Journal of Modern Applied Statistical Metods Volume 5 Issue Article --06 Estimation of Population Mean on Recent Occasion under Non-Response in -Occasion Successive Sampling Anup Kumar Sarma Indian Scool of Mines, Danbad, India, aksarma.ism@gmail.com Garib Nat Sing Indian Scool of Mines, Danbad, India Follow tis additional works at: ttp://digitalcommons.wane.edu/jmasm Part of te Applied Statistics Commons, Social Beavioral Sciences Commons, te Statistical Teor Commons Recommended Citation Sarma, Anup Kumar Sing, Garib Nat (06) "Estimation of Population Mean on Recent Occasion under Non-Response in - Occasion Successive Sampling," Journal of Modern Applied Statistical Metods: Vol. 5 : Iss., Article. DOI: 0.37/jmasm/4780000 Available at: ttp://digitalcommons.wane.edu/jmasm/vol5/iss/ Tis Regular Article is brougt to ou for free open access b te Open Access Journals at DigitalCommons@WaneState. It as been accepted for inclusion in Journal of Modern Applied Statistical Metods b an autorized editor of DigitalCommons@WaneState.
Estimation of Population Mean on Recent Occasion under Non- Response in -Occasion Successive Sampling Cover Page Footnote A. K. Sarma G. N. Sing Department of Applied Matematics, Indian Scool of Mines, Danbad-86004, India. E-mail: gnsing_ism@aoo.com aksarma.ism@gmail.com Tis regular article is available in Journal of Modern Applied Statistical Metods: ttp://digitalcommons.wane.edu/jmasm/vol5/ iss/
Journal of Modern Applied Statistical Metods November 06, Vol. 5, No., 49-70. doi: 0.37/jmasm/4780000 Coprigt 06 JMASM, Inc. ISSN 538 947 Estimation of Population Mean on Recent Occasion under Non-Response in -Occasion Successive Sampling Anup Kumar Sarma Indian Scool of Mines Danbad, India Garib Nat Sing Indian Scool of Mines Danbad, India In tis article, an attempt as been made to stud on general estimation procedures of population mean on recent occasion wen non-response occurs in -occasion successive sampling. Suggested estimators ave advantageousl influenced te estimation procedures in te presence of non-response. Detailed properties of te suggested estimation procedures ave been eamined compared wit te estimation process of te same circumstances but in te absence of non-response. Empirical studies ave been carried out to demonstrate te performances of te estimates suitable recommendations ave been made. Kewords: Non-response, successive sampling, stud variable, variance Introduction Successive sampling was developed for estimation of population parameters on recent point of time (occasion), wen te population parameters canges over successive points of time (occasion). It is a sampling metod to provide reliable fruitful estimates of population parameters over different desire points of time (occasion). Jessen (94) initiated a tecnique wit te elp of past information to provide te effective estimates on current occasion in two-occasion successive sampling. Later, tis tecnique was etended b Yates (949), Patterson (950), Tikkiwal (95), Eckler (955), Rao Graam (964), Gupta (979), Binder Hidiroglou (988), Kis (998), McLaren Steel (000), Sing, Kenned Wu (00), Steel McLaren (00) among oters. Sen (97, 973) applied tis teor in designing te estimators of population mean using information on two or more auiliar variables wic was readil available on A. K. Sarma is in te Department of Applied Matematics. Email at: aksarma.ism@gmail.com. G. N. Sing is in te Department of Applied Matematics. Email at: gnsing_ism@aoo.com. 49
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE previous occasion in two-occasion successive sampling. Sing, Sing Sukla (99), Sing Sing (00) made an efficient use of auiliar variable on current occasion subsequentl Sing (003) uses tis metodolog for - occasion successive sampling in estimation of current population mean. In man situations, information on an auiliar variable ma be readil available on te first as well as on te second occasion. Utilizing te auiliar information on bot occasions, Feng Zou (997), Biradar Sing (00), Sing (005), Sing Karna (009), Sing Prasad (00), Sing, Prasad, Karna (0), Sing, Maji, Maura, Sarma (05) Sing Sarma (04, 05) ave proposed several estimators of population mean on current (second) occasion in two-occasion successive sampling. Non-response is a common problem almost encountered in all sample surves successive sampling is more prone to tis problem because of its repetitive nature. For eample, in agriculture ield surves, it migt be possible tat crop on certain plots are destroed due to some natural calamities or disease so tat ield on tese plots are impossible to be measured. Hansen Hurwitz (946) suggested a metod of sub sampling of non-respondents to address te problems of non-response in mail surves. Later on Cocran (977) Okafor Lee (000) etended tis tecnique for te case wen besides te information on caracter under stud, information is also available on one auiliar caracter. More recentl, Coudar, Batla, Sud (004), Sing Prianka (007), Sing Kumar (008) used te Hansen Hurwitz (946) tecnique for te estimation of population mean on current occasion in contet of sampling on two occasions. Motivated wit te above arguments using Hansen Hurwitz (946) metod, te aim of te present work is to suggest te estimation procedure for population mean at t (recent) occasion wen te non-response occurs on t occasion, (-) t (previous) occasion simultaneousl on bot t (-) t occasions in -occasion successive (rotation) sampling. Te properties of te proposed estimation procedure ave been eamined compared wit te similar estimation but under complete response. Empirical studies are carried out suitable recommendations ave been made. Notations Let U = (U, U, - - -, U N ) be te finite population of N units, wic as been sampled over occasions. Te caracter under studies are denoted b - on te t (-) t occasions respectivel. Assume tat te non-response occur 50
SHARMA & SINGH on t occasion, (-) t occasion simultaneousl on bot t (-) t occasions, so tat te population can be divided into two classes, tose wo will respond at te first attempt tose wo will not. Let te sizes of tese two classes be N N on te t occasion te corresponding sizes on (-) t occasion be N - N -. Let a simple rom sample (witout replacement) of size n be selected on te t occasion wic consist of n n units common to te units observed on te (-) t occasion n n units drawn afres on te t occasion i.e. n n n. Here λ μ (λ + μ = ) are te fractions of matced unmatced samples, respectivel, on te t occasion. Te values of λ μ sould be cosen optimall. Assume tat in te unmatced portion of te sample on te t occasion n units respond n units do not respond. Let n denote te size of sub sample drawn from te non-response class in te s unmatced portion of te sample on te t occasion teir response collected b direct contact or interview. Similarl, n units respond n units do not respond in te sample of matced units let n s denote te size of sub sample drawn from te non-response class in te matced portion of te sample on te t occasion teir response collected b direct contact or interview. Following are te list of notations, wic are considered for teir furter use: Y : : : Te population mean of te stud variable on te t occasion. Te sample mean of te stud variable based on n units common to te units observed on te (-) t occasion. Te sample mean of te stud variable based on n units drawn afres on te t occasion., : Te correlation between te stud variables -. S : N W : N W N : N Te population variance of te variable on te t occasion. Te proportion of non-responding units in te population on te (-) t occasion. Te proportion of non-responding units in te population on te t occasion. 5
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE n f : Te sampling fraction. N f n. n s f n. n s Formulation of Estimator For estimating te population mean Y on te t occasion, a sample mean a regression tpe estimator are suggested. First is te Hansen Hurwitz (946) tpe estimator, sa, wic is based on n sample units drawn afres on t occasion suc tat out of tese n units, n units respond remaining n n n units do not respond. Hence, is defined as () were n n s n Te second estimator is based on te sample of size n, wic is common to te units observed on te (-) t occasion. Because non-response is occurred on te previous occasion, terefore, again Hansen Hurwitz (946) tpe estimator are considered. Te second estimator,, for estimating te population mean on t occasion is a regression tpe estimator, is defined as were, () 5
SHARMA & SINGH n n n, n n n, is population regression coefficient between te stud variable -. Te resulting estimator is a conve linear combination of te estimators. Te estimator is defined as were 0 criterion. - (3) is te unknown constant to be determined under certain Remark : For estimating te mean on t occasion te estimator is suitable, wic implies tat more belief on could be sown b coosing as (or close to ), wile for estimating te cange from one occasion to te net, te estimator could be more useful so migt be cosen as 0 (or close to 0). For asserting bot te problems simultaneousl, te suitable (optimum) coice of is required. Remark : (i) Assume tat te correlation between variables observed on two occasions, more tan one occasion apart is zero. (ii) For practical application te population regression coefficient will be estimated b teir respective sample estimates. Properties of te Estimator Because are sample mean difference tpe estimators respectivel, Y. Terefore, te resulting estimator te are unbiased for population mean defined in equation (3) is also an unbiased estimator of Y. Te variance of te estimator is sown in following teorem. Teorem : Variance of te estimator to te first order of approimations is obtained as 53
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE were, V V V C (4) V W f S n N V W f S,, n n N (5) (6) C (7) N, S Remark 3: Following Hansen Hurwitz (946) tecnique, some variances wic are used in Teorem, are evaluated as given below:,, V V E n n E V n n were Furter we assume tat n V E f S n n f N S S N n N n N S N is te population variance of non response class on t occasion. S N S, ence V W S n N n f (8) Similarl 54
SHARMA & SINGH V V W S n N n W f S n N n f (9) (0) were f n ; f n n s n s Minimum Variance of te Estimator Substituting te values of variances covariance from equations (5), (6) (7) in equation (4) we ave te epression of te eact variance of te proposed estimator. Now, minimize te variance of, wic is sown in equation (4). Define a function f (, ), were te variables are interpreted as respectivel, wic represents te epression of te variance of given in equation (4). Tus, variance of is reduce to following equation S f, f n () were S S,, t, W f,,, n W f, t,, f. N To find te minimum variance, we differentiate te equation () wit respect to respectivel ten equate to zero, () 55
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE (3) From equations () (3), Again, from equations (3) (4), if ten (4) (5) were t r t (6) r (7) Because te values of α depend on te values of correlation. Terefore, 0 consequentl r is real. After iteration, t rk (8) j k j Hence, minimum variance of is obtained from equations () () wic is as follows S V f, t opt opt f (9) n 56
SHARMA & SINGH Special Cases Case : Wen non-response occurs onl on (-) t (previous) occasion. For te case wen non-response occurs onl on (-) t occasion, te estimator for population mean Y on recent occasion ma be structured as were,. (0) determined so as to minimize te variance of te estimator. Properties of te estimator Because is unknown constant to be are sample mean difference tpe estimators respectivel, te are unbiased for population mean Y. Terefore, te resulting estimator defined in equation (0) is also unbiased estimator of Y. is Teorem : variance of te estimator is obtained as, V V V C () were V n N - S V W f S -,, n n N () (3) 57
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE C (4) N, S Minimum Variance of te estimator Similarl, represent te epression of te variance of in equation () as S f, f n (5) To find te minimum variance, (6) (7) From equations (6) (7), Furter, (8) (9) r t t (30) were r (3) 58
SHARMA & SINGH rk j k j t (3) From (5) (6) minimum variance of is epressed as S V, f t opt opt f n (33) Case : Wen non-response occurs onl on t (recent) occasion Te estimator for te population mean Y on recent occasion for tis case ma be given as were (34) is is defined in equation (), unknown constant to be determined so as to minimize te variance of te estimator. Properties of te estimators Because are sample mean difference tpe estimators respectivel, Y. Terefore, te resulting estimator te are unbiased for population mean defined in equation (34) is also unbiased estimator of Y. Teorem 3: Variance of estimators is obtained as, V V V C (35) were V is sown in equation (5), V,, S n n N (36) 59
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE Minimum Variance of te estimator C, S (37) N Te epression of te variance of sown in equation (35) is reduced to te following form S f, f n (38) To find te minimum variance, (39) (40) From equations (39) (40) ten (4) (4) (43) r t t were 60
SHARMA & SINGH r (44) rk j k j t (45) Tus, from (38) (39) minimum variance of is obtained as S V f, t f n opt opt (46) Efficienc Comparison To eamine te loss in precision of te estimators, due to nonresponse, te percent relative loss in precision of estimator, wit. respect to te estimator, ave been computed for different coices of, Te estimator is defined under te similar circumstances as te estimator but in te absence of non-response. Hence te estimator is given as (47), is unknown constant to be determined b te minimization of te variance of. Following Sukatme, Sukatme, Sukatme, Asok (984) te optimum variance of is given b were, S V ˆ opt t f n (48) were tˆ j k j rˆ k rˆ k. 6
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE Remark 4: To compare te performance of te estimators, wit respect to, te assumptions W W (sa W ) are introduced. Te percent relative losses in precision of te estimators, under teir respective optimalit conditions are given b to V V opt V V V V opt opt opt opt L 00 L 00 L V opt V V opt 00 opt opt wit respect Te epressions of opt, opt, opt te percent relative losses are given in terms of te population correlation coefficients. Terefore, te ave been computed for different coices of correlation,. Percent relative losses in precision of te estimators, coices of f, f, f, W, W,. ave been computed for different Presented in Tables - 3 are te optimum values of opt, opt, opt te percent relative losses wit respect to. 6
SHARMA & SINGH Table. Percent relative loss L in precision of wit respect to for f = 0.. Occasions (), 0.5 0.7 0.9 3 4 f W f opt L opt L opt L.0 0..5 0.3668 4.873 0.5 4.84 0.670 5.467.0 0.053 6.5394 0.4443 8.798 0.645 9. 0.4.5 0.053 6.5394 0.4443 8.798 0.645 9..0 0.364.594 0.5978 5.98.0 0..5 0.60 3.0095 0.5745.0443 0.663 9.6643.0 0.443 7.0978 0.500 5.879 0.6357 3.6380 0.4.5 0.700.645 0.568.450 0.6309 7.409.0 0.3503 8.938 0.467 6.8537 0.5750 3.0665.0.0.0.0 0. 0.4 0. 0.4 0. 0.4 0. 0.4.5 0.8 3.53 0.4008.8605 0.57.5974.0 0.0379 3.39 0.794 3.849 0.4545.9800.5 0.0379 3.39 0.794 3.849 0.4545.9800.0 0.0094.478 0.305 0.064.5 0.6048 3.38 0.58.6047 0.575 9.3567.0 0.395 6.9 0.4000 5.0408 0.4640 0.94.5 0.704.9963 0.58.90 0.4778 6.7377.0 0.849 8.6079 0.806 5.354 0.343 6.6839.5 0.709 3.53 0.3753.8605 0.446.5974.0 0.007 3.39 0.305 3.849 0.35.9800.5 0.007 3.39 0.305 3.849 0.35.9800.0 0.94 0.064.5 0.604 3.38 0.5039.6047 0.478 9.3567.0 0.390 6.9 0.3800 5.0408 0.3830 0.94.5 0.704.9963 0.5057.90 0.44 6.7377.0 0.784 8.6079 0.488 5.354 0.7 6.6839 Note: indicate opt does not eist. 63
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE Table. Percent relative loss L in precision of wit respect to for f = 0.. Occasions (), 0.5 0.6 0.8 3 4 f.5.0.5.0.5.0 Note indicate opt does not eist. W opt L opt L opt L 0. 0.6668 3.9706 0.663 3.6895 0.6936.470 0.4 0.8053 6.3585 0.654 6.7430 0.690 4.654 0. 0.8053 6.3585 0.654 6.7430 0.690 4.654 0.4 0.737.73 0.697 8.595 0. 0.6556 4.3688 0.5670 4.6766 0.58 3.888 0.4 0.803 6.849 0.688 8.330 0.588 7.990 0. 0.803 6.849 0.688 8.330 0.588 7.990 0.4 0.796 3.3894 0.604 3.085 0. 0.655 4.406 0.5607 4.8963 0.546 4.6755 0.4 0.803 6.8849 0.656 8.6488 0.553 8.688 0. 0.803 6.8849 0.656 8.6488 0.553 8.688 0.4 0.789 3.7590 0.5766 5.0 Table 3. Percent relative loss L in precision of wit respect to for f = 0.. Occasions (), 0.5 0.6 0.7 3 4 f W opt L opt L opt L.5 0. 0.3668 4.873 0.5 4.84 0.670 5.467 0.4 0.053 6.5394 0.4443 8.798 0.645 9..0 0. 0.053 6.5394 0.4443 8.798 0.645 9. 0.4 0.364.594 0.5978 5.98.5.0.5.0 Note: indicate opt does not eist. 0. 0.8 3.53 0.4008.8605 0.57.5974 0.4 0.0379 3.39 0.794 3.849 0.4545.9800 0. 0.0379 3.39 0.794 3.849 0.4545.9800 0.4 0.0094.478 0.305 0.064 0. 0.709.9384 0.3753.653 0.446-0.676 0.4 0.007.475 0.305.86 0.35-3.638 0. 0.007.475 0.305.86 0.35-3.638 0.4 0.94-3.065 64
SHARMA & SINGH Results Beavior of Estimator, From Table, (a) For te fied values of, f, f W, te value of opt are mostl increased wile te values of L are almost decreased wen te values of, are increased. (b) For te fied values of, f, W,, te values of opt decrease wile L increases wit te increasing value of f. Tis trend sows te larger fres sample is required to be replaced on te recent occasion. (c) For te fied values of, f, W,,, te values of opt L are increasing wit te increasing values of f. (d) For te fied values of, f, f,, te values of opt almost decrease wile L increases wit te increasing value of W. Tis beavior sows tat te iger te non-response rate, te larger fres sample is required to be replaced on te recent occasion. (e) For te fied values of, f, W,, te values of opt L are almost decreasing wit te increasing values of number of occasions (). Tis penomenon suggests tat smaller fres sample is required on te recent occasion wic leads in reducing te cost of te surve. Beavior of Estimator From Table, (a) For te fied values of, f, W, no patterns are visible in te values of opt L wit te increasing value of,. 65
ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE (b) For te fied values of, W,,, te values of opt L are increasing wit te increasing values of f. (c) For te fied values of, f,,, te values of opt L increase wit te increasing values of W. (d) For te fied values of f, W,, te values of opt are decreasing wile te values of L are increasing wit te increasing values of number of occasions (). Tis event suggests tat smaller fres sample is required on te recent occasion so tat cost of te surve is reduced. Beavior of Estimator From Table 3, (a) For te fied values of, f, W te values of opt L are almost increased wen te value of, is increased. (b) For te fied values of,,, W te values of opt decrease wile L increases wit te increasing values of f. Tis penomenon indicates tat if a igl correlated auiliar variable is available it pas in terms of reducing te cost of te surve smaller fres sample is required at te recent occasion. (c) For te fied values of, f,,, te values of opt decreases wile te values of L does not follow an certain pattern wit te increasing value of W. (d) For te fied values of f, W,, te values of opt L are decreasing wit te increasing values of number of occasions (). Tis beavior suggests tat lower te non-response is useful smaller fres sample is required at te recent occasion wic leads in te minimizing te surve cost. 66
SHARMA & SINGH Conclusion On te basis of preceding interpretations, it ma be concluded tat te proposed estimation procedure is more useful fruitful in te estimate of population mean wen non-response occur on t occasion, (-) t occasion simultaneousl on bot t (-) t occasions in te -occasion successive sampling. It is also visible from te empirical studies tat te percent relative loss in precision is not so ig. Hence, te proposed estimators,, are performing well in terms of precision even in te presence of non-responses. Tus te are reliable ma be recommended to te surve statisticians practitioners for its practical applications. Acknowledgement Autors are tankful to te Universit Grants Commission, New Deli Indian Scool of Mines, Danbad for providing te financial assistance necessar infrastructure to carr out te present work. References Binder, D. A., & Hidiroglou. (988). Sampling in Time. In P. R. Krisaia C. R. Rao (Eds.), Sampling (pp. 87-). Amsterdam: Nort Holl. Biradar, R. S., & Sing, H. P. (00). Successive sampling using auiliar information on bot occasions. Bulletin of te Calcutta Statistical Association, 5(03-04), 43-5. Coudar, R. K, Batla, H. V. L., & Sud, U. C., (004). On non-response in sampling on two occasions. Journal of te Indian Societ of Agricultural Statistics 58(3), 33-343. Cocran, W. G. (977). Sampling tecniques (3 rd ed.). New York, NY: Jon Wile & Sons. Eckler, A. R. (955). Rotation sampling. Te Annals of Matematical Statistics, 6(4), 664-685. doi:0.4/aoms/777847 Feng, S., & Zou, G. (997). Sample rotation metod wit auiliar variable. Communications in Statistics - Teor Metods, 6I6), 497-509. doi:0.080/03609970883996 Gupta, P. C. (979). Sampling on two successive occasions. Journal of Statistical Researc, 3, 7-6. 67
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ESTIMATION OF POPULATION MEAN UNDER NON-RESPONSE Tikkiwal, B. D. (95). Teor of successive sampling (Unpublised dissertation). Institute of Agricultural Researc Statistics, New Deli, India. Yates, F. (949). Sampling metods for census surves. London: Carles Griffin Compan Limited. 70