Sampling Theory MODULE V LECTURE - 17 RATIO AND PRODUCT METHODS OF ESTIMATION

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1 Samplng Theory MODULE V LECTURE - 7 RATIO AND PRODUCT METHODS OF ESTIMATION DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPUR

2 Propertes of separate rato estmator: Note that there s an analogy between w and w R. We already have derved the bas of E f C C C n ( R + ( ρ. R y as So for R, we can wrte f E C C C ( R + ( ρ n N N where y, j j N j N j S f, C, C, N n y S y N S (, S (, N N y j j N j N j ρ C : : correlaton coeffcent between the observaton on and n th stratum coeffcent of varaton of values n th sample.

3 Thus E ( we ( R f w + ( C ρ CC w f + C ρ CC y n ( y n Bas( E( w f C ( C ρ C y. n Assumng fnte populaton correcton to be appromately, C, C and ρ respectvely, we have y n n/ and C, C and ρ y are same for the strata as Bas( ( C ρccy. n Thus the bas s neglgble when the sample sze wthn each stratum should be suffcently large and C ρ C. y s unbased when 3

4 Now we derve the MSE of. We already have derved the MSE of earler as f R ρ y MSE( ( C C C C n N f ( R nn ( where R. Thus for th stratum f MSE C C C C ( R ( ρ n( N N f ( j R j n ( N R and so MSE ( ( w MSE R w f ( C + C ρ C C n N f w ( j R j. n( N j 4

5 MSE( An estmate of can be found by substtutng the unbased estmators of S, S and S as s, s and s respectvely for th stratum and R / can be estmated by r y /. y y Also w f MSE( ( sy + r s rs y. n n w f MSE( ( yj r j. n( n j 5

6 Propertes of combned rato estmator: Here wy y R st RC c. w st It s dffcult to fnd the eact epresson of bas and mean squared error of Defne, so we fnd ther appromate epressons. ε ε E( ε 0 E( ε 0 E( ε E( ε y st st Nn n n w f S E( εε. n N n ws f ws f ws 6

7 Thus assumng RC ε <, ( + ε ( + ε ( + ε ( ε + ε... ( + ε ε εε + ε... Retanng the terms upto order two due to same reason as n the case of R, RC ( + ε ε εε + ε ( ε ε εε + ε. RC The appromate bas of Bas( E( upto second order of appromaton s E( ε ε εε + ε 0 0 E( εε + E( ε f S S n w f S ρ S S n w f S ρ S ws n f R w S C ρc n ( 7

8 where R ρ s the correlaton coeffcent between the observatons on and n the th stratum, C and C are the, coeffcents of varaton of and respectvely n th stratum. The mean squared error up to second order of appromaton s y MSE ( E ( E( ε ε εε + ε E( ε + ε εε f S S + S w n f S S + S S w ρ n f + n w S S ρ S S f w ( R S + S ρ RS S. n 8

9 An estmate of MSE( can be obtaned by replacng S by ther unbased estmators, S and S y respectvely whereas R s replaced by r as follows: Thus the followng estmate s obtaned: s, s and s y y w f s sy s y MSE( y + n + w f n ( r s sy rsy where s nown. 9

10 Comparson of combned and separate rato estmators An obvous queston arses that whch of the estmates or s better. So we compare ther MSEs. Note that the only dfference n the term of these MSEs s due to the form of rato estmate. It s y * R n ( MSE * R n MSE(. Thus MSE( MSE( w f ( R R S + ( R R ρs S n w f ( R R S + ( R R ( RS ρs S. n 0

11 The dfference depends on. The magntude of the dfference between the strata ratos ( and whole populaton rato (R.. The value of ( RS ρ S S s usually small and vanshes when the regresson lne of y on s lnear and y passes through orgn wthn each stratum. In such a case R but MSE( > MSE( Bas( < Bas(. So unless vares consderably, the use of would provde an estmate of wth neglgble bas and precson as good as. R If R R, can be more precse but bas may be large. If R R, can be as precse as but ts bas wll be small. It also does not requre nowledge of,,...,.