Chapter 8 Double Sampling (Two Phase Sampling)
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- Beryl Prudence Spencer
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1 Chapter 8 Double Samplg (Two Phase Samplg The rato ad regresso methods of estmato requre the kowledge of populato mea of aular varable ( to estmate the populato mea of stud varable (. If formato o the aular varable s ot avalable, the there are two os oe o s to collect a sample ol o stud varable ad use sample mea as a estmator of populato mea. A alteratve soluto s to use a part of the budget for collectg formato o aular varable to collect a large prelmar sample whch aloe s measured. The purpose of ths samplg s to fursh a good estmate of. Ths method s approprate whe the formato about s o fle cards that have ot bee tabulated. After collectg a large prelmar sample of sze uts from the populato, select a smaller sample of sze from t ad collect the formato o. These two estmates are the used to obta a estmator of populato mea. Ths procedure of selectg a large sample for collectg formato o aular varable ad the selectg a sub-sample from t for collectg the formato o the stud varable s called double samplg or two phase samplg. It s useful whe t s cosderabl cheaper ad qucker to collect data o tha ad there s hgh correlato betwee ad. I ths samplg, the radomzato s doe twce. Frst a radom sample of sze s draw from a populato of sze N ad the aga a radom sample of sze s draw from the frst sample of sze. So the sample mea ths samplg s a fucto of the two phases of samplg. If SRSWOR s utlzed to draw the samples at both the phases, the - umber of possble samples at the frst phase whe a sample of sze s draw from a N populato of sze N s M, sa. - umber of possble samples at the secod phase where a sample of sze s draw from the frst phase sample of sze s M, sa.
2 Populato of (N uts Sample (Large uts M samples Subsample (small uts M samples The the sample mea s a fucto of two varables. If s the statstc calculated at the secod phase such that j,,,..., M, j,,..., M wth Pj beg the probablt that th sample s chose at frst phase ad j th E( E E ( sample s chose at secod phase, the where E ( deotes the epectato over secod phase ad E deotes the epectato over the frst phase. Thus E( M M j M M j M M Pj/ j j st stage d stage P P j j PP (usg P( A B P( A P( B/ A j/ j
3 Varace of Var( E E( ( ( ( ( ( E E E E E E ( E ( E( EE E( ] [ E( E( ( ( ( EE E EE E E ( ( ( ( ( V E ( E V E E E E E V costat for E Note: The two phase samplg ca be eteded to more tha two phases depedg upo the eed ad objectve of the epermet. Varous epectatos ca also be eteded o the smlar les. Double samplg rato method of estmato If the populato mea tal sample of sze s ot kow the double samplg techque s appled. Take a large b SRSWOR to estmate the populato mea as. The a secod sample s a subsample of sze selected from the tal sample b SRSWOR. Let ad be the meas of ad based o the subsample. The E(, E(, E(. The rato estmator uder double samplg ow becomes. The eact epressos for the bas ad mea squared error of are dffcult to derve. So we fd ther appromate epressos usg the same approach metoed whle descrbg the rato method of estmato. 3
4 Let,, E( E( E( E( C N E( E( ( E E( ( E ( S N C N E(. where E( Cov(, Cov E E E Cov Cov, ECov(, Cov(, S N S S N CC N s the sample mea of (, ( (, s based o the sample sze. 4
5 E( Cov(, S N S S N CC N E( Var( V E E V V ( E s S N C N ( ( S S N where s s the mea sum of squares of based o tal sample of sze. E( Cov(, Cov E(, E( Var( where Var( s the varace of mea of based o tal sample of sze. 5
6 Estmato error of Wrte as ( ( ( ( ( ( ( ( (... ( o upto the terms of order two. Other terms of degree greater tha two are assumed to be eglgble. Bas of E ( E( E( E( E( Bas( ( E E( E( E( E( CC CC C C N N N N C C C C( C C. The bas s eglgble f s large ad relatve bas vashes f passes through org. MSE of : MSE( E( ` E( ( retag the terms upto order two 6 C C.e., the regresso le, E E C C C CC CC N N N N N C C CC C( C C N N MSE (rato estmator CC C.
7 The secod term s the cotrbuto of secod phase of samplg. Ths method s preferred over rato method f or CC C C C Choce of ad Wrte V V MSE( where V ad V cota all the terms cotag ad respectvel. The cost fucto s C C C where C ad C are the costs per ut for selectg the samples ad respectvel. Now we fd the mum sample szes ad for fed cost C. The Lagraga fucto s V V ( C C C V C V C. Thus C V or V C or C VC. Smlarl C V C. Thus ad so VC V C C 7
8 C Optmum VC Optmum, sa Var ( V C VC V V V C C V C ( VC V C C V C, sa Comparso wth SRS C If s gored ad all resources are used to estmate b, the requred sample sze =. C S CS Var( C / C C Var( CS Relatve effec = Var ( ( VC V C Double samplg regresso method of estmato Whe the populato mea of aular varable s ot kow, the double samplg s used as follows: - A large sample of sze s take from of the populato b SRSWOR from whch the populato mea s estmated as,.e.. - The a subsample of sze s chose from the larger sample ad both the varables ad are measured from t b takg place of ad treat t as f t s kow. The E(, E(, E(. The regresso estmate of ths case s gve b ( ( ( where s S s a estmator of s S ( based o the sample of sze. 8
9 It s dffcult to fd the eact propertes lke bas ad mea squared error of, so we derve the appromate epressos. Let ( ( s S 3 s ( 3 S S s S 4 s ( 4 S S E(, E(, E( 3, E( 4 Defe E ( ( 3 E 3 Estmato error: The ( S S ( 3 ( ( 4 S ( ( ( S 3 4 ( ( ( Retag the powers of s upto order two assumg 3, (usg the same cocept as detaled the case of rato method of estmato (
10 Bas: The bas of upto the secod order of appromato s E ( E( E( E( E( Bas( E( ( ( s S N N S ( ( s S N N S ( ( s S N N S ( ( s S N N S N S N S N S N S 3. S S 3 3 Mea squared error: MSE( E( Retag the powers of appromato s ( E ( ( 3( ( s upto order two, the mea squared error upto the secod order of
11 Clearl, MSE( E ( ( E( E( E[( ( ] S S S Var( N N N S S N N Var( S S Var( S S S S Var( S 4 S S S S S N S S S (usg S SS N ( S S. (Igorg the fte populato correcto s more effcet tha sample mea SRS,.e. whe o aular varable s used. Now we address the ssue that whether the reducto varablt s worth the etra epedture requred to observe the aular varable. Let the al cost of surve s C C C where C ad C are the costs per ut observg the stud varable ad aular varable respectvel. Now mmze the MSE ( for fed cost C usg Lagraga fucto wth Lagraaga multpler as
12 S( S ( C C C S ( C S C Thus S ( C ad S. C Substtutg these values the cost fucto, we have C CC S( S C C C C or C CS( C S or S C( S C. C Thus the mum values of ad are SC C S C( S C CS. C S C( S C The mum mea squared error of MSE( s obtaed b substtutg S ( C C S ( S C C S ( S C S C( S C SC S C( S C C S C( C C ad as
13 The mum varace of uder SRS for SRS where o aular formato s used s Var( SRS CS C whch s obtaed b substtutg, C MSE( SRS. The relatve effcec s Var( SRS CS RE MSE( S C( C C C. Thus the double samplg regresso estmator wll lead to ga precso f C C. Double samplg for probablt proportoal to sze estmato: Suppose t s desred to select the sample wth probablt proportoal to aular varable but formato o s ot avalable. The, ths stuato, the double samplg ca be used. A tal sample of sze s selected wth SRSWOR from a populato of sze N, ad formato o s collected for ths sample. The a secod sample of sze s selected wth replacemet ad wth probablt proportoal to from the tal sample of sze tal sample of sze. Let deote the mea of for the, Let ad deote meas respectvel of ad for the secod sample of sze. The we have the followg theorem. Theorem: ( A ubased estmator of populato mea s gve as, where deotes the al for the frst sample. 3
14 ( N ( Var( S, where ad deote the als N N( N of ad respectvel the populato. (3 A ubased estmator of the varace of s gve b ( A B ( Var N ( ( ( where A ad B Proof. Before dervg the results, we frst meto the followg result proved varg probablt scheme samplg. Result: I samplg wth varg probablt scheme for drawg a sample of sze from a populato of sze N ad wth replacemet. ( z z s a ubased estmator of populato mea where z, p beg the Np probablt of selecto of th ut. Note that ad wth tal probabltes PP,,..., PN, respectvel. p ca take aoe of the N values,,..., N N N ( Var( z. N P N P N P. ( A ubased estmator of varace of z s Var( z z.. ( Np Let E deote the epectato of, whe the frst sample s fed. The secod s selected wth probablt proportoal to, hece usg the result ( wth P, we fd that 4
15 where E E E s the mea of for the frst sample. Hece E( E E E (, whch proves the part ( of the theorem. Further, S EV Var( VE EV V ( EV. N Now, usg the result (, we get ad hece V j j j j, N ( j EV j, N( N j j usg the probablt of a specfed par of uts beg selected the sample s epress (. N( N So we ca 5
16 N ( / EV. N( N Substtutg ths V, we get ( Var S N (. N N( N Ths proves the secod part ( of the theorem. We ow cosder the estmato of Var(. Gve the frst sample, we obta E p where p. Also, gve the frst sample,, Hece E V E ( (. ( p E. ( p Substtutg ad p the epresso becomes E ( Usg E p, we get 6
17 E A B ( ( where A, ad B whch further smplfes to where ( A B E s ( ( s s the mea sum of squares of for the frst sample. Thus, we obta ( A B EE E( s S ( ( (, whch gves a ubased estmator of S. Net, sce we have N ( EV, N( N ad from ths result we obta Thus E V. ( EE N ( ( N ( whe gves a ubased estmator of ( ( N( N N Usg ( ad ( a ubased estmator of the varace of. s obtaed as ( A B ( Var N ( ( ( Thus, the theorem s proved. 7
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