USE OF DOUBLE SAMPLING SCHEME IN ESTIMATING THE MEAN OF STRATIFIED POPULATION UNDER NON-RESPONSE

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1 STATISTICA, anno LXXV, n. 4, 015 USE OF DOUBLE SAMPLING SCHEME IN ESTIMATING THE MEAN OF STRATIFIED POPULATION UNDER NON-RESPONSE Manoj K. Chaudhary 1 Department of Statstcs, Banaras Hndu Unversty, Varanas, Inda Amt Kumar Department of Statstcs, Banaras Hndu Unversty, Varanas, Inda 1. Introducton Auxlary nformaton can be used to mprove the effcency of the estmator of a partcular populaton parameter. The effectveness of the estmaton procedure wth the use of auxlary nformaton closely depends upon the method n whch the estmator has been proposed,.e., the form n whch the functons of auxlary nformaton have been consdered. There s a lot of works n whch the auxlary nformaton are used to enhance the precson of the estmator. Upadhyaya and Sngh (1999 have suggested a class of estmators for estmatng the populaton mean smple random samplng. Kadlar and Cng (003 and Shabbr and Gupta (005 expanded these estmators for estmatng the populaton mean of stratfed populaton. Sngh et al. (01 have proposed a general famly of estmators for estmatng the populaton mean systematc samplng. We know that the problem of non-response s nherent n the populaton of every survey. The non-response error s not so mportant f the characterstcs of the non-respondng unts are smlar to those of the respondng unts. But, t s notced that such smlarty of characterstcs between two types of unts (respondng and non-respondng unts s not always obtaned n practce. Hansen and Hurwtz (1946 ntroduced a technque of sub-samplng of non-respondents n order to adjust the non-response n mal surveys. Khoshnevsan et al. (007 have proposed a general famly of estmators of populaton mean usng known coeffcents of some populaton parameters n smple random samplng. Chaudhary et al. (009 have proposed a combned-type famly of estmators wth the use of an auxlary varable for mean of a stratfed populaton the presence of nonresponse adoptng Khoshnevsan et al. (007. Recently, Chaudhary and Sngh (013 have proposed factor type famles of estmators of populaton mean two-stage samplng wth equal sze clusters under non-response. 1 Correspondng Author e-mal: rtamanoj15@gmal.com

2 406 M. K. Chaudhary and A. Kumar It s well known fact that when the parametrc values of auxlary varable are not known, one can utlze the double samplng (or two-phase samplng scheme mprovng the estmaton procedure. Two-phase samplng s very effectve n terms of cost as well as applcatons. Ths samplng scheme s used to obtan the nformaton about auxlary varable nexpensvely from a larger sample at frst phase and comparatvely small sample at the second phase. Okafor and Lee (000 have dscussed the method of double samplng scheme for estmatng the famly expendture n a household survey under non-response. Khare and Snha (004 have suggested the estmators for populaton rato usng two-phase samplng scheme n the presence of non-response. In the lght of above context, we have suggested a famly of combned-type estmators of populaton mean stratfed random samplng usng two-phase samplng scheme under non-response whenever the populaton mean of auxlary varable s unknown. The propertes of the famly along wth ts optmum property have been dscussed.. Notatons, Samplng Strategy and Estmaton Procedure y j : Observaton on the j th unt n the th stratum under study varable. ( = 1,,..., k; j = 1,,..., N. x j : Observaton on the j th unt n the th stratum under auxlary varable. ( = 1,,..., k; j = 1,,..., N. N 1 : Populaton sze of the response group n the th stratum. N : Populaton sze of non-response group n the th stratum. Y = k p Y : Populaton mean under study varable. X = k p X : Populaton mean under auxlary varable. Y = 1 N N j=1 y j: Populaton mean of the th stratum under study varable. X = 1 N N j=1 x j: Populaton mean of the th stratum under auxlary varable. Y = 1 N N j y j : Populaton mean of the non-response group n the th stratum under study varable. x = 1 n j x j : Sample mean of the th stratum under auxlary varable. x = 1 n n j x j : Mean based on frst phase sample n the th stratum under auxlary varable.

3 Use of Double Samplng Scheme n Estmatng the Mean etc. 407 y n1 = 1 n1 1 j y j : Sample mean of the response group n the th stratum under study varable. y h = 1 h h j y j : Mean based on h non-respondent unts n the th stratum under study varable. SY = 1 N : N 1 j=1 yj Y Populaton mean square of the th stratum under study varable. SX = 1 N : N 1 j=1 xj X Populaton mean square of the th stratum under auxlary varable. SY = 1 N : N 1 j yj Y Populaton mean square of the non- response group n the th stratum under study varable. W 1 = N 1 N : Response rate n the th stratum. W = N N : Non-response rate n the th stratum. Let us suppose that a populaton of sze N s dvded nto k strata. Let the sze of the th stratum be N ( = 1,,..., k such that k N = N. Let a sample of sze n be selected from the entre populaton such a way that unts are selected from the th stratum and we have k = n. Let Y be the study varable wth populaton meany = k p Y (Y beng the mean of the th stratum based on N unts and p = N N and we assume that the non-response s observed on study varable. It s observed that there are 1 respondent unts and non-respondent unts n unts. Usng Hansen and Hurwtz (1946 technque of sub-samplng of non-respondents, we select a sub-sample of h unts out of non-respondent unts such that = L h, L 1and collect the nformaton from all h unts. Thus, an unbased estmator of populaton mean Y s gven by y st = p y (1 where y = 1y n1 + y h, y n1 and y h are the means based on 1 respondent unts and h sub-sampled non-respondent unts respectvely. The varance of y st s gven as V (y st = ( 1 1 N p S Y + (L 1 W p S Y ( where SY and S Y are the populaton mean squares of the entre group and nonresponse group respectvely th stratum for study varable. W s the populaton non-response rate n the th stratum. In order to mprove the effcency of an estmator, one can utlze the auxlary nformaton at the estmaton stage. In ths sequence, Chaudhary et al. (009

4 408 M. K. Chaudhary and A. Kumar have proposed a famly of combned-type estmators of populaton mean usng an auxlary varable X n stratfed random samplng under the condton that the non-response s observed on study varable and auxlary varable s free from non-response, gven as [ ] g T C = y ax + b st α (ax st + b + (1 α ( ax + b (3 where x st = k p x, X = k p X (populaton mean of auxlary varable, a 0 and bare ether real numbers or functons of known parameters of auxlary varable. α and gare the constants and to be determned. x and X are respectvely the mean based on unts and mean based on N unts n the th stratum for auxlary varable. The bas and mean square error (MSE of T C up to the frst order of approxmaton are respectvely gven by and B (T C = 1 Ȳ MSE (T C = [ ] g(g + 1 f p α λ R SX αλgrρ S X S Y f p [ S Y + α λ g R SX ] αλgrρ S X S Y + (L 1 W p SY 1 where f = 1 N, λ = ax, R = Y, ax+b X S X s the populaton mean square of auxlary varable for the th stratum and ρ s the populaton correlaton coeffcent between Y and X n the th stratum. (4 (5 3. Proposed Famly of Estmators If the populaton mean of auxlary varable X s known, one can easly use the famly of estmators shown equaton (3 for estmatng the populaton mean of study varabley. But f the populaton mean X s unknown, t s not easy to adopt the present form of the consdered famly of estmators. Thus the double samplng scheme (or two-phase samplng scheme can be utlzed to estmate the populaton meany. Adoptng double samplng scheme frst the estmate of Xmay be generated from a large frst phase sample of sze n drawn from the N unts by smple random samplng wthout replacement (SRSWOR scheme for the th stratum. Secondly, a smaller second phase sample of sze s drawn from n unts by SRSWOR. It s noted that the non-response s observed on study varable at second phase sample. Out of unts, let 1 unts respond and unts do not respond at the second phase. Applyng Hansen and Hurwtz (1946 technque of sub-samplng of non-respondents, a sub-sample of h (= /L, L 1 unts s

5 Use of Double Samplng Scheme n Estmatng the Mean etc. 409 selected from the sample of non-respondents and the nformatos collected from all of them. Let us assume that the full nformatos suppled on the n unts at the frst phase for the auxlary varable Xand the study varable goes through the nonresponse. Thus the usual combned rato and product estmators of populaton mean Y usng two-phase samplng scheme n stratfed random samplng under non-response are respectvely gven by and T 1 = y st x st x st (6 T = y st x x st (7 st where x st = k p x and x s the mean based on n unts for auxlary varable. The mean square errors (MSEs of the estmators T1 and T up to the frst order of approxmaton are respectvely gven by and where f = ( MSE (T 1 = MSE (T = 1 n f p SY + f p SY + ( 1 N and f = f p ( S Y + R SX Rρ S X S Y + (L 1 W p SY f p ( S Y + R SX + Rρ S X S Y 1 1 n. + (L 1 W p SY In the presence of above crcumstances, a famly of combned-type estmators of mean Y of stratfed populaton usng two-phase samplng n the presence of non-response s gven by T C = y st [ ax st + b α (ax st + b + (1 α ( ax st + b (8 (9 ] g (10 In order to obtan the bas and MSE of T C, we use large sample approxmaton. Let us assume that y st = Y (1 + e 0, x st = X (1 + e 1, x st = X 1 + e 1. Form the above, we have E (e 0 = E (e 1 = E e 1 = 0,

6 410 M. K. Chaudhary and A. Kumar E e 1 = 1 X E e 0 e 1 = 1 XY E ( e 1 0 = Y ( 1 n ( 1 [( 1 1 N E ( e 1 1 = X ( 1 p SY + (L ] 1 W p SY, 1 N 1 N p S X, E (e 0 e 1 = 1 XY n p S X, ( 1 1 p ρ S Y S X, E e 1 e 1 = 1 N X 1 N p ρ S Y S X, ( 1 1 p S N X. Puttng the above assumptons nto equaton (10, T C can be expressed as g [ { g T C = Y (1 + e λe λ αe 1 + (1 α e 1}] (11 On expandng equaton (11 and neglectng the terms ofe 0, e 1 and e 1havng power greater than two, we get [ { } T C Y = Y gλ αe 1 + (1 α e g (g + 1 { 1 + λ α e 1 + (1 e e 1 } { } +α (1 α e 1 e 1 + gλe 1 g λ e 1 αe 1 + (1 α e 1 (1 + g (g 1 λ e 1 + e 0 gλe 0 {αe 1 + (1 α e 1 n } + gλe 1e 0 ] Takng expectaton both the sdes of equaton (1, we get bas of T C the frst order of approxmaton as B T C = 1 [ { } (g + 1 Y gλ p f λα R SX Rρ S Y S X {( + f (g + 1 λ (1 α R + (g + 1 λα (1 α R gλr }] (g 1 + λr SX + Rαρ S Y S X up to (13 Squarng both the sdes of the equaton (1 and then takng the expectaton by neglectng the terms ofe 0, e 1 and e 1havng power greater than two, we get MSE of T C up to the frst order of approxmaton as MSE T C = g λ R α f p SX gλrα + f p SY + (L 1 W p SY f p ρ S X S Y (14

7 Use of Double Samplng Scheme n Estmatng the Mean etc. 411 or MSE T C = f p SY + f p ( S Y + g λ R α SX gλrαρ S X S Y + (L 1 W p SY ( Optmum Choce of α In ths secton, we choose the optmum value of α for whch the MSE of the proposed famly would remats mnmum. On dfferentatng M SE T C wth respect to α and equatng the dervatve to zero, we get MSE T C = g λ R α f p SX gλr f p ρ S X S Y = 0 α (16 k α opt = f p ρ S X S Y gλr k f p S X On substtutng the value of α opt from equaton (17 nto equaton (14 or equaton (15, we get the mnmum MSE of T C. (17 3. Cost of the Survey and Optmum,n, L Let c be the cost per unt assocated wth the frst phase sample of sze and n c 0 be the unt cost of frst attempt on study varable wth second phase sample of sze. Let c 1 and c be respectvely the cost per unt of enumeratng the 1 respondent unts and h non-respondent unts. Then the total cost for the th stratum s gven by C = c n + c 0 + c c h = 1,,..., k. Now, we obtan the expected cost per stratum as ( E (C = c n W + c 0 + c 1 W 1 + c. L Thus the total cost over all the strata s represented as = C 0 = E (C [c n ( ] W + c 0 + c 1 W 1 + c. (18 L

8 41 M. K. Chaudhary and A. Kumar Let us consder the Lagrange functon ϕ = MSE T C + µc 0 (19 where µ s Lagrange s multpler. In order to get the optmum values of, n andl, we dfferentate ϕ wth respect to, n and L respectvely and equate the dervatves to zero. Thus, we have and ϕ = p n ( S Y + g λ R α SX (L 1 gλrαρ S X S Y W p SY ( W +µ c 0 + c 1 W 1 + c = 0 (0 L ϕ = p ( n n g λ R α S X gλrαρ S X S Y + µc = 0 (1 ϕ = p W S W Y µ c L L = 0 ( From equatons (0, (1 and (, we respectvely get = p S Y + g λ R α SX gλrαρ S X S Y + (L 1 W SY ( (3 W µ c 0 + c 1 W 1 + c L n = p gλrαρ S X S Y g λ R α SX µc n (4 µ = p L S Y c (5 Puttng the value of µ from equaton (5 nto equaton (3, we get where A = c 0 + c 1 W 1 and L (opt = c B S Y A (6 B = S Y + g λ R α S X gλrαρ S X S Y W S Y. On substtutng the value of L (opt from equaton (6 nto equaton (3, we can express as = p B + c B W S Y A µ A + (7 c A W S Y B In obtanng the value of µ n terms of total costc 0, we put the values of n, L (opt and from equatons (4, (6 and (7 nto equaton (18. Thus, we have

9 Use of Double Samplng Scheme n Estmatng the Mean etc. 413 TABLE 1 Partculars of Data Stratum No. N n Y X SY SX ρ SY µ = 1 C 0 [ c gλrαρ S X S Y g λ R α SX + p (A B + ] c W S Y (8 Substtutng the value of µ from equaton (8 nto equatons (7 and (4, we respectvely get the optmum values of and n (opt = C 0 p B + c B W S Y A A + c A W S Y B [ c gλrαρ S X S Y g λ R α SX + p (A B + ] 1 c W S Y (9 n (opt = C 0p gλrαρ S X S Y g λ R α S X ( c [ c gλrαρ S X S Y g λ R α SX + p (A B + ] 1 c W S Y (30 4. Emprcal Study The theoretcal study of the proposed famly can easly be comprehended by an emprcal analyss. To support the theoretcal results, we have used the data consdered by Chaudhary et al. (01. There are 84 muncpaltes dvded nto four strata havng respectve szes 73, 70, 97, and 44. The data relate to the populaton sze (n thousands n the year 1985 as study varable and the populaton sze (n thousands n the year 1975 as auxlary varable. Partculars are gven Table 1. Table represents the MSE and percent relatve effcency (PRE of T1, T and T C (at α (opt, a = 1, b = 1 and g = 1wth respect to y st for the dfferent choces of W and L.

10 414 M. K. Chaudhary and A. Kumar TABLE MSE and PRE of T1, T and T C wth respect to y st W L V (y st MSE (T1 MSE (T MSE T C P RE (T1 P RE (T P RE T C

11 Use of Double Samplng Scheme n Estmatng the Mean etc. 415 From the Table, t s revealed that the optmum estmator of the proposed famly certanly provdes the better estmates as compared to y st, T 1 andt. It s obvous that the ncrement n populaton non-response rate W or nverse samplng rate L would cause a further ncrement n the MSE of the estmators. Thus, from the above table (Table, t s also revealed that the effcency of the estmators decreases wth ancrease n non-response rate W as well as wth an ncrease nverse samplng rate L. 5. CONCLUSION The use of auxlary nformaton certanly mproves the precson of the estmator at the estmaton stage. The stuatons n whch the parametrc values of auxlary varable(s are not known, one can use ts estmates by usng double samplng scheme n estmatng the parameters of study varable. To cope up the stuatons, we have proposed a famly of combned-type estmators of populaton mean n stratfed random samplng usng double samplng scheme under non-response. The optmum combned-type estmator of the famly has been obtaned and comparson of t wth the usual mean, usual combned rato and usual combned product estmators has also been made. The optmum values of frst phase sample szen, second phase sample sze and nverse samplng rate L for the dfferent strata through the proposed famly, have been determned under the cost survey. From Table, t s observed that the optmum combned-type estmator of the proposed famly provdes better estmates than the usual mean, usual combned rato and usual combned product estmators. It s also observed that precson of the optmum combned-type estmator as well as usual mean, usual combned rato and usual product estmators decreases wth ncrease n non-response rate and nverse samplng rate. The results are ntutvely expected. Acknowledgements The authors are thankful to the learned referees for ther constructve suggestons regardng mprovement of the present paper. References M. K. Chaudhary, R. Sngh, R. K. Shukla, M. Kumar (009. A famly of estmators for estmatng populaton mean stratfed samplng under nonresponse. Pakstan Journal of Statstcs and Operaton Research, 5, no. 1, pp M. K. Chaudhary, V. K. Sngh (013. Estmatng the populaton mean two- stage samplng wth equal sze clusters under non-response usng auxlary characterstc. Mathematcal Journal of Interdscplnary Scences,, no. 1, pp M. K. Chaudhary, V. K. Sngh, R. K. Shukla (01. Combned-type famly of estmators of populaton mean stratfed random samplng under non-

12 416 M. K. Chaudhary and A. Kumar response. Journal of Relablty and Statstcal Studes (JRSS, 5, no., pp M. H. Hansen, W. N. Hurwtz (1946. The problem of non-response n sample surveys. Journal of The Amercan Statstcal Assocaton,, no. 41, pp C. Kadlar, H. Cng (003. Rato estmators n stratfed random samplng. Bometrcal journal,, no. 45, pp B. B. Khare, R. R. Snha (004. Estmaton of fnte populaton rato usng two-phase samplng scheme n the presence of non-response. Algarh J. Stat., 4, , 4, pp M. Khoshnevsan, R. Sngh, P. Chauhan, N. Sawanand, F. Smarandache (007. A general famly of estmators for estmatng populaton mean usng known value of some populaton parameter(s. Far East Journal of Theoretcal Statstcs,, pp F. C. Okafor, H. Lee (000. Double samplng for rato and regresson estmaton wth sub-samplng the non-respondents. Survey Methodology, 6, pp J. Shabbr, S. Gupta (005. Improved rato estmators n stratfed samplng. Amercan Journal of Mathematcal and Management Scences, 5, no. 3-4, pp R. Sngh, S. Malk, M. K. Chaudhary, H. K. Verma, A. A. Adewara (01. A general famly of rato-type estmators n systematc samplng. Journal of Relablty and Statstcal Studes, 5, no. 1, pp L. N. Upadhyaya, H. P. Sngh (1999. Use of transformed auxlary varable n estmatng the fnte populaton mean. Bometrcal Journal, 41, no. 5, pp Summary The present paper focuses on the use of double samplng scheme n stratfed random samplng for estmatng the populaton mean the presence of non-response. Motvated by Khoshnevsan et al. (007, we have proposed a famly of combned-type estmators of populaton mean utlzng the nformaton on an auxlary varable wth the use of double samplng scheme under non-response. The optmum property of the proposed famly has been dscussed. An emprcal study has also been carred out n the support of theoretcal results. Keywords: Double samplng scheme; stratfed random samplng; auxlary varable; populaton mean; non-response