RATIO-TO-REGRESSION ESTIMATOR IN SUCCESSIVE SAMPLING USING ONE AUXILIARY VARIABLE

Transcription

1 SAISICS IN RANSIION ew series, Suer 5 83 SAISICS IN RANSIION ew series, Suer 5 Vol. 6, No., pp. 83 RAIO-O-REGRESSION ESIMAOR IN SUCCESSIVE SAMPLING USING ONE AUXILIARY VARIABLE Zorathaga Ralte, Gitasree Das ABSRAC he proble of estiatio of fiite populatio ea o the curret occasio based o the saples selected over two occasios has bee cosidered. I this paper, first a chai ratio-to-regressio estiator was proposed to estiate the populatio ea o the curret occasio i two-occasio successive (rotatio) saplig usig ol the atched part ad oe auxiliar variable, which is available i both the occasios. he bias ad ea square error of the proposed estiator is obtaied. We proposed aother estiator, which is a liear cobiatio of the eas of the atched ad uatched portio of the saple o the secod occasio. he bias ad ea square error of this cobied estiator is also obtaied. he iu ea square error of this cobied estiator was copared with (i) the iu ea square error of the estiator proposed b Sigh (5) (ii) ea per uit estiator ad (iii) cobied estiator suggested b Cochra (977) whe o auxiliar iforatio is used o a occasio. Coparisos are ade both aalticall as well as epiricall b usig real life data. Ke words: ratio-to-regressio estiator, auxiliar variable, successive saplig, bias, ea square error, iu replaceet polic.. Itroductio he successive ethod of saplig cosists i selectig saples of the sae size o differet occasios such that soe uits are coo to saples selected o previous occasios. I successive saplig, the ratio estiator is aog the ost cool aded estiators of the populatio ea or total of soe variables of iterest of a fiite populatio with the help of a auxiliar variable whe the correlatio coefficiet betwee the two variables is positive. Patterso (95) ad Cochra (977) suggested a uber of estiatio procedures o Departet of Statistics, North-Easter Hill Uiversit, Shillog. Idia. E-ail: araaralte7@gail.co. Departet of Statistics, North-Easter Hill Uiversit, Shillog. Idia. E-ail: gitasree@gail.co.

2 84 Z. Ralte, G. Das: Ratio-to-regressio estiator i saplig over two occasios. Rao ad Graha (964), Gupta (979), Das (98), Se (97) developed estiators for the populatio ea o the curret occasio usig iforatio o two auxiliar variables available o the previous occasio. Se (97, 973) exteded his work for several auxiliar variates. Sigh et al. (99) ad Sigh ad Sigh () used the auxiliar iforatio o curret occasio for estiatig the curret populatio ea i two occasio successive saplig. Sigh (3) exteded their work for h-occasios successive saplig. Utilizig the auxiliar iforatio o both the occasios, Sigh (5), Sigh ad Priaka (6, 7a, 8) proposed varieties of chai-tpe ratio, differece ad regressio estiators for estiatig the populatio ea o the curret (secod) occasio i two occasio successive saplig. Sigh (5) suggested two estiators for populatio ea usig the iforatio o a auxiliar variable i successive saplig over two occasios. Cosider a character uder stud o the first (secod) occasio is deoted b x() respectivel ad the auxiliar variable z is available o both the occasios. A siple rado saple (without replaceet) of uits is take o the first occasio. he suggested chai-tpe ratio estiator based o the saple of size coo to both the occasios is give b x Z () x z he bias B(.) ad ea square error M(.) of up to the first order of approxiatio ad for large populatio of size N of equatio () are derived as ad x x x z z z B Y C C C C C C C M Y C C C C C C x x x z z z A classical ratio estiator based o a saple of size u (fractio of a saple) draw afresh o the secod occasio is give b u Z () u Cobiig ad, the resultig estiator of Y is z where is a ukow costat to be deteried uder certai iu criterio. After iizig ad (saplig fractio), the iu ea square error of the estiator is give b

3 SAISICS IN RANSIION ew series, Suer 5 85 where M A Y C A C A CB C A C B C, B Y C x xcxc, C Y C z zccz z z x z x ad Followig the chai-tpe ratio estiator proposed b Sigh (5), i the preset work we propose a chai ratio-to-regressio estiator for estiatig populatio ea o the curret occasio usig auxiliar iforatio that is available o both the occasios. he behaviour of the proposed estiator have bee exaied aalticall ad also through epirical eas of copariso.. Proposed estiator Cosider a populatio cosistig of N uits. Let a character uder stud o the first (secod) occasio be deoted b x(), respectivel. It is assued that the iforatio o a auxiliar variable z is available o the first as well as o the secod occasio. We cosider the populatio to be large eough, ad the saple size is costat o each occasio. Usig siple rado saplig without replaceet (SRSWOR) we select a saple of size o the first occasio. Of these uits, a sub-saple of size is retaied o the secod occasio. his subsaple is suppleeted b selectig SRSWOR of u ( ) uits afresh fro the uits that were ot selected o the first occasio. B odifig the estiator i (), we propose a chai ratio-to-regressio estiator for Y o the secod occasio which is based o a saple of size coo to both the occasios ad is give b where p x b z z x Z z (3) Z : Populatio ea of z. x,,z,z : Saple eas of the respective variates with saple sizes as show i the subscript. b : Regressio coefficiet of x o z.

4 86 Z. Ralte, G. Das: Ratio-to-regressio estiator i he bias of the proposed estiator p up to the secod order of approxiatio as obtaied i Appedix A is: Bias( p) = z z Cov z,s Cov z,s Cov z,s Cov z,s β Y Sz S Sz S Y +β Cov,z -Cov,z - Var x X X Y XY -β Covx,z -Covx,z -XCov,z + Var z X Z (4) Ad the ea square error of p cosiderig E( p ) up to the first order of approxiatio obtaied i Appedix B is: C M Y C C C C C C p x z x z z z (5) Followig Sigh (5), we cobie the estiators ad p ad the fial estiator of Y o the secod occasio is give as p p (6) where is a ukow costat to be deteried uder certai criterio. 3. Bias ad ea square error of a cobied estiator Sice both ad p are biased estiators of Y, therefore, the resultig estiator is also a biased estiator of Y. Usig the otatios i Sectio, the bias B ( ) ad ea square error M ( ) up to the first order of approxiatio are give below: p p B B B (7) ad Mp M Mp (8)

5 SAISICS IN RANSIION ew series, Suer 5 87 where, fro Cochra (977) Y z z z B C C C (9) u Y ad M C Cz zccz 4. Miiu ea square error of () u Sice M ( ) i (8) is a fuctio of ukow costat, therefore, it is iiized with respect to ad subsequetl the iu value of obtaied i Appedix C is M M p Mp () Now, substitutig the value of i equatio (8) we obtai the iu ea square error of (give i Appedix C) as M p p Mp M M M () Usig equatio (5) ad (), let x z x z z z A Y C, B Y C C C, C Y C C C he, fro Appedix C M p or M A C A C B C A C B C (3) where A C, B C

6 88 Z. Ralte, G. Das: Ratio-to-regressio estiator i 5. Optiu replaceet polic o deterie the iu value of µ (fractio of a saple to be take afresh at the secod occasio) so that populatio ea Y a be estiated with axiu precisio, we iiize ea square error of with respect to µ which results i quadratic equatio i µ ad solutio of (fro Appedix D) is give below: z z z z z (4) Fro equatio (4), it is obvious that the real value of exists if the quatities uder square root are greater tha or equal to zero. o choose the adissible value of, it should be reebered that. All other values of are iadissible. Substitutig the value of fro (4) i (3), we have M p (5) 6. Copariso of ea square error ad efficiec Now we copare the iu MSE of the proposed estiator iu MSE of the estiator proposed b Sigh (5), (ii) uit estiator, ad (iii) with (i) the, i.e. ea per (Cochra, 977) whe o auxiliar iforatio is used at a occasio. I each case, we obtai coditios uder which the estiator is better tha the three estiators etioed above. (i) Copariso with Sigh s (5) estiator. First, we cosider the estiator which is due to Sigh (5). he variace of this estiator, to the first order of approxiatio ad for large populatio, is give as M

7 SAISICS IN RANSIION ew series, Suer 5 89 where A C, B C, A Y C, B Y Cx xcc x, C Y Cz zccz he variace of is iiized for hus, the resultig iiu variace of is M Uder the assuptio of Cx C Cz C, iiizig M with respect to, the iu value of (fractio of a saple to be take afresh at the secod occasio) is give b z z x z x herefore, M he proposed cobied estiator proposed b Sigh (5) if (sa) is better tha the cobied estiator M M p (See Appedix E(i) for derivatio of the above result) Sice, is alwas positive ad is alwas positive, which idicates that the above coditio is alwas valid. hat is, is a better estiator tha the estiator proposed b Sigh (5).

8 9 Z. Ralte, G. Das: Ratio-to-regressio estiator i (ii) Copariso with ea per uit estiator. Next, we cosider the ea per uit estiator i i estiator is give b, uder the assuptio of C V he proposed cobied estiator as follows: C, the variace of ea per uit YC YC is better tha this estiator if p V M (See Appedix E(ii) for derivatio of the above result) (6) Hece, is a better estiator tha ea per uit estiator wheever equatio (6) is satisfied. Ad it ca be easil verified that the above coditio is true for.5. (iii) Copariso with Cochra s (977) estiator. Now we cosider the Cochra s (977) estiator as follows: u, i.e. cobied estiator suggested b Cochra (977) whe o auxiliar iforatio is used at a occasio. Here, u, are the uatched ad atched portios of the saple at the secod occasio respectivel, ad where Wu W W W u ad W are the iverse variaces, i.e. u u CY u C Y x C Y x V, V W u W

9 SAISICS IN RANSIION ew series, Suer 5 9 B least square theor, the variace of Miiizig V W u is W x C Y u u x V with respect to u, the iu value of u is give b u Uder the assuptio of C is give as x x ad C, the iu variace of the above estiator YC x V he, the proposed cobied estiator p x is better tha this estiator if V M 4 (See Appedix E(iii) for derivatio of the above result) (7) Hece, is a better estiator tha Cochra s (977) estiator wheever equatio (7) is satisfied. Ad it ca be easil verified that the above coditio is true for.5. Further, coditios (6) ad (7) are true for all values of.5. his is justified sice x expresses the relatioship betwee curret occasio ad previous occasio stud variable, for previous occasio stud variable ad auxiliar variable ad for curret occasio stud variable ad auxiliar z variable. As a exaple, the percet relative efficiec of the proposed estiator with respect to (i), the estiator proposed b Sigh (5), (ii), i.e. ea per

10 9 Z. Ralte, G. Das: Ratio-to-regressio estiator i uit estiator, ad (iii) (Cochra, 977) whe o auxiliar iforatio is used o a occasio, have bee coputed for soe assued values of x, z ad zx. Sice ad are ubiased estiators of Y, their variaces for large N are respectivel give b ad V YC YC x V Here, E, E, E 3 are desigated as the percet relative efficiecies of the proposed estiator respectivel. with respect to Sigh s (5) estiator, ad Further, the expressio of the iu, i.e. ad the percet relative efficiecies E, E ad E 3 are i ters of populatio correlatio coefficiets. We assued that all the correlatios ivolved i the expressios of, E, E ad E 3 are equal, i.e. x z. Accordigl, coputed values of, E, E ad E 3 for differet choices of high positive correlatios are show i able. able. Relative Efficiec (%) of with.respect to estiators, ad E E E Fro able, it is clear that the proposed estiator is ore efficiet tha, ad. Also, whe the correlatio coefficiet.5 is icreasig, the gai i precisio of the proposed estiator over, ad. is also icreasig. Further, it a be oticed that the axiu gai i efficiec occurs while coparig it with ea per uit estiator, which is ver obvious.

11 SAISICS IN RANSIION ew series, Suer Illustratio usig real life data I order to illustrate the copariso of relative efficiec of the proposed estiator with respect to (i), the estiator proposed b Sigh (5), (ii), i.e. ea per uit estiator, ad (iii) (Cochra,977) whe o auxiliar iforatio is used o a occasio, usig real life approach, the data fro the Cesus of Idia () ad () was cosidered. We defie the variables x() as the total uber of workers i villages i the state of Mizora, Idia i () ad z is defied as a auxiliar variable which is the total uber of literate people i villages i the state of Mizora, Idia. Usig successive saplig as defied i sectio for the above data set we take = 7, = 35 ad u = 35. he followig table shows the values of the differet estiators as coputed fro the saple alog with their correspodig ea square errors ad efficiec of the proposed estiator with respect to, ad. able. Relative Efficiec (%) of usig real life data. with.respect to estiators, ad Estiators Estiates MSE Efficiec % he above table shows that the coclusios are the sae as those of able, that is the proposed estiator is ore efficiet tha, ad with axiu gai i efficiec occurrig while coparig it with ea per uit estiator, which is ver obvious. 8. Coclusio I this stud we have proposed a ew chai ratio-to-regressio estiator i successive saplig. he bias of the proposed estiator was coputed up to the secod order of approxiatio. he iu replaceet polic of the saplig fractio was obtaied ad cosiderig bias up to the first order of approxiatio, the iu ea square error of the proposed estiator was also obtaied. he iu ea square error of the proposed estiator was copared with that of

12 94 Z. Ralte, G. Das: Ratio-to-regressio estiator i Sigh s (5) estiator ad it was foud that the proposed estiator is alwas better tha Sigh s (5) estiator. Further, the proposed estiator was copared with ea per uit estiator ad Cochra s (977) estiator whe o auxiliar iforatio is used o a occasio. It was foud that the proposed estiator is better tha both of these estiators for.5, which is etirel justified. A exaple was cosidered b assuig differet values of.5 to illustrate the above facts. At the ed, a real life stud was also doe to deostrate that the proposed estiator is ore efficiet tha the other three existig estiators. Hece, the proposed estiator is recoeded for further use. REFERENCES CENSUS OF INDIA, (). CENSUS OF INDIA, (). COCHRAN, W. G., (977). Saplig echiques. hird editio. Wile Easter Ltd. DAS, A. K., (98). Estiatio of populatio ratio o two occasios. Joural of the Idia Societ of Agricultural Statistics. 34: 9. GUPA, P. C., (979). Saplig o two successive occasios. Joural of Statistical Research. 3: 7 6. PAERSON, H. D., (95). Saplig o successive occasios with partial replaceet of uits. Joural of the Roal Statistical Societ. (B): RAO, J. N. K., GRAHAM J., E., (964). Rotatio desigs for saplig o repeated occasios. Joural of Aerica Statistical Associatio. 59: SEN, A. R., (97). Successive saplig with two auxiliar variables. Sakha. 33(B): SEN, A. R., (97). Successive saplig with p auxiliar variables, he Aals of Matheatical Statistics. 43: SEN, A. R., (973). heor ad applicatio of saplig o repeated occasios with several auxiliar variables. Bioetrics. 9: SINGH, G. N., (3). Estiatio of populatio ea usig auxiliar iforatio o recet occasio i h occasios successive saplig. Statistics i rasitio. 6(4): SINGH, G. N., (5). O the use of chai-tpe ratio estiator i successive saplig. Statistics i rasitio. 7(): 6.

13 SAISICS IN RANSIION ew series, Suer 5 95 SINGH, G. N., PRIYANKA, K., (6). O the use of chai-tpe ratio to differece estiator i successive saplig. Iteratioal Joural of Applied Matheatics ad Statistics. 5(S6): SINGH, G. N., PRIYANKA, K., (7). O the use of auxiliar iforatio i search of good rotatio patters o successive occasios. Bulleti of Statistics ad Ecooics. (A7): 4 6. SINGH, G. N., PRIYANKA, K., (8). O the use of several auxiliar variates to iprove the precisio of estiates at curret occasio. Joural of the Idia Societ of Agricultural Statistics. 6(3): SINGH, G. N., SINGH, V. K., (). O the use of auxiliar iforatio i successive saplig. Joural of the Idia Societ of Agricultural Statistics.54():. SINGH, H. P., VISHWAKARMA, G. K., (7). A geeral class of estiators i successive saplig. Metro. LXV(): 7. SINGH, P., ALWAR, H. K., (99). Estiatio of populatio regressio coefficiet i successive saplig. Bioetrical Joural. 33:

14 96 Z. Ralte, G. Das: Ratio-to-regressio estiator i APPENDICES Appedix A Bias of the proposed estiator Bias ( p) = E( p) - Y Let =Y+ε, S Sz p = x +b z -z x z = Z+ε, z Z, Z z x =X+ε, s =S +ε 3, s z =S z +ε 4, such that Eε i = assuig the ters havig order higher tha i s are egligible, the Y+ε S +ε Z X+ε S+ε z 4 Z+ε 3 p = X+ε + Z+ε-Z-ε ε (ε ) ε4 ε 4 ε ε Y+ε - + X+ε β ε -ε - + X X X S Sz Sz Z Z ε3 = = = Y Y Y Y Y+ε - ε - ε ε + (ε ) -Yε -ε ε + ε ε + ε X X X X X Z εε 4 εε3 εε 4 εε 3 X+ε +β ε- + -ε + + Sz S Sz S YX+β Y β ε ε -ε ε X Y XY -β εε-εε - -Xε ε + ε X Z εε 4 εε3 εε 4 εε3 Sz S Sz S

15 SAISICS IN RANSIION ew series, Suer 5 97 herefore, E( p) = Hece, Bias z z Cov z,s Cov z,s Cov z,s Cov z,s β Y Sz S Sz S Y Y +β Cov,z -Cov,z - Var x X X Y XY -β Covx,z -Covx,z -XCov,z + Var z X Z (8) p = p E -Y = z z Cov z,s Cov z,s Cov z,s Cov z,s β Y Sz S Sz S Y +β Cov,z -Cov,z - Var x X X Y XY -β Covx,z -Covx,z -XCov,z + Var z X Z Appedix B Mea Square Error of the proposed estiator p = x +b z -z x Z z Fro (8), we ca see that up to the first order of approxiatio E p =Y. herefore, ea square error up to the first order of approxiatio is give b M p =E p -Y. Now, Y+ε S +ε3 Z p = X+ε + Z+ε-Z-ε X+ε S+ε z 4 Z+ε

16 98 Z. Ralte, G. Das: Ratio-to-regressio estiator i herefore, ε ε 4 ε Y+ε - X+ε β ε -ε - X X S Sz Z = ε3 ε ε ε Y+ε - ε X+ε β ε -ε - Y p = Now agai, X 3 4 S X Sz Z E p = XY E XY+Xε +βyε -βyε - ε X Z Assuig the ters up to the secod order of approxiatio ad eglectig ters higher tha d power of ε s, we have E p = XY XY +X ε +β Y ε +β Y ε + ε +β XYε ε Z E X X Y XY XY -β XYε ε - ε ε-β Y ε ε-β ε +β ε ε Z Z Z = C ρcx ρcx C ρ z ρz CxC ρ ρz Cx C Y ρ zccz ρcx - - herefore, Appedix C C = Y ρcx -ρρ zcxc + Cz -ρzc Cz M( p) = E( p) - Y = C Y + - ρ C -ρ ρ C C + C -ρ C C Miiu Mea Square Error of Fro equatio (8), we have x z x z z z p p M =ψ M + -ψ M (9)

17 SAISICS IN RANSIION ew series, Suer 5 99 Sice M ( ) is a fuctio of a ukow costat ψ, therefore, it is iiized with respect to ψ, i.e. differetiatig M ( ) with respect to ψ ad equatig the to zero, we get he, substitutig For siplificatio, let herefore, ψ M p p = M +M ψ fro equatio (9), the iiu MSE of M p p p M M = M +M as x z x z z z A =Y C, B =Y ρ C -ρ ρ C C, C =Y C -ρ C C A +C A +C M = = u μ A C A B μ C M p = + - B + = + + -μ -μ Now, sice =-u=-μ= -μ A +C A +B μ+c -μ M p = A +C -μ +A μ+bμ +C μ -μ or M where -μ A +C + A +C B -C A +C + B -C μ α +α α μ = α +α μ () α =A +C, α =B -C

18 Z. Ralte, G. Das: Ratio-to-regressio estiator i Appedix D Optiu Replaceet Polic Differetiatig M to zero, we get fro equatio () with respect to ad equatig the α +αμ αα -α +αα μαμ α +αμ α α μ + α α μ- α α = = he, -α ± α + α α μ= α Now, assuig C x =C =C z =C(sa) Cz -ρ zccz α =A +C = Y C +Y z z z z =Y C -ρ x z x z z z α =B -C =Y ρ C -ρ ρ C C -Y C -ρ C C =Y C -ρ ρ --ρ α α = A +C B -C z z = Y C -ρ Y C -ρ ρ --ρ = Y C -ρ -ρ ρ --ρ herefore, -α ± α + α α = α = z z z z -ρ ρz -ρ - -ρ ± -ρ -ρ + -ρ ρ -ρ μ (sa)

19 SAISICS IN RANSIION ew series, Suer 5 Appedix E Copariso of ea square error (i). M -M > p α +α αμ α +α α μ - > α +α μ α +α μ z z -ρ z + ρz -ρ μ -ρ z + -ρ ρz --ρ μ - > -ρ z + ρz -ρ μ -ρ + -ρ ρ --ρ μ Let us assue that ρ herefore, =ρ =ρ sa, equatio () will becoe z -ρ + μ -ρ + -ρ ρ- μ - > -ρ + μ -ρ + -ρρ- μ - -ρ μ - > --ρμ p M -M > -ρ -μ () (ii). p V -M > -ρ z -ρ z + -ρ ρz --ρ μ - > -ρ z + -ρ ρz --ρ μ ρ =ρ =ρ sa, equatio () will becoe Assuig that z -ρ - -ρ μ - > --ρμ - -ρ μ - -ρ - -ρ μ > ()

20 Z. Ralte, G. Das: Ratio-to-regressio estiator i herefore, p V -M > - -ρ μ > -ρ - -ρ μ (iii). p V -M > Y C -ρ z + -ρ ρ z --ρ μ Y C -ρ z -ρ z + -ρ ρ z --ρ μ YC + -ρ x - > (3) Assuig ρ =ρ =ρ sa, equatio (3) will becoe z herefore, + -ρ - -ρ μ - 4 -ρ - -ρ μ > p V -M > + -ρ - -ρ μ >4 -ρ - -ρ μ