Sampling Theory MODULE X LECTURE - 35 TWO STAGE SAMPLING (SUB SAMPLING)
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1 Samplg Theory ODULE X LECTURE - 35 TWO STAGE SAPLIG (SUB SAPLIG) DR SHALABH DEPARTET OF ATHEATICS AD STATISTICS IDIA ISTITUTE OF TECHOLOG KAPUR
2 Two stage samplg wth uequal frst stage uts: Cosder two stage samplg whe the frst stage uts are of uequal sze ad SRSWOR s employed at each stage Let y j : value of j th secod stage ut of the th frst stage ut umber of secod stage uts th : frst stage ut 0 : total umber of secod stage uts the populato m umber of secod stage uts to be selected from th : frst stage uts, f t s the sample m0 m : total umber of secod stage uts the sample y ( m ) m j m j y y j y j j j y u u
3 The pctoral scheme of two stage samplg wth uequal frst stage uts case s as follows: Populato ( uts) uts uts clusters uts Populato clusters uts uts clusters (small) uts Frst stage sample clusters (small) m uts m uts m uts Secod stage sample clusters (small) 3
4 ow we cosder dfferet estmators for estmato of populato mea Estmator based o frst stage ut meas the sample: Bas ˆ y y S ( m ) E( ys ) E y( m ) E E ( y( m )) E m So ys s a based estmator of ad ts bas s gve by [ Sce a sample of sze s selected out of uts by SRSWOR] Bas ( y ) E( y ) S S ( )( ) 4
5 Ths bas ca be estmated by Bas( ys ) ( m)( y( m) ys ) ( ) whch ca be see as follows: E Bas( y ) S E E {( m)( y( m) ys ) / } E ( m)( y) ( )( ) y A ubased estmator of populato mea s thus obtaed as y + ( m)( y y ) S ( m) S ote that the bas arses due to equalty of szes of the frst stage uts ad probablty of selecto of secod stage uts vares from oe frst stage to aother 5
6 Varace: [ ] [ ] Var( y ) E Var( y ) + Var E( y ) S S S Var y E Var( y ( m) ) + S + E S m b S + m ( ) Sb S y j j ( ) b S The SE ca be obtaed as [ ] SE( y ) Var( y ) + Bas( y ) S S S 6
7 Estmato of varace: Cosder mea square betwee cluster meas the sample It ca be show that s y y ( ) ( ) b m S Also Es ( ) S S b b + m s y y m ( j ( m) ) m j E( s ) S ( y ) j j So E s S m m Thus Es ( b) Sb + E s m ad a ubased estmator of S b s S s s ˆ b b m So a estmator of varace ca be obtaed by replacg ˆ ˆ Var( ys) Sb + S m S ad S b by ther ubased estmators as 7
8 Estmato based o frst stage ut totals: ˆ ys y uy ( m) ( m) u Bas E( ys ) E uy ( m) y S E ue ( y( m) ) E u u Thus s a ubased estmator of 8
9 Varace: Var( ys) Var E( ys ) + E Var( ys ) Var u E u ( Var y( m) ) + S + u S m S ( y ) j j S ( u ) b j b 9
10 3 Estmator based o rato estmator: ˆ y uy y ( m) ( m) S S u u y u, u u Ths estmator ca be see as f arsg by rato method of estmato as follows: Let y uy ( m) x,,,, be the values of study varable ad auxlary varable referece to rato method of estmato The y y y S x x u X X The correspodg rato estmator of ˆ y ys R X ys x u s 0
11 So the bas ad mea squared error of y S ca be obtaed drectly from the results of rato estmator Recall that rato method of estmato, the bas of rato estmator upto secod order of approxmato s Bas : The bas of ˆ Bas ( yr) ( Cx ρcxcy) Var( x) Cov( x, y) X X ˆ SE ( R ) Var( y) + R Var( x) RCov( x, y) R X y S up to secod order of approxmato s Var( xs) Cov( xs, ys) Bas( ys ) X X x S y S s the mea of auxlary varable smlar to as x S x ( m) ow we fd Cov( xs, ys) Cov( xs, ys ) Cov E ux ( m), u y( m) E Cov ux ( m), u y( m) + Cov ue( x ( m) ), ue( y( m) ) E u ( Cov x ( m), y( m) ) + Cov u X, u E u S + m xy S + u S m bxy xy
12 S ( u X X)( u ) bxy S ( x X )( y ) xy j j j Smlarly, Var( x S ) ca be obtaed by replacg x place of y Cov( xs, ys) as Var( x ) S u S S bx + x m S ux X bx ( ) S ( x X ) x j Substtutg Cov( x, y ) ad Var( x S ) Bas( y S ), we obta the approxmate bas as S S S S bxy bx S S x xy Bas( ys) + u X X m X X
13 ea squared error SE( y ) Var( y ) R Cov( x, y ) + R Var( x ) S S S S S Var( y ) S u S S by + y m Var( x ) S u S S bx + x m Cov( x, y ) S u S S S bxy + xy m S u by ( ) S y R y ( j ) j X Thus Also SE( ys ) ( Sby RSbxy + R Sbx ) + u ( Sy RSxy+ R Sx ) m SE( ys ) u ( R X ) + u ( Sy R Sxy + R Sx ) m 3
14 Estmate of varace Cosder s uy y ux x It ca be show that ( ( ) )( ( ) ) bxy m S m S s x x y y ( ( ))( ( )) xy j m j m m j So E( s ) S u S bxy bxy + xy m Es ( ) S xy xy E u sxy u Sxy m m Thus Sˆ s u s bxy bxy xy m Sˆ s u s bx bx x m S s u s ˆ by by y m 4
15 Also E u sx u Sx m m E u sy u Sy m m A cosstet estmator of SE of SE( y S ) as y S ca be obtaed by substtutg the ubased estmators of respectve statstcs m SE( ys ) ( sby rsbxy + r sbx ) + u ( sy rsxy+ r sx ) + + m ( y( m) r x ( m) ) u ( sy rsxy r sx ) r y S xs 5
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