Chain ratio-to-regression estimators in two-phase sampling in the presence of non-response

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1 ProbStat Forum, Volume 08, July 015, Pages ISS ProbStat Forum is a e-joural. For details please visit Chai ratio-to-regressio estimators i two-phase samplig i the presece of o-respose B.B. hare a, Sajay umar b a Departmet of statistics, Baaras Hidu Uiversity Varaasi U.P.)-1005, Idia b Departmet of Statistics, Cetral Uiversity of Rajastha ishagarh , Ajmer, Rajastha Abstract. Usig double samplig, this paper presets covetioal ad alterative chai ratio-toregressio estimators for the fiite populatio mea whe the populatio of the mai auxiliary variable x is ukow but that of a additioal auxiliary variable z is kow. The proposed estimators are foud to be more efficiet tha the relevat estimators i case of fixed cost. A empirical study has bee made for the support of the problem. 1. Itroductio Iformatio o variables correlated with the mai variable uder study is popularly kow as auxiliary iformatio which may be utilized either at plaig stage or at desig stage or at the iformatio stage to arrive at improved estimator compared to those, ot utilizig auxiliary iformatio. Use of auxiliary iformatio for formig ratio ad regressio method of estimatio were itroduced durig 1930s with a comprehesive theory provided by Cochra 194). Chaed 1975) ad iogyera 1980, 1984) have proposed chai ratio type estimators usig additioal variable with kow populatio mea. If the populatio mea of the auxiliary variable x) is ot kow but the populatio mea of a additioal auxiliary variable z) is kow which is cheaper ad less correlated to the study variable y) i compariso to the mai auxiliary variable i.e. ρ yx > ρ yz ) sometimes it may ot be possible to collect the compete iformatio for all the uits selected i the sample due to o-respose. Estimatio of the populatio mea i sample surveys whe some observatios are missig due to o-respose ot at radom has bee cosidered by Hase ad Hurwitz 1946) ad Rao 1986, 1987). I this paper, we have proposed covetioal ad alterative ratio-to-regressio estimators for the populatio mea of the study variable i the presece of o-respose. The expressios for mea square error of the proposed estimators are obtaied ad a compariso of the proposed estimators has bee made with the relevat estimators.. The estimators Let X i, Y i ad Z i be the o egative values for the i th uit of the populatio U = U 1, U,..., U o the study variable y, the auxiliary variable x ad the additioal auxiliary variable z with their populatio meas Y, X ad Z. Here X is ukow, but Z, the populatio mea of additioal auxiliary variable z closely 010 Mathematics Subject Classificatio. Primary 6D05 eywords. Chai ratio-to-regressio estimator, Populatio mea, Study variable, Auxiliary variable, Additioal variable. Received: 13 August 014; Accepted: 9 Jue 015 addresses: bbkhare56@yahoo.com B.B. hare), sajay.kumar@curaj.ac.i Sajay umar)

2 hare ad umar / ProbStat Forum, Volume 08, July 015, Pages related to x) is kow, which may be cheaper ad less correlated to the study variable z) i compariso to the mai auxiliary variable x). ow a first phase sample of size is selected from the populatio of size usig SRSWOR ad we estimate the populatio mea X usig additioal auxiliary character with kow populatio mea Z ad observatios o x ad z. Agai subsample of size is selected from first phase sample by usig SRSWOR scheme of samplig ad it has bee observed that 1 uits respod ad uit do ot respod i the sample of size for the study variable y. It is also assumed that the populatio of size is composed of 1 respodig ad o-respodig uits, though they are ukow. Further a sub-sample of size r = > 1) from o-respodig uits has bee draw by usig SRSWOR method of samplig by makig extra effort. I some situatios, X is ukow, the we select a larger sample of size > ) from the populatio of size by usig SRSWOR method of samplig ad we estimate X by x = 1 x i ad agai draw a sub sample of size ad observe y variable. ow, it has bee observed that 1 uits respod ad uits do ot respod ad the we select a sub-sample of size r = > 1) from o respodig uits by usig SRSWOR method of samplig ad cosequetly the covetioal ad alterative two phase samplig ratio t 1, t ) for populatio mea i the presece of o-respose proposed by hare ad Srivastava 1993) are give as follows: where, t 1 = y x x ad t = y x x, y = 1 y 1 + y, x = 1 x 1 + x, x = x i, The estimator y is ubiased ad the V y ) is give by V y ) = f S y + W 1) Sy) where, f =, W i = i i = 1, ), S y ad Sy) are the populatio mea squares of the character y for the whole populatio ad for the o respodig part of the populatio. If X is ot kow, but Z, the populatio mea of the additioal auxiliary character z closely related to x) is kow, which may be cheaper ad less correlated to the study charactery) i compariso to mai auxiliary character x). I this situatio, we take a first phase sample from the populatio of size with SRSWOR ad we estimate X by x = 1 x i by usig the sample meas x ad z = 1 z i based o uits ad the kow additioal populatio mea Z. It is well kow result that ratio type estimator of X usig auxiliary iformatio z is optimum if the regressio of x o z is liear ad passes through the origi ad the variace of x about this regressio lie is proportioal to z. However, the regressio of x o z is liear but does ot go through the origi. I such situatio, iogyera 1980) suggested a chai ratio-to-regressio e 1 ) estimator, which is give as follows: e 1 = y x { x + b Z z ) } where b = Ŝxz s z, S xz = 1 1 Zi Z ) X i X) ads z = 1 1 z i z). The estimate Ŝxz is based o the available data uder the give samplig desig. Sigh ad umar 010) proposed improved estimators of populatio mea uder two-phase samplig with sub-samplig the o-respodets, which are give as follows: t 1 = {y + b yx x x )} Z { Z + δ1 z Z) } ad t = {y + b yx x Z x)} { Z + δ 1 z Z) },

3 hare ad umar / ProbStat Forum, Volume 08, July 015, Pages where M δ 1opt.) = ) Y Z D, δ 1opt.) = Q { 1 ) }, 1 Y Z { 1 D = 1 ) Sz + W } 1) Sz), { 1 M = 1 ) 1 β 1 Sz + 1 ) A Sz + W } 1) B Sz), { 1 A = β 1 ββ ), B = β 3 ββ 4 ), Q = 1 ) 1 A + 1 β = S yx Sx, β 1 = S yz Sz β = S xz Sz, β 3 = S yz) Sz), β 4 = S xz) Sz), x = 1 x i, b yx =Ŝyx Ŝ x S xz = 1 1 S yx = 1 1, b yx = Ŝyx s, s x = 1 x 1 x i x), S z = 1 1 { Xi X)Z i Z) },S x = 1 1 { Yi Y )X i X) }, S yz = 1 1 X i X), ) β 1 }, Z i Z), { Yi Y )Z i Z) } ad S z), S yz), S xz) ) deotes populatio mea squares of the character z, the covariace y, z ), covariace x, z ) for the o-respose group of the populatio. The expressios for the MSEt i ) ad MSE t 1) mi.,) up to the terms of order 1 ad 1 are give as follows: MSE t 1 ) = 1 ) [ ] 1 S y + R Sx Rρ yx S y S x + 1 ) 1 S y ] [S y) + R S x) Rρ yx)s y) S x), + W 1) MSE t ) = 1 1 ) [ S y + R S x Rρ yx S y S x ] ) S y + W 1) S y), MSE t 1) mi. = MSE t 1 ) M D, { } MSE t Q ) mi. = MSE t ) 1 ) 1 Sz, where, ρ yx = Syx S ys x, ρ yx) = S yx) S y) S x), ρ yz = Syz S ys z, R 1 = Y, R Z = X, Z S y = 1 1 y i Y ), β = Syx S x, R = Y ad S X yx), ρ yx)) deotes the covariace ad correlatio betwee y ad x variables for the o-respose group of the populatio. 3. The proposed estimators ad their mea square error MSE) Followig the estimator i iogyera 1980), we propose the covetioal ad the alterative chai ratio-to-regressio estimators T 1, T ) for populatio mea Y usig two phase samplig i the presece of o-respose, which are give as follows: T 1 = y { x x + b 3 Z z ) } ad T = y { x + b 4 Z z ) } x

4 where b 3 = Ŝxz Ŝ z hare ad umar / ProbStat Forum, Volume 08, July 015, Pages , b 4 = Ŝxz s, S xz = 1 z 1 Zi Z ) X i X), Sz = 1 1 Z i Z) The estimate Ŝ z are based o the available data uder the give samplig desig. The expressios for the MSE of the proposed estimators T i, i = 1, ) up to the terms of order 1 ad 1 are give as follows: 1 MSE T 1 ) = MSE t 1 ) + 1 ) Rρ xz S x Rρ xz S x ρ yz S y ), 1 MSE T ) = MSE ) + 1 ) Rρ xz S x Rρ xz S x ρ yz S y ). May estimators tur out as special cases of T i,, which are give as follows: 1. T 1 reduces to the estimators t 1 for b 3 = 0.. T reduces to the estimators t for b 4 = Compariso of the proposed estimators with the relevat estimators O comparig the MSE T i ), with MSEy ), MSEt i ) ad MSE t i ) mi. i = 1, ). we see that MSE T 1 ) < MSE t 1 ) < MSEy ) if ρ yx > 1 C x, ρ yx) > 1 C y MSE T ) < MSE t ) < MSEy ) if ρ yx > 1 C x ad ρ yz > 1 C y ρ xz R C x) ad ρ yz > 1 R C y) ρ C x xz, C y MSE T 1 ) < MSE T ) if ρ yx) > 1 R C x) R C y) MSE T 1 ) < MSE t M 1 1) if D + 1 ) 1 E + 1 ) F + W 1) G < 0 { } MSE T ) < MSE t M 1 ) if 1 ) 1 Sz + 1 ) E < 0 where E = R S x + ρ yx S y RS x ) ρ yx S y, F = Rρ xz S x Rρ xz S x ρ yz S y ) ad G = R β ) S x) R β) ρ yx) S y) S x). C x C y 5. Determiatio of, ad for fixed cost C C 0 Let C 0 be the total cost fixed) of the survey apart from overhead cost. The expected total of the survey apart from overhead cost is give by ) C = C 1 +C ) + W C 1 + C W 1 + C 3, 1) where C 1 is the cost per uit of idetifyig ad observig mai auxiliary character x, C is he cost per uit of idetifyig ad observig additioal auxiliary character z, C 1 is the cost per uit of mailig questioaire/visitig the uit at the secod phase, C is the cost per uit of collectig ad processig data from 1 respodig uits, C 3 is the cost per uit of obtaiig ad processig data after extra effort from the sub-sampled uits, W i = i i = 1, ) deotes the respose ad o-respose rate i the populatio respectively. The expressio for MSET i ) ca be writte as follows: MSE T i ) = M 0i + M 1i + M i + terms idepedet from ad,

5 hare ad umar / ProbStat Forum, Volume 08, July 015, Pages where M 0i M 1i ad M i are the coefficiets of the terms of 1, 1 ad respectively i the expressio of MSE T i ). Let us defie a fuctio ϕ which is give by )} ϕ = MSE T i ) + α i {C 1 +C ) W + C 1 + C W 1 + C 3, where α i is the Lagrage s multiplier. ow, differetiatig ϕ with respect to, ad ad equatig them to zero, we get, M 1i = α i C 1 + C ) ) ad = M 0i + M i α i C1 + C W 1 + C 3 W = Mi α i C 3 W ow, puttig the value of i 4), we get, C 3 W M 0i opt. = C 1 + C W 1 ) M i ) 3) 4) 5) Puttig the values of, ad opt. from ), 3) ad 5) i 1), we get [ [ αi = 1 C 1 C + C )M 1i + C 1 + C W 1 + C ] ] 3W [M 0i + opt. M i ] 0 opt. 6) It has also bee see that the determiat of the matrix of secod order derivative of ϕ with respect to, ad opt. is egative for the optimum values of, ad opt., which shows that the solutio for, give by ) ad 3) usig 5), 6) ad the optimum value of for C for C C 0 miimizes the V T i ). The miimum value of MSE T i ) for the optimum value of, ad opt. are give by MSE T i ) mi. [ C 1 + C )M 1i + 1 [ C 1 + C W 1 + C 3W [ S y + Rρ xz S x Rρ xz S x ρ yz S y ) ] opt. opt. ] ] [M 0i + opt. M i ] ow, eglectig the term of order 1, we have [ [ MSE T i ) mi. = 1 C 1 C + C )M 1i + C 1 + C W 1 + C ] ] 3W [M 0i + opt. M i ]. 7) 0 ow, puttig the value of opt. from 5) i 7), we have MSE T i ) mi. [C 1 + C )M 1i + M 0i C 1 + C W 1 ) + C 3 W M i ]. I the case of y, we have MSE y ) mi. [ M0 C 1 + C W 1 ) + C 3 W M ] S y.

6 hare ad umar / ProbStat Forum, Volume 08, July 015, Pages ow, eglectig the term of order 1, we have MSE y ) mi. [ M0 C 1 + C W 1 ) + C 3 W M ], where, M 0 ad M are the coefficiets of the terms of 1 ad respectively i the expressio of Vy ) = f S y + W 1) Sy. Similarly, for the fixed cost C 0, the expressio for MSE t i ) mi. is give by MSE t i ) mi. [ C 1 + C )V 1i + V 0i C 1 + C W 1 ) + C 3 W V i ] S y. ow, eglectig the term of order 1, we have MSE t i ) mi. [C 1 + C )V 1i + V 0i C 1 + C W 1 ) + C 3 W V i ], where V 0i V 1i ad V i are the coefficiets of the terms of 1, 1 ad MSE t i ). The value of C 1 for which MSE t i ) < MSEy ) is give by C 1 < 1 [ C1 + C W 1 ) M0 ) V 0i + C 3 W M. V i )] V 1i For the fixed value of C 0, the value of C for which MSE T i ) < MSE t i ) is give by C C 1 < V 1i M 0i. M 1i respectively i the expressio of 6. Empirical study The data from the populatio of 100 records of resales of homes from Feb 15 to Apr 30, 1993 from the files maitaied by the Albuquerque Board of Realtors o Sellig price $hudreds) as a study variable y), Square feet of livig space as a auxiliary variable x) ad aual taxes $) as a additioal variable z) have bee take. The values of the parameters of the populatio are give as follows: X = , Y = , Z = , S x = , = 5, S y = , R = Y X, S z = 316.6, ρ yx = 0.84, ρ yz = 0.64, ρ xz = 0.86, = 60. The o-respose rate i the populatio is cosidered to be 5%. So, the values of the populatio parameters based o the o-respodig parts, which are take as the last 5% uits of the populatio are give as follows: X = , Y = , S x) = , S y) = , ρ yx) = 0.84, R = Y /X. From the table 1, we see that the proposed estimator T 1 is more efficiet tha y, t 1, t, t 1, t ad T. The estimator T is also more efficiet tha y, t, t 1 ad t. It has bee also observed that the mea square error decreases as decreases. From figure 1, we see that the proposed estimator T 1 is more efficiet tha y, t 1, t, t 1, t ad T. The estimator T is also more efficiet tha y, t, t 1 ad t. It has bee also observed that the mea square error decreases as decreases. Cost study has ot be doe for t 1, t ) due to the lack of iformatio o coefficiets of 1, 1 ad. From the table, we see that the proposed estimator T 1 is more efficiet tha y, t 1, T ad t for fixed cost C C 0. The estimator T is also more efficiet tha y ad t for fixed cost C C 0.

7 hare ad umar / ProbStat Forum, Volume 08, July 015, Pages Table 1: Relative efficiecy of the proposed estimators w.r.t. y. MSE.)R.E..) i % w.r.t. y ) Estimator =.5 =.0 = 1.5 y ) ) ) T ) ) ) t ) ) ) T ) ) ) t ) ) ) t ) ) ) t ) ) ) Figure 1: A graph for mea square errors of differet estimators for differet values of = 1.5,.0,.5. Table : Relative efficiecy of the proposed estimators w.r.t. y for fixed cost. C 0 = Rs fixed) Estimator opt. opt. opt. R.E..) y T t T t

8 7. Coclusios hare ad umar / ProbStat Forum, Volume 08, July 015, Pages I this work, we have proposed covetioal ad alterative chai ratio-to-regressio type estimators for the populatio mea usig a additioal auxiliary variable i the presece of o-respose. Here, we coclude that the usig iformatio o a additioal auxiliary variable are fruitful i icreasig the precisio of the estimators compared to those, ot utilizig such iformatio. For the support of the problem, a empirical study has bee made. O the basis of the empirical study, we observe that use of a additioal iformatio i the proposed estimators for populatio mea i the presece of o-respose is foud to be more useful i icreasig the precisio of the proposed estimators with respect to the relevat estimators for the fixed cost C C 0.The proposed estimators may be used i the field of Medical, Agricultural ad Biological Scieces etc. Refereces Cochra, W.G. 194) Samplig thepry whe the samolig uits are of uequal sizes, Joural of America Statistical Associatio 37, Chaed, L. 1975) Some ratio-type Sstimotors based o two ar more auxiaiery variables, PhD Thesis submitted to Iowa etata Uiversity, Ames, IOWA. hare, B.B., Srivastava, S. 1993) Estimatio of populatio mea usig auxiliary character i presece of o-respose, atioal Academy Sciece Letters, Idia, 163), iogyera, B. 1980) A chai ratio type estimator i fiite populatio double samplig usig two auxiliary variabres, Metrika 7, iogyera, B.1984) Regressio type estimators usig two auxiliary variables ad the model of double samplig from fiite populatios, Metrika 31, Hase, M.H.A., Hurwitz, W ) The problem of o-respose i sample surveys, Joural of America Statistical Associatio 41, Rao, P.S.R.S. 1986) Ratio estimatio with sub-samplig the o-respodets, Survey Methodology 1), Rao, P.S.R.S. 1987) Ratio ad regressio estimates with sub-samplig the o-respodets, Paper preseted at a special cotributed sessio of the Iteratioal Statistical Associatio Meetigs, September, Tokyo, Japa, 16. Sigh, H.P., umar, S. 010) Improved estimatio of populatio mea uder two-phase samplig with subsamplig the o-respodets, Joural of Statistical Plaig ad Iferece 1409),

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