amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR
Estmato of populato mea ad populato varace Oe of the ma objectves after the selecto of a sample s to kow about the tedecy of the data to cluster aroud the cetral value ad the scatterdess of the data aroud the cetral value. Amog varous dcators of cetral tedecy ad dsperso, the popular choces are arthmetc mea ad varace. o the populato mea ad populato varablty are geerally measured by arthmetc mea (or weghted arthmetc mea) ad varace. There are varous popular estmators for estmatg the populato mea ad populato varace. Amog them, sample arthmetc mea ad sample varace are more popular tha other estmators. Oe of the reaso to use these estmators s that they possess ce statstcal propertes. Moreover, they are also obtaed through well establshed statstcal estmato procedures lke maxmum lkelhood estmato, least squares estmato, method of momets etc. uder several stadard statstcal dstrbutos. Oe may also cosder other dcators lke meda, mode, geometrc mea, harmoc mea for measurg the cetral tedecy ad mea devato, absolute devato, Ptma earess etc. for measurg the dsperso. The propertes of such estmators ca be studed by umercal procedures lke bootstrapg.
. Estmato of populato mea y = Let us cosder the sample arthmetc mea y = y ad verfy f s a ubased estmator of Y uder the two cases. as a estmator of populato mea Y = = Y R W O R E( y) = E y E( t) t ( where t y) = = = = = = =. y. = = Whe uts are sampled from uts by wthout replacemet, the each ut of the populato ca occur wth ( - ) other uts selected out of the remag ( - ) uts the populato ad each ut occurs so ow of the possble samples. o y = = = = y ( )!!( )! E( y) =. y = y = Y. ( )!( )!! = = Thus y s a ubased estmator of Y. 3
Alteratvely, the followg approach ca also be adopted to show that the sample mea s a ubased estmator of populato mea E( y) = E( yj ) j= = YP j() j= = = Y. j= = = Y = Y j= where Pj () deotes the probablty of selecto of th ut at j th stage. R W R E( y) = E y = = E( y) = ( YP +... + YP) = Y = Y = = = where P s the probablty of selecto of a ut. Thus s a ubased estmator of = for all =,,..., y populato mea uder RWR also. 4
Varace of the estmate Assume that each observato has same varace σ. The V( y) = E( y Y) = E ( y Y) = = E ( ) ( )( ) y Y + y Y yj Y = j = E( y Y) + E( y Y)( y Y) j = j = + σ = where = + K = E( y Y)( y Y). j K K j ow we fd the value of K uder the setups of RWR ad RWOR. 5
R W O R K = E( y Y)( y Y) Cosder ce j j E( y Y)( yj Y) = ( yk Y)( yl Y). ( ) k k k l k= k= k l k ( y Y) = ( y Y) + ( y Y)( y Y) l 0 = ( ) + ( yk Y)( yl Y) k l ( yk Y)( yl Y) = ( ) ( ) =. Thus K= ( ) k l 6
ad substtutg the value of K, the varace of y uder RWOR s V( ywor ) = ( ) =. R W R ow we obta the value of K uder RWR. K = E( y Y)( y Y) = E( y Y) E( y Y) = 0 j j j j because th ad j th draws ( j) are depedet. Thus the varace of y uder RWR s V y ( WR ) =. 7
It s to be oted that f s fte (large eough), the V( y) = both the cases of RWOR ad RWR. o the factor s resposble for chagg the varace of y whe the sample s draw from a fte populato comparso to a fte populato. Ths s why fte populato correcto (fpc). s called as It may be oted that =, so s close to f the rato of sample sze to populato sze s very small or eglgble. I such a case, the sze of the populato has o drect effect o the varace of y. The term s called as samplg fracto. I practce, fpc ca be gored wheever 5% < ad for may purposes eve f t s as hgh as 0%. Igorg fpc wll result the over estmato of varace of y. 8
Effcecy of y uder RWOR over RWR: ow we compare the varaces of sample meas uder RWOR ad RWR. V( y) V( y) WOR WR = = = + = V( y) + a postve quatty WOR V( y) > V( y) WR WOR ad so, RWOR s more effcet tha RWR. 9
Estmato of varace from a sample ce the expressos of varaces of sample mea volve whch s based o populato values, so these expressos ca ot be used real lfe applcatos. I order to estmate the varace of y o the bass of a sample, a estmator of (or equvaletly ) s eeded. Cosder s as a estmator of (or σ ) ad we vestgate ts basedess for the cases of RWOR ad RWR. σ Cosder s y y = ( ) = = ( y Y) ( y Y) = = ( ) ( ) y Y y Y = E( s ) = E( y Y) E( y Y) = = Var( y ) Var( y) = σ = Var( y) 0
I case of RWOR ad so Var( y) WOR = Es ( ) = σ = = I case of RWR ad so Var( y) WR = Es ( ) = σ = = = σ Hece Es ( ) = σ RWOR RWR
A ubased estmate of Var( y) Var ( y) WOR = s s case of RWOR. Var( y) WR =. s s = case of RWR.