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 s the vaable ude study ad X s a auxlay vaable elated to, the the most commoly used vayg pobablty scheme, the uts ae selected wth pobablty popotoal to the value of X, called as sze. Ths s temed as pobablty popotoal to a gve measue of sze (pps) samplg. If the samplg uts vay cosdeably sze, 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 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 sze cosdeed s the value of auxlay vaable X ad ot the value of study vaable. 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 sze of auxlay vaable. Also, the cultvated aea fo a pevous peod ca also be take as the sze whle estmatg the yeld of cop. Smlaly, a dustal suvey, the umbe of wokes a factoy ca be cosdeed as the measue of sze 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 stage by takg the pobabltes of selecto to accout, the t s possble to obta ubased estmatos. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

2 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 sze 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 sze 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: Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

3 Uts Sze 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 sze. 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 szes 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 sze, 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 sze. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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 sze sample. X. So ths method geeates a pps Advatage:. It does ot eque wtg dow all cumulatve totals fo each ut.. Szes of all the uts eed ot be kow befoe had. We eed oly some umbe geate tha the maxmum sze ad the szes 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 Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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 szes (X) ( thousads) ( metc tos) () 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 64.. Secod daw: - Suppose t s 3 -T 4 < 3 < T 5 - Ut s selected ad 5 8 etes the sample. - Daw a adom umbe betwee ad 64 - Suppose t s 38 - T7 < 38 < T8 - Ut 8 s selected ad 8 7 etes the sample - ad so o. - Ths pocedue s epeated tll the sample of equed sze s obtaed. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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 4 7 7> 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 : value of study vaable fo the th ut of the populato,,,,. X : kow value of auxlay vaable (sze) fo the th ut of the populato. P : pobablty of selecto of th ut the populato at ay gve daw ad s popotoal to sze X. Cosde the vayg pobablty scheme ad wth eplacemet fo a sample of sze. Let value of th obsevato o study vaable the sample ad Defe the y z,,,...,, p z z y be the p be ts tal pobablty of selecto. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 6

7 s a ubased estmato of populato mea, vaace of z s σ z whee σ z P P ad a ubased estmate of vaace of z s Poof: sz ( z z). ote that z ca take ay oe of the values out of Z, Z,..., Z wth coespodg tal pobabltes P, P,..., P, espectvely. So E( z) ZP P P. Thus Ez ( ) Ez ( ). So z s a ubased estmato of populato mea. The vaace of z s Va( z ) Va z ow Va z [ ] Va( z ) E z E( z ) E z ( ) Z P σ z P (say). ' ( ) ( zs ae depedet WR case). P Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 7

8 Thus Va( z ) σ z. σ z To show that s z s a ubased estmato of vaace of z, cosde ( ) Es ( z) E ( z z) o Es ( z ) E z z Ez ( ) Ez ( ) Va( z) { E( z )} Va( z ) { E( z )} z usgva( z) P σz P ( ) σ z + + σ z ( σ + ) ( + ) z s z σ z E σ Va( z ) s z y Va( z ) z. ( ) p ote: If P, the z y, σ Va( z ). whch s the same as the case of SRSWR. y Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 8

9 Estmato of populato total: A estmate of populato total s ˆ y tot z. p. Takg expectato, we get ˆ E ( tot ) P+ P P P P P tot. tot Thus ˆtot s a ubased estmato of populato total. Its vaace s Va ˆ ( tot ) Va( z ) P P P tot P tot P. A estmate of the vaace ˆ sz Va( tot ). 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. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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 - U + s selected at daw ad U s selected at daw 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. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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 p y estmato of. I geeal, ( ) wll ot povde a ubased estmato of. p wll povde a ubased 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 geealze 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 z p z y y + p / ( p) y + y z+ z z. ( p ) p Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

12 p ote that p s the pobablty PU ( () U () ). Estmato of Populato Mea: Fst we show that z s a ubased estmato of. Ez ( ). ote that Cosde P. y y p p P P P ( ) E ote that ca take ay oe of out of the values,,..., Ez P+ P P P P P Ez ( ) E y+ y ( p ) p ( P ) E( y ) + E E y U() ( Usg E ( ) EX[ E( 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) P wth pobablty P, P so ( P) y P E y U P E U P * () ( ) () ( ). p p P P. whee the summato s take ove all the values of except the value y whch s selected at the fst daw. So ( P ) * E y U() tot y. p Substtutg t Ez ( ), we have Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

13 Thus Ez ( ) Ey ( ) + E ( tot y) [ ] + [ E( y) E ( y) ] tot tot E ( tot ). Ez ( ) + Ez ( ) Ez ( ) +. Vaace: The vaace of z fo the case of two daws s gve as Va( z ) P P P 4 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 z s [ ] Vaz Ez Ez ( ) ( ) ( ) y y( p) E + y + p p y ( + p ) y ( p ) E + 4 p p atue of atue of vaable vaable depeds depeds st oly o upo ad st d daw daw Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 3

14 ( + ) ( ) P P PP 4 P P P ( ) ( ) ( ) + P PP P PP P PP P P P P PP P 4 Usg the popety + ( + P) ( ) P P P P P P P ab a b b, we ca wte ( P). ( ) + P Va( z ) + ( ) + ( + )( )] P P P P P 4 P( P) P P (+ P + ) + ( P P P) + ( + P)( ) 4 P P P P P P P P P + P + P + P ] P tot totp P P P + tot tot 4 P 4 tot totp + 4 P P tot totp tot tot P ( + 4 ) 4 + P P tot P tot ( totp + tot tot + P tot ) P 4 + P 4 P P tot ( totp P tot ) P P tot P tot P 4 P P P tot P tot P 4 P 4 P ` Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 4

15 tot tot P P P Va( z ) P P P 4 4 vaace of WR educto of vaace case fo WR wth vayg pobablty Estmato of Va( z ) Vaz ( ) Ez ( ) ( Ez ( )) Ez ( ) Sce Ezz ( ) EzEz ( u) [ ] E z E( z). Cosde E z zz E( z ) E( zz ) Ez ( ) Va( z ) Va( z ) z z z s a ubased estmato of Va( z ) Alteatve fom Va( z ) z z z z+ z ( z z) 4 zz y y y p 4 p p y y( p) ( p ) 4 p p ( p ) y y 4 p p. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 5

16 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. () () ( ) ( ) Futhe, let y z p y z y+ y y + ( p... p ) fo,3,...,. p Cosde z z as a estmato of populato mea. We aleady have show case that Ez ( ). ow we cosde Ez ( ),,3,...,. We ca wte E( z ) EE z 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 Sce codtoally ca take ay oe of the -( -) values,,,..., wth pobabltes p P P, so P P... P () () ( ) Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 6

17 whee y P (... ) (... ) * E p p E P P P. p () () ( ) P ( P P... P ) () () ( ) * E * 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. Thus ow we ca expess y E( z ) EE y+ y y + ( p... p ) p * E () ()... ( ) E () () ( ) ( (), (),..., ( )) { tot ( )} E () () ( ) () () ( ) E tot tot fo all,,...,. + The ( ) E( z ) E z. Thus z s a ubased estmato of populato mea. The expesso fo vaace of z geeal case s complex but ts estmate s smple. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 7

18 Estmate of vaace: Va( z ) E( z ). Cosde fo < s, [ ] Ezz ( ) EzEz ( U, U,..., U ) s s s E z E( z ) because fo < s, z wll ot cotbute ad smlaly fo s< z wll ot cotbute the expectato., s Futhe, fo s<, Cosde Substtutg [ ] Ezz ( ) EzEz ( U, U,..., U ) s s E z s E( z ). s E zz s Ezz ( s) ( ) ( s) s ( ) ( s) s ( ) ( ). Va( z ), we get Va( z ) E( z ) Ez ( ) E zz s ( ) ( s) s Va( z ) z zzs ( ) ( s) s z z zz s ( s) s zz s z z, ( s) s Usg + Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 8

19 The expesso of Va ( z ) ca be futhe smplfed as Va( z ) z z z ( ) ( ) z z ( ) ( z z). 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 sze, thee ae Coespodg to ay uodeed sample(s) of sze 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 ). uodeed sample(s) of sze. 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 uodeedsample 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), Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 9

20 Let zs, s,,..,,,,...,!( M) be a estmato of populato paamete θ 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 sze wth uts deoted as,,,, the samples of sze ae tuples. I the th daw, the sample space wll cosst of ( )...( + ) uodeed sample pots. p so P[ selecto of ay odeed sample] ( )...( + )! selecto of ay psu P[ selecto of ay uodeed sample ] P! ( )...( + ) odeed sample M(!)!( )! the p s p so.! ˆ M Theoem: If θ ˆ 0 z,,,..., ;,,..., (!) ad s s M θ u zs ps ae the odeed ad uodeed estmatos of θ epectvely, the () E( ˆ θ ) ( ˆ u E θ0) () Va( ˆ θ ) Va( ˆ θ0) u whee z s s a fucto of th s odeed sample (hece a adom vaable) ad p s s the pobablty of selecto of th p s odeed sample ad p s s p. s Poof: Total umbe of odeed sample! () E( ˆ θ ) 0 s M ( ˆ E θ ) z p p u s s s s p s zs p s ps s E( ˆ θ ) M 0 z p s s z p s s s Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 0

21 () Sce ˆ θ 0 z s, so ˆ θ 0 z s wth pobablty ps,,,..., M, s,,...,. Smlaly, M M ˆ ˆ θ, so u zs ps θ u zs ps wth pobablty p s Cosde Va( ˆ θ ˆ ˆ 0) E( θ0) ( 0) E θ z ( ˆ 0) s ps s E θ ˆ ˆ ˆ u u u Va( θ ) E( θ ) ( ) E θ ( ˆ z 0) s ps ps E θ s ˆ θ u s s s s s s s Va( ˆ θ0) Va( ) z p z p p 0 z + s ps zs ps ps s s z p z p p s z + s ps zs ps ps zs ps zs ps ps s s s s s s z + s ps zs ps ps zs ps zs ps s Va( ˆ θ ) Va( ˆ θ ) 0 o Va( ˆ θ ) Va( ˆ θ ) u ( zs zs ps ) ps 0 s u 0 Estmate of Va ˆ θ Sce ( ) ˆ ˆ Va( θ0) Va( θ ) ( z z p ) p u u s s s s s ˆ ˆ Va( θ ) ( 0) ( u Va θ zs zs ps ) ps s ˆ p Va( θ ) p ( z z p ). s 0 s s s s Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

22 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. Muthy s uodeed estmato coespodg to Des Ra s odeed estmato fo the sample sze 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 sze 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 z( s ) ( + p) + ( p) p p whee the coespodg Des Ra estmato s gve by ad y y( p) y + + p p * y y z( s) ( + p) + ( p) p p whee the coespodg Des Ra estmato s gve by y y( p) y + +. p p The pobabltes coespodg to z( s ) * ad z( s ) ae * pp * ps ( ) p pp * ps ( ) p ps () ps ( ) + ps ( ) * * pp( p p) ( p )( p ) Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page

23 p * p'( s ) p p * p p'( s). p p Muthy s uodeed estmate zu ( ) coespodg to the Des Ra s odeed estmate s gve as z( u) z( s ) p'( s ) + z( s ) p'( s ) * * zs ( ) ps ( ) + zs ( ) 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,,..., ad P, P,..., P, espectvely. The y ad p ca take ay oe of the emag values out of,,..., ad P, P,..., P, espectvely,.e., all the values except the values take at the fst daw. ow Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 3

24 [ ( )] Ezu PP PP ( P) + ( P) + P P P P P P < PP PP ( P) + ( P) + P P P P P P < PP PP ( P) + ( P) + P P P P P P PP ( P) + ( P) P P ( P)( P) P P + P P Usg esult ab a b b, we have Ezu [ ( )] ( P P) + ( P P) P P ( P) + ( P) P P + +. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 4

25 Vaace: The vaace of zuca ( ) be foud as [ ( )] Va z u P P P P PP P P ( P P) P P ( P)( P) ( )( )( ) ( ) PP ( P P) ( P P) P P Usg the theoem that Va( ˆ θ ) ( ˆ u Va θ0) we get [ ] Va [ z u ] Va z ( u) ( ) * Va z s ad ( ) ( ) * Va z s Ubased estmato of V[ zu ( )] A ubased estmato of Va ( z u ) s Va [ z u ] ( p p)( p)( p) y y ( ). ( p p) p p Hovtz 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 sze s eve modeately lage. The HT estmate s smple tha othe estmatos. Let be the populato sze ad y, (,,..., ) be the value of chaactestc ude study ad a sample of sze 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 α (,,.., ) as f s cluded a sample ' s' of sze α 0 othewse. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 5

26 y Let z,... assumg E( α) > 0 fo all E( α ) whee E( α ). P ( s) + 0. P ( s) π s the pobablty of cludg the ut the sample ad s called as cluso pobablty. The HT estmato of based o y, y,..., y s ˆ z z HT α z. Ubasedess ˆ E ( ) Ez ( α ) HT ze( α ) y E( α ) E( α ) y whch shows that HT estmato s a ubased estmato of populato mea. Vaace Cosde V ( ˆ ) Vz ( ) HT [ Ez ] ( ) ( ) Ez ( ). Ez Ez ( ) E α z E α z + αα zz ( ) z ( ) ( ). Eα + zze αα ( ) Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 6

27 If S { s} the s the set of all possble samples ad π s pobablty of selecto of th ut the sample s E( α ) Py ( s) + 0. Py ( s). π + 0.( π ) π ( α ). ( ) + 0. ( ) π. E Py s Py s So E( α) E( α ) E( z ) z π + πzz (# ) whee π s the pobablty of cluso of th ad th ut the sample. Ths s called as secod ode cluso pobablty. ow [ Ez ( )] E α z z [ E( ) ] zze ( ) E( ) α + α α ( ) + zπ ππ zz ( ). Thus ˆ Va( HT ) πz + π zz ( ) + π z ππ zz ( ) + π ( π) z ( π ππ ) zz ( ) + y y y ( ) ( ) π π π ππ π ( ) ππ π π ππ + y yy π ( ) ππ Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 7

28 Estmate of vaace ˆ ˆ y ( π ) π ππ yy V Va( HT ). + π ( ) π ππ Ths s a ubased estmato of vaace. Dawback: It does ot educes to zeo whe all y π ae same,.e., whe y π. Cosequetly, ths may assume egatve values fo some samples. A moe elegat expesso fo the vaace of y ˆHT has bee obtaed by ates ad Gudy. ates ad Gudy fom of vaace Sce thee ae exactly values of α. Takg expectato o both sdes α whch ae ad ( ) values whch ae zeo, so Also E( α ). α α αα ( ) E E( ) + E( ) ( ) α + αα J α α ( ) E E( ) E( ) (usg E( ) E( )) E + ( ) ( ) ( αα ) E( αα ) ( ) J J Thus E( αα ) P( α, α ) P( α ) P( α α ) E( α ) E( α α ) Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 8

29 Theefoe Smlaly ( ) E( αα ) E( α) E( α ) E( α) E( α α ) E( α) E( α ) ( ) E( α ) E( α α ) E( α ) ( ) [ α ] [ E α ] E( α ) ( ) ( E( ) E( α ) ( ) π ( π ) () ( ) E( αα ) E( α) E( α ) π ( π ). () We had eale deved the vaace of HT estmato as ˆ Va( HT ) ( ) ( ) π π z + π ππ zz ( ) Usg () ad () ths expesso, we get ˆ Va( HT ) ( ) ( ) ( ) π π z + π π z ππ π z z E( αα ) E( α) E( α) z ( ) E( αα ) E( α) E( α) z { E( α) E( α) E( αα ) } zz ( ) ( ) ( π ππ ) z ( π ππ ) z ( π ππ ) zz ( ) ( ) ( ) + ( ππ π )( z z zz ). ( ) The expesso fo π ad π ca be wtte fo ay gve sample sze. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 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( α ) Pobablty of selectg a sample of two P + P whee P s the pobablty of selectg at th daw (, ). If P th ut at fst daw (,,..., ) the we had eale deved that So P P d y s ot selected y s selected at daw P P P st at daw y st s ot selected at daw PP P ( ) P P P. P P E( α ) P P P + P P P Aga E( αα ) Pobablty of cludg both y ad y a sample of sze two PP + PP P P P + P P P PP + + P. P P s the pobablty of selectg the Estmate of Vaace The estmate of vaace s gve by ππ π ( ) ( ). ˆ Va HT z z ( ) π Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 30

31 Mdzuo 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( α ) π P ( ut ( U ) s cluded the sample) 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( αα ) 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) + ( ) π ( P + P ) +. Smlaly, E( αα α ) π Pobablty of cludg U, U ad U the sample k k k ( )( ) 3 ( P + P + Pk) +. ( )( ) 3 3 Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 3

32 By a exteso of ths agumet, f U, U,..., U ae the uts the sample of sze ( < ), the pobablty of cludg these uts the sample s ( )( )...( + ) E( αα... α ) π... ( P + P P ) + ( )( )...( + ) Smlaly, f U, U,..., U q be the uts, the pobablty of cludg these uts the sample s ( )( )... E( αα... αq ) π... q ( P + P Pq ) ( )( )...( + ) ( P + P Pq) whch s obtaed by substtutg. Thus f P' s ae popotoal to some measue of sze of uts the populato the the pobablty of selectg a specfed sample s popotoal to the total measue of the sze of uts cluded the sample. Substtutg these π, π, π 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 ππ π ˆ Va( HT ) ( ) z z π ππ π ( ) 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 sze. Samplg Theoy Chapte 7 Vayg Pobablty Samplg Shalabh, IIT Kapu Page 3