)!( )! p so.! M ˆ ˆ ' Theoem : If 0 s, s,,..., ;,,..., M(!) ad u s ps ae the odeed ad uodeed estmatos of epectvely, the () E( ˆ ) ( ˆ u E 0) () Va( ˆ ) Va( ˆ 0) u whee s s a fucto of th s odeed sample (hece a adom vaable) ad p s s the pobablty of selecto of th s odeed sample ad p ' s p p s. s Poof: Total umbe of odeed sample =! () E( ˆ ) 0 s M ' u s s s s E( ˆ ) p p E( ˆ ) M 0 s s s s ps s p s s p p s p s () Sce ˆ 0, so s ˆ 0 s wth pobablty ps,,,..., M, s,,...,. Smlaly, M M ˆ ' ˆ ' u sps,sou sps wth pobablty p s 0

21 Cosde Va( ˆ ˆ ˆ 0) E( 0) E( 0) ( ˆ s ps E 0) s ˆ ˆ ˆ u u u Va( ) E( ) E( ) ' ( ˆ s ps ps E 0) s ' u s s s s s s s Va( ˆ ˆ 0) Va( ) p p p 0 s ps s p s ps s s p p p ' s s s s s s ' ' s ps s ps ps s ps s ps ps s ' ' s ps s ps ps s ps s ps s Va( ˆ ) Va( ˆ ) 0 o Va( ˆ ) Va( ˆ ) u ' ( s s ps ) ps 0 s u 0 Estmate of Va ( ˆ ) Sce ˆ ˆ Va( 0) Va( ) ( p ) p u ' u s s s s s ˆ ˆ ' Va( u) Va( 0) ( s sps) ps s ' ˆ ' ' p Va( ) p ( p ). s 0 s s s s Based o ths esult, ow we use the odeed estmatos to costuct a uodeed estmato. It follows fom ths theoem that the uodeed estmato wll be moe effcet tha the coespodg odeed estmatos.

22 Muthy s uodeed estmato coespodg to Des Ra s odeed estmato fo the sample se Suppose y ad y ae the values of uts U ad U selected the fst ad secod daws espectvely wth vayg pobablty ad WOR a sample of se ad let p ad p be the coespodg tal pobabltes of selecto. So ow we have two odeed estmates coespodg to the odeed samples s ad s as follows * * s ( y, y ) wth ( U, U ) * s ( y, y ) wth ( U, U ) * whch ae gve as * y y ( s ) ( p) ( p) p p whee the coespodg Des Ra estmato s gve by y y ( p) y p p ad * y y ( s) ( p) ( p) p p whee the coespodg Des Ra estmato s gve by y y( p) y. p p The pobabltes coespodg to ( s ) * ad ( s ) ae * pp * ps ( ) p pp * ps ( ) p ps () ps ( ) ps ( ) * * p p ( p p ) ( p )( p )

23 p * p'( s ) p p * p p'( s). p p Muthy s uodeed estmate u ( ) coespodg to the Des Ra s odeed estmate s gve as ( u) ( s ) p'( s ) ( s ) p'( s ) * * s ( ) ps ( ) s ( ) ps ( ) * * * * * * ps ( ) ps ( ) y y pp y y pp ( p) ( p) ( p) ( p) p p p p p p pp pp p p y y y y ( p) ( p) ( p) ( p) ( p) ( p) p p p p ( p ) ( p ) y y ( p) ( p) ( p) ( p) ( p) ( p p p p p y y ( p) ( p) p p ( p p ). Ubasedess: ote that y ad p ca take ay oe of the values out of Y, Y,..., Y ad P, P,..., P, espectvely. The y ad p ca take ay oe of the emag values out of Y, Y,..., Y ad P, P,..., P, espectvely,.e., all the values except the values take at the fst daw. ow 3

24 E ( u) Y Y PP PP ( P) ( P) P P P P P P Y Y PP PP ( P) ( P) P P P P P P Y Y PP PP ( P) ( P) P P P P P P Y Y PP ( P) ( P) P P ( P)( P) YP YP P P Usg esult ab a b b, we have Y Y E( u) ( P P) ( P P) P P Y Y ( P) ( P) P P Y Y Y Y Y. 4

25 Vaace: The vaace of uca ( ) be foud as Va ( u) P P P P Y Y PP P P ( P P) P P ( P)( P) PP ( P P) Y Y ( P P) P P ( )( )( ) ( ) Usg the theoem that Va( ˆ ) ( ˆ u Va 0) we get Va ( u) Va ( s ) * ad Va ( u) ( ) * Va s Ubased estmato of V( u ) A ubased estmato of Va u s ( p p)( p)( p ) y y Va ( u). ( p p) p p Hovt Thompso (HT) estmate The uodeed estmates have lmted applcablty as they lack smplcty ad the expessos fo the estmatos ad the vaace becomes umaageable whe sample se s eve modeately lage. The HT estmate s smple tha othe estmatos. Let be the populato se ad y, (,,..., ) be the value of chaactestc ude study ad a sample of se s daw by WOR usg abtay pobablty of selecto at each daw. Thus po to each succeedg daw, thee s defed a ew pobablty dstbuto fo the uts avalable at that daw. The pobablty dstbuto at each daw may o may ot deped upo the tal pobablty at the fst daw. Defe a adom vaable (,,.., ) as f Y s cluded a sample ' s' of se 0othewse. 5

26 y Let,... assumg E( ) 0 fo all E( ) whee E( ). PY ( s) 0. PY ( s) s the pobablty of cludg the ut the sample ad s called as cluso pobablty. The HT estmato of Y based o y, y,..., y s ˆ Y HT. Ubasedess ˆ EY ( ) E ( ) HT E( ) y E( ) E( ) y Y whch shows that HT estmato s a ubased estmato of populato mea. Vaace Cosde VY ( ˆ ) V ( ) HT E E E ( ) ( ) Y ( ). E ( ) E E ( ) E( ) ( ). E ( ) 6

27 If the So whee s S s the set of all possble samples ad s pobablty of selecto of th ut the sample s E( ) P( ys) 0. P( ys). 0.( ) E P y s P y s. ( ). ( ) 0. ( ) E( ) E( ) E ( ) (# ) s the pobablty of cluso of th ad th ut the sample. Ths s called as secod ode cluso pobablty. ow Y E( ) E ( ) ( ) ( ) E E E ( ) ( ). Thus ˆ Va( YHT ) ( ) ( ) ( ) ( ) ( ) y y ( ) ( ) ( ) y yy ( ) 7 y

28 Estmate of vaace ˆ ˆ y ( ) y y V Va( YHT ). ( ) Ths s a ubased estmato of vaace. Dawback: It does ot educes to eo whe all y ae same,.e., whe y. Cosequetly, ths may assume egatve values fo some samples. A moe elegat expesso fo the vaace of y ˆHT has bee obtaed by Yates ad Gudy. Yates ad Gudy fom of vaace Sce thee ae exactly values of. Takg expectato o both sdes whch ae ad ( ) values whch ae eo, so Also E( ). ( ) E E( ) E( ) J ( ) E E( ) E( ) (usg E( ) E( )) E ( ) ( ) ( ) E( ) ( ) J J Thus E( ) P(, ) P( ) P( ) E( ) E( ) 8

29 Theefoe Smlaly ( ) E( ) E( ) E( ) E( ) E( ) E( ) E( ) ( ) E( ) E( ) E( ) ( ) E( ) ( ) ( E( ) E( ) E( ) ( ) () ( ) E( ) E( ) E( ) ( ). () We had eale deved the vaace of HT estmato as ˆ Va( YHT) ( ) ( ) ( ) Usg () ad () ths expesso, we get ˆ Va( YHT ) ( ) ( ) ( ) E( ) E( ) E( ) ( ) E( ) E( ) E( ) E( ) E( ) E( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ). ( ) The expesso fo ad ca be wtte fo ay gve sample se. 9

30 Fo example, fo, assume that at the secod daw, the pobablty of selectg a ut fom the uts avalable s popotoal to the pobablty of selectg t at the fst daw. Sce E( ) Pobablty of selectg Y a sample of two P P whee P s the pobablty of selectg Y at th daw (,). If P th ut at fst daw (,,..., ) the we had eale deved that P P d y s ot selected y s selected at daw P P st P st at daw y s ot selected at daw PP ( ) P P P P. P P So P P E( ) P P P Aga E( ) Pobablty of cludg both y ad y a sample of se two PP PP P P P P P P = PP. P P s the pobablty of selectg the Estmate of Vaace The estmate of vaace s gve by ( ) ( ). ˆ Va YHT ( ) 30

31 Mduo system of samplg: Ude ths system of selecto of pobabltes, the ut the fst daw s selected wth uequal pobabltes of selecto (.e., pps) ad emag all the uts ae selected wth SRSWOR at all subsequet daws. Ude ths system E( ) P ( ut ( U ) s cluded thesample) P( U s cluded st daw ) + P( U s cluded ay othe daw) Pobablty that U s ot selected at the fst daw ad P s selected at ay of subsequet ( -) daws P ( P) P. Smlaly, E( ) Pobablty that both the uts U ad U ae the sample Pobablty that U s selected at the fst daw ad U s selected at ay of the subsequet daws ( ) daws Pobablty that U s selected at the fst daw ad U s selected at ay of the subsequet ( ) daws Pobablty that ethe U o U s selected at the fst daw but both of them ae selected dug the subsequet ( ) daws ( )( ) P P ( P P) ( )( ) ( ) ( P P) ( ) ( P P ). Smlaly, E( ) Pobablty of cludg U, U ad U the sample k k k ( )( ) 3 ( P P Pk). ( )( ) 3 3 3

32 By a exteso of ths agumet, f U, U,..., U ae the uts the sample of se ( ), the pobablty of cludg these uts the sample s ( )( )...( ) E(... )... ( P P... P) ( )( )...( ) Smlaly, f U, U,..., U q be the uts, the pobablty of cludg these uts the sample s ( )( )... E(... q)... q ( P P... Pq) ( )( )...( ) ( P P... Pq) whch s obtaed by substtutg. Thus f P' s ae popotoal to some measue of se of uts the populato the the pobablty of selectg a specfed sample s popotoal to the total measue of the se of uts cluded the sample. Substtutg these,, k etc. the HT estmato, we ca obta the estmato of populato s mea ad vaace. I patcula, a ubased estmate of vaace of HT estmato gve by whee ˆ Va( YHT ) ( ) ( ) PP ( ). P P ( ) The ma advatage of ths method of samplg s that t s possble to compute a set of evsed pobabltes of selecto such that the cluso pobabltes esultg fom the evsed pobabltes ae popotoal to the tal pobabltes of selecto. It s desable to do so sce the tal pobabltes ca be chose popotoal to some measue of se. 3