An Efficient Class of Exponential Estimator of Finite Population Mean Under Double Sampling Scheme in Presence of Non-Response

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1 Global Journal of Pure an Applie Mathematics. ISSN Volume 3, Number 9 (07), pp Research Inia Publications An Efficient Class of Eponential Estimator of Finite Population Mean Uner Double Sampling Scheme in Presence of Non-Response Yater Tato Department of Mathematics, NERIST, Nirjuli-7909, Arunachal Praesh, Inia Corresponing author Abstract The present stu eals with the problem of estimating the population mean in simple ranom sampling in the presence of non-response. Using an auiliar variable an eponential ual to ratio cum ual to prouct estimator of the population mean of the stu variable in ouble sampling is efine. The bias an mean square error (MSE) of the propose estimator has been obtaine for both the cases. The asmptoticall optimum estimator (AOE) of the propose estimator has also been obtaine along with its bias an MSE. Comparisons have been mae with the eisting similar estimators theoreticall an numericall to emonstrate the superiorit of the propose estimator. Kewors: Non-response, Bias, Mean square error (MSE), optimum estimator, Efficienc.. INTRODUCTION All sample surves on large population are susceptible to a variet of errors that affect parts of the surve process an results. Non response of surve forms is one tpe of such errors that are cause in selecte sample surve as a result of the failure to measure information from some of the units in the selecte sample. The result of nonresponse in the surve makes the sample size to be smaller than the epecte quantum an in cases of high non-response rates, the results ma show high variances. This situation ma cause biase estimates of the population parameters. Hansen an

2 5300 Yater Tato Hurwitz (946) were the first to recognize that non-response coul lea to biase estimates of population characteristics while conucting mail surves. The propose a technique of sub-sampling the non-responents to eal with the problem of nonresponse an its ajustments. The evelope an unbiase estimator for population mean in the presence of non-response b iviing the population into responing an non-responing group an b taking a sub-sample of the non-responing units in orer to avoi bias ue to non-response. The problem of estimation of population mean of the stu character utilizing auiliar information have been emonstrate b Srivastava (97), Re (974), Ra an Sahai (980), Srivenkataramana (980), Srivastava an Jhajj (98), Khare an Srivastava (993), Khare an Sinha (004) an Singh an Kumar (0). For the estimation of population mean X of the auiliar variable, a large first phase U U U U of N units b :,,..., N sample of size n is selecte from a finite population simple ranom sampling without replacement (SRSWOR). A smaller secon phase sample of size n is selecte from n b SRSWOR. Non-response occurs on the secon phase sample of size n in which n units respon an n units o not. From the n nonresponents, b SRSWOR a sample of r r n k, k units is selecte where k is the inverse sampling rate at the secon phase sample of size n. All the r units respon this time roun. The auiliar information can be use at the estimation stage to compensate for units selecte for the sample that fail to provie aequate responses an for population units missing from the sampling frame. An unbiase estimator for the population meany of the stu variable propose b Hansen an Hurwitz (946) is efine as n n n n r where an r enote the sample means of variable base on n an r units respectivel. The estimators is unbiase an has variance where, n N V S S () ( ) N ( i ) N i W k N, W n N S N Y an S ( i Y) N i are the population mean square of for the entire population an for the non-responing part of the population.

3 An Efficient Class of Eponential Estimator of Finite Population Mean 530 Similarl, the estimator for population mean X in the presence of non-response base on corresponing ( n r) observations is given b n n n n r where an r are the sample means of variable base on n an r units respectivel. We have V S S ( ) where N S ( ) i X an N S ( i X ) are population mean N N i squares of for the entire population an non-responing part of the population. In case, when population mean X is not known, then, it is estimate b taking a preliminar sample of size n n Nfrom the population of size N b using simple ranom sampling without replacement (SRSWOR) sampling scheme. Neman (938) was the first to give the concept of ouble sampling in estimating the population parameters an then several authors have use the concept to foun more precise estimates of population parameter. Khare an Srivastava (995) propose conventionalt an alternative T ouble sampling ratio estimators for population mean Y in the two ifferent cases of nonresponse, i.e. when there is non-response on both the stu variable as well as on the auiliar variable an when there is non-response in the stu variable onl, which are given as R an R i where n i n i, n i n i. The MSE of R an R are given respectivel as MSE( ) R Y C C k C n n () MSE( R ) Y C C k C C k n n (3)

4 530 Yater Tato where C, C S Y, C S X C are coefficient of variation, k C C, k C C, S S S is correlation coefficient between the stu variable variables an the auiliar variable C, C are the variances for the whole population for variables an C, C are the population variances for the stratum of non-response for the variables an respectivel an S, S are the covariances for the whole population an the population of non-responents respectivel. Singh an Vishwakarma (007) suggeste the eponential ratio an prouct-tpe estimators for Y in ouble sampling respectivel as er ep an ep ep The MSE of er an ep in the two cases of non-response are given respectivel as MSE er Y C C 4k C 4 n n MSE er Y C C 4k C C 4k 4 n n 4 MSE ep Y C C 4k C 4 n n (4) (5) (6) MSE ep Y C C 4k C C 4k 4 n n 4 In the present paper, we have suggeste an efficient class of eponential estimator for the population mean uner ouble sampling scheme in the presence of non-response where population mean of auiliar variable is not known. The optimum bias an MSE of the propose estimators are obtaine. The properties of the suggeste estimator have been stuie an. theoretical comparisons of the optimum MSE are mae with others eisting estimators an illustrate with the help of an empirical stu. (7)

5 An Efficient Class of Eponential Estimator of Finite Population Mean THE PROPOSED ESTIMATOR Utilizing information on the auiliar variable with unknown population mean X, we have suggeste an efficient class of eponential estimator in ouble sampling scheme in presence of non-response. The following two cases will be consiere separatel. Case I: When non-response occurs onl on. Case II: When non-response occurs on both an.. Case I: Non-Response onl on The propose estimator is ep ep (8) erp where N n N n. an, are unknown constants such that To obtain the bias an MSE of erp, we write, X e an X e Y e 0 Epressing erp in terms of e s, we obtain g erp e e e e e n where g N n. 3g e e e e 8 g e e ee e gee ge ge Epaning the above equation, multipling out an ignoring terms of e's greater than two, we get g Y Y e e e e e e e e e e erp g e e e e 8

6 5304 Yater Tato g e e e e e e e e e ge e ge ge 0 0 (9).. Bias, MSE an Optimum Value of In this case, we have 0 3 erp in Case I E( e ) E( e ) E( e ) E( e ) 0 E( e ) C C 0 E( e ) C E( e ) C n N E( e3 ) C z n N E e e ( 0 ) kc E e e n N ( 0 ) kc E e e n N ( ) C On taking epectations on both sies of the equation (9) an using the results of (0), the bias of erp to the first orer of approimation is given b 3 B( erp ) Y g C gk C gk C g C () 8 n n n n n n n n Squaring both sies of the above equation (9), taking epectations an using the results of (0), we obtain the MSE of erp MSE erp Y C C gc g 4k 4 n n to the first orer of approimation as (0) gc g k g C n n n n () Differentiating () in terms of gives its optimum value as g k g = opt (sa) (3) Substituting the value of (3) in (), we get the optimum MSE of erp MSE( erp ) opt Y C C kc n n (4) as

7 An Efficient Class of Eponential Estimator of Finite Population Mean 5305 Remarks. When, the propose estimator reuces to eponential ual to ratio estimator er uner ouble sampling. The bias an MSE of in () an () as follows er are obtaine b putting B( er ) Y g C gk C 8 n n n n MSE er Y C C gc g 4k 4 n n (5). When 0, the propose estimator reuces to ouble sampling eponential ual to prouct estimator. The bias an MSE of ep in () an () as follows 3 B( ep ) Y g C gk C 8 n n n n ep are obtaine b putting 0 MSE ep Y C C gc g 4k 4 n n.. Efficienc Comparisons of erp in Case I: opt (6) (i) Comparison with Mean Per Unit Estimator: From () an (4), we have V MSE( erp ) opt Y kc 0, (7) n n (ii) Comparison with Usual Ratio Estimator in Double Sampling: From () an (4), we have ( ) ( ) MSE R MSE erp opt Y C k 0 n n (8) (iii) Comparison with Eponential Ratio Estimator in Double Sampling: From (4) an (4), we have

8 5306 Yater Tato ( ) MSE er MSE erp opt Y C k 0 4 n n (iv) Comparison with Eponential Prouct Estimator in Double Sampling: From (6) an (4), we have (9) ( ) MSE ep MSE erp opt Y C k 0 4 n n (v) Comparison with Eponential Dual to Ratio Estimator in Double Sampling: From (5) an (4), we have ( ) MSE er MSE erp opt Y C g k 0 4 n n (vi) Comparison with Eponential Dual to Prouct Estimator in Double Sampling: From (7) an (4), we have (0) () ( ) MSE ep MSE erp opt Y C g k 0 4 n n. Case II: Non-Response on Both an () The suggeste estimator in case II is given as follows erp ep ep (3) where N n N n. an, are scalar constants such that.. Bias, MSE an Optimum Value of In this case, we have 0 3 erp in Case II E( e ) E( e ) E( e ) E( e ) 0 E( e ) C C 0 E( e ) C C E( e ) C n N

9 An Efficient Class of Eponential Estimator of Finite Population Mean 5307 E( e3 ) C z n N 0 E( e e ) k C k C E e e n N ( 0 ) kc where X e. E e e n N ( ) C (4) e e Replacing b an taking epectations on both sies of the equation (9) an using the results of (4), we obtaine the bias of given b erp to the first orer of approimation is 3 3 B( ) Y g C g C gk C gk C 8 n n 8 n n erp gk C gk C g C g C n n n n (5) e e Replacing b, squaring both the sies of the equation (9), taking epectations an using the results of (4), we obtain the MSE of the estimator approimation as erp to first orer of MSE erp Y C C gc g 4k gc g 4k 4 n n gc g k gc g k n n g C g C n n Differentiating (6) in terms of gives its optimum value as (6) gc ( g k ) gc ( g k) n n g C g C n n A (sa) (7) B = opt

10 5308 Yater Tato where A k C k C n n an B gc gc n n. Substituting the value of (7) in (6), we get the optimum MSE of erp ( ga MSE erp ) opt Y C C B (8) Remarks. When, the propose estimator reuces to eponential ual to ratio estimator uner ouble sampling. The bias an MSE of er in (5) an (6) as follows as er are obtaine b putting B( er ) Y g C g C gk C gkc 8 n n n n MSE er Y C C gc g 4k gc g 4k 4 n n (9). When 0, the propose estimator reuces to ouble sampling eponential ual to prouct estimator. The bias an MSE of ep in (5) an (6) as follows ep are obtaine b putting 0 3 B( ep ) Y g C g C gk C gk C 8 n n n n MSE ep Y C C gc g 4k gc g 4k 4 n n.. Efficienc Comparisons of erp (i) Comparison with Mean per Unit Estimator: From () an (8), we have opt in Case II: (30) ga erp opt V MSE( ) Y 0 B (3)

11 An Efficient Class of Eponential Estimator of Finite Population Mean 5309 (ii) Comparison with Usual Ratio Estimator in Double Sampling: From (3) an (8), we have B g MSE MSE Y A g B ( R ) ( erp ) opt 0 (iii) Comparison with Eponential Ratio Estimator in Double Sampling: From (5) an (8), we have B g er erp opt MSE MSE( ) Y A 0 g B (iv) Comparison with Eponential Prouct Estimator in Double Sampling: From (7) an (8), we have (3) (33) B g ep erp opt MSE MSE( ) Y A 0 g B (v) Comparison with Eponential Dual to Ratio Estimator in Double Sampling: From (9) an (8), we have g er erp opt MSE MSE( ) Y gb A 0 B (vi) Comparison with Eponential Dual to Prouct Estimator in Double Sampling: From (30) an (8), we have (34) (35) g ep erp opt MSE MSE( ) Y gb A 0 B (36) 3. EMPIRICAL STUDY In this section, we have illustrate the relative efficienc of the estimators with respect to. For this purpose, we have use the ata consiere b Khare an Sinha (007). The ata are base on the phsical growth of upper-socio-economic group of 95 school chilren of Varanasi istrict uner an ICMR stu, Department of Paeiatrics, BHU, Inia uring The escription of the population is given below: : Chest circumference of the chilren (in cm)

12 530 Yater Tato : Weights of the chilren (in kg) N 95, n 70, n 35, N 7, N 4, Y 9.5, X 55.86, S 9.46, S 5.547, S 0.758, S 6.300, S 0.85, 0.79, , S , 0.08, 0.04, S 4.648, S Table : PRE of the ifferent estimators ofy with respect to W k * R er 7. ep er 5.06 ep 87.0 erp

13 An Efficient Class of Eponential Estimator of Finite Population Mean 53 Table : PRE of the ifferent estimators ofy with respect to W k R er ep er ep erp RESULTS AND DISCUSSIONS In the present stu, the stu proposes an eponential ual to ratio cum ual to prouct estimators an erp in two phase sampling in the presence of nonresponse. The bias an MSE of the propose estimators have been obtaine in the two erp ifferent cases. The MSE equations have been compare with the MSEs of the estimators,,,,, R er ep er ep, R, er, ep, er, ep on a theoretical basis an the situations uner which the propose estimator at its optimum is more efficient than the estimators uner consierations have been obtaine. The same is seen an iscusse in relation to other estimators in terms of percent relative efficienc (PRE) from Table an Table.

14 53 Yater Tato REFERENCES [] Cochran, W.G., 977, Sampling techniques, 3 r e., John Wile an sons, New York. [] Hansen, M. H. an Hurwitz, W. N., 946, The problem of non-response in sample surves, Journal of the American Statistical Association, 4, pp [3] Khare, B. B., an Srivastava, S., 993, Estimation of population mean using auiliar character in presence of non-response, National Acaem Science Letters-Inia, 6, pp. -4. [4] Khare, B. B., an Srivastava, S., 995, Stu of conventional an alternative two phase sampling ratio, prouct an regression estimators in presence of nonresponse, Proceeings-National Acaem Of Sciences Inia Section A, 65, pp [5] Khare, B. B., an Srivastava, S., 997, Transforme ratio tpe estimators for the population mean in the presence of non-response, Comm. Statist. Theor Methos, 6, pp [6] Khare, B.B., an Sinha, R.R., 00, General class of two phase sampling estimators for the population means using an auiliar character in the presence of non-response, Proc. of Vth International Smposium on Optimization an Statistics, AMU, Aligarh, pp [7] Khare, B.B., an Sinha, R.R., 004, Estimations of finite population ratio using two phase sampling scheme in the presence of non-response, Aligarh Journal of Statistics, 4, pp [8] Khare, B.B., an Sinha, R.R., 007, Estimation of the ratio of the two population means using multi auiliar characters in the presence of nonresponse, Statistical Technique in Life testing, Reliabilit, Sampling Theor an Qualit Control, Eite b B.N. Pane, Narosa publishing house, New Delhi, pp [9] Neman, J., 938, Contribution to the theor of sampling human populations, Journal of the American Statistical Association, 33(0), pp [0] Rao, P. S. R. S., 986, Ratio estimation with sub sampling the nonresponents, Surve Methoolog,, pp [] Ra, S.K., an Sahai, A., 980, Efficient families of ratio an prouct tpe

15 An Efficient Class of Eponential Estimator of Finite Population Mean 533 estimators, Biometrika, 67, pp [] Re, V.N., 974, On a transforme ratio metho of estimation, Sankha, 36C, pp [3] Singh, H.P., an Vishwakarma, G.K., 007, Moifie eponential ratio an prouct estimators for finite population mean in ouble sampling, Austrian Journal of Statistics, 36(3), pp [4] Singh, R., an Kumar, M., 0, A note on transformation on auiliar variable in surve sampling, Moel Assiste Statistics an Applications, 6(), pp oi 0.333/MASA [5] Srivastava, S.K., 97, A generalize estimator for the mean of a finite population using multi-auiliar information, Journal of the American Statistical Association, 66 (334), pp [6] Srivastava, S. K., an Jhajj, H.S., 98, A class of estimators of the population mean in surve sampling using auiliar information, Biometrika, 68(), pp [7] Srivenkataramana, T., 980, A ual to ratio estimator in sample surves, Biometrika, 67(), pp

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