ESTIMATION OF A POPULATION MEAN OF A SENSITIVE VARIABLE IN STRATIFIED TWO-PHASE SAMPLING

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1 Pak. J. Stati. 06 Vol. 3(5), ESTIMATION OF A POPUATION MEAN OF A SENSITIVE VARIABE IN STRATIFIED TWO-PHASE SAMPING Nadia Muaq, Muammad Noor-ul-Amin and Muammad Hanif National College of Business Adminiration and Economics aore. Pakian. dia_qau@yaoo.com drmiananif@gmail.com COMSATS Initute of Information and Tecnology aore, Pakian. nooramin.ats@gmail.com ABSTRACT In tis udy, we present te eimation of population mean of a sensitive variable in ratified sampling using two-pase sampling based on randomized response tecnique. We introduce a ratio, a regression and general class of eimators for te mean of sensitive variable using non-sensitive auxiliary variable based on randomized response tecnique in ratified two-pase sampling. Under ratified two pase sampling, te expression of bias and mean square error (MSE) up to te fir-order approximations are derived. Simulation udies and real data are presented to demonrate te performance of proposed eimators.. INTRODUCTION In social science researc, survey respondents esitate to answer sensitive queions. Tis explains wy traditional self-report surveys often suffer from ig levels of nonresponse and disone answers. To overcome tese problems, Warner (965) introduced Randomized Response Tecnique (RRT), wic elps interviewers extract reliable data corresponding to sensitive queions wile maintaining respondent anonymity. RRT models allow respondents to mask teir actual response by giving a scrambled response wic te researcer is later able to unscramble at an aggregate level but not at an individual level. Te ratified random sampling is used to eimate te parameters if te population consi of eterogeneous units. Many autors ave presented te ratio and regression eimators suc as Kadilar and Cingi (003), Kadilar and Cingi (005), Kadilar et al. (007), Sabbir and Gupta (007, 00), Koyuncu and Kadilar (00), Sousa et al. (00), Gupta et al. (0) and Sousa et al. (04). Two-pase sampling is a procedure in wic we obtain te information about auxiliary variable(s) from a larger sample at fir pase and relatively small sample from te second pase. Many autors suggeed some improved ratio, regression and product type eimators included as Sukatme (96), Sing and Viswakarma (007), Saoo et al. (00), Noor-ul-Amin and Hanif (0) Sanaulla et al. (04), Rasid et al. (05) etc. 06 Pakian Journal of Statiics 393

2 394 Eimation of a Population Mean of a Sensitive Variable In tis udy we presents ratio, regression eimator of population mean of a sensitive variable using non-sensitive auxiliary variable using RRT metodology in ratified two pase sampling. We also propose a general class of eimators for eimating population mean of a sensitive variable using non-sensitive auxiliary variable using RRT metodology in ratified two-pase sampling.. TERMINOOGY Consider a finite population of size N wic is ratified into omogenous rata. t et N be te size of ratum (,..., ) suc tat N N. et Y be te udy variable, a sensitive variable wic cannot be observed directly due to respondent bias and be non-sensitive auxiliary variable wic is correlated wity. et S be scrambling variable independent of Y and. So te reported response is given as Z Y S. And ( y, x ) be te observations of te udy variable (y) and te auxiliary variable ( x ) on te i i t i unit of t ratum respectively. Wen information on is unknown, a fir large sample of size n is selected from eac t ratum to eimate. To eimate Y N W Y, we eimate tat i N W is known, were W i N. N N et Z W Z be te population mean for te reported variable Z. To obtain te bias i and mean square error (MSE) under ratified two-pase sampling, let us define N Z W Z, z Z e, i 0 e 0 W Z e 0 Z N W, x e, i W e e, x e, e W e. Te expectations are defined given as: E e E e E e, V z 0 r s E z Z E x r s r, s W r s Z, E e W S V Z x 0 x 0 E e W S V, E e W S V 0 zx Z E e e W S V

3 Nadia, Noor-ul-Amin and Hanif zx Z E e e W S V, 0 V0 V 0, V V.,, n N n N, yx zx s s S S S. zx zx z x Te procedure of ratified two-pase sampling is as follows: s y, i. Select a sample of size n from te witout replacement t ratum using simple random sampling SRSWOR suc tat n n and observe auxiliary caracteriic for tese units. Tis is called s ratified fir- pase sample. ii. Select anoter ratified random sample of size n from eac n n n using SRSWOR suc tat n n and collect information on sensitive variable of te intere. Tis is called a second-pase sample. Under ratified two-pase sampling, usual unbiased eimator for population mean of te sensitive variable is given by t Yd W z (.) wic is unbiased eimator of population mean Y and using Z Te Var of t Yd is given by Y. W Sy Ss W Sz 0 Var tyd Y Y Y V Y Y (.) were n N N, S y yi Y N and S s si S N i N. We Consider ratio and regression eimators under ratified two-pase sampling as, i and x Wx trd z W z x Wx t z x x W z W x W x (.3) regd

4 396 Eimation of a Population Mean of a Sensitive Variable Using Z Y and te mean square errors of te above eimators, up to fir order approximation are MSE t Y V rd regd zx zx MSE t Y V. 3. Proposed Eimators Sousa et al. (04) ave introduced ratio and regression eimators for te mean of sensitive variables based on a Randomized Response Tecnique (RRT) in ratified sampling. Motivated by tis, we propose a general family of eimators for te mean of sensitive variable based on a Randomized Response Tecnique (RRT) in ratified twopase sampling, is given by te following expression x b x x tsid kz kx x exp x b x x b (3.) were k and k are weigts wose values are to be determined, 0 or, and b is te real number or known parameters of te auxiliary variable suc as WC x and Wx were 4 E x x E x Some special cases of te general family of eimators t Sid given in (3.). Generalized Exponential Type Eimators 0 t k z k x x S0d exp x x x x t k z k x x Sd exp x x x x t k z k x x S d exp x x x x Generalized Ratio Type Eimators t k z k x x S3d x x t k z k x x S 4d x x t k z k x x S5d x x b 0

5 Nadia, Noor-ul-Amin and Hanif Te Bias and Mean Square Error of te General Family of Eimators Using notations from section, te general family of eimators given in (3.) may be expressed as given below: were tsid k Z e k e e g b 0 ge ge exp g e e g e e tsid Y k Y k Y e g e e g e e 8 Using (3.3), te Bias and MSE of t Sid 0 0 g e e e e (3.) k e e g e ee. (3.3), are given by Bias t k Y k Y g 3 5 g 8 sid 0 k g 0 (3.4) MSE tsid Y k 7 4 k V0 g 0 g k g 5 3 g k 8 Y k g 0 kk g 0 Y Y And optimum values of k and k, respectively, are found as, (3.5)

6 398 k opt Eimation of a Population Mean of a Sensitive Variable g V zx g Y V kopt g k opt g V0 Subituting tese optimum values in (3.5), te minimum MSE of t Sid is given by g 0 4 MSE tsid Y min g V0 V0 zx g (3.6) By using (3.6), for different values of minimum MSEs t 0,,,3, 4,5 of Sid i. 4. SIMUATION STUDY b and 0 or, we can get te We use te simulation udies for efficiency comparison by empirically and teoretically. Two populations for simulation udies of size 000 eac from bivariate normal populations for (Y, ), wit different covariance matrices are used. Te Scrambling variable S N0, 0. x and Z Y S is te reported response. Mean of Y, given as 5,5 Population : Y ; Y Population :, Y 0.95; Y For eac population we considered tree sample sizes for fir pase: n = 60,50 and 300. Te population is divided in two rata according to a certain criteria set for te auxiliary variable. Te sample size from eac ratum is based on Neyman allocation and for second pase given as: n = 5, 55, 5 respectively. Table and gives te empirical and teoretical MSE s for te various eimators based on order approximation. Te empirical MSE s are computed by using t and teoretical MSE s are computed by using MSE t.

7 Nadia, Noor-ul-Amin and Hanif 399 We eimate te empirical MSE using 5000 samples of size n for fir pase and n for second pase. We use te following expression to find te percent relative efficiency PRE of udy eimators as compared to te ordinary sample mean: MSE t Yd PRE 00 MSE t were R, Re g, S 0, S, S, S3, S 4, S5. d d d d d d d d 5. NUMERICA EAMPE For tis analysis, we consider te real population used by Sousa et al. (04). Te data come from a sample from te survey on Information and Communication Tecnologies (ICT) usage in enterprises in 00 wit seat in Portugal (Smilily and Storm, 00). et Y be te purcase orders in 00, is te enterprises of turnover. And S N0, 0. x so te reported scrambled responses on Y is given by Z Y S (te purcase order value plus a random quantity). Sampling Information: N 698, , 0.884, 4.44, 7.97, Y.39, 5.3. Y Y Te variable and Y are expressed in millions of Euros. We te our ratified sample eimators wit random sample of sizes for fir pase n =00, 50, 500 and for second pase n = 45, 00, 05. Te proportional allocation as been used for allocating sample size of eac ratum. Y Stratum N Y Y Y Table 3 presents te empirical and teoretical results of MSE eimates and PRE of te various eimators in te ratified sample. We eimate te empirical MSE using 5000 samples wit random sample of sizes for fir pase n =00, 50, 500 and for second pase n = 45, 00, 05. According to te MSE and PRE results in Table 3, te proposed a general family of eimators for eimating sensitive mean eimator based on randomized response tecnique in ratified two pase sampling is considerably better tan te exiing

8 400 Eimation of a Population Mean of a Sensitive Variable eimators i.e., usual eimator, ratio eimator and regression eimator in ratified two pase sampling. 6. CONCUSION In tis udy, we consider a ratio, a regression and propose a general class of eimators for mean of sensitive variable based on randomized response tecnique in ratified two-pase sampling. Te expression for bias and MSE are derived. From Tables -3, it is observed tat te teoretical and empirical MSE and PRE of te family of eimators are performed better tan te usual eimator, ratio and regression eimator in ratified two pase sampling based on randomized response tecnique. Tese results are computed wit a simulation udies and using a real data set. REFERENCES. Gupta, S., Sabbir, J. and Sera, S. (00). Mean and sensitivity eimation in optional randomized response models. Journal of Statiical Planning and Inference, 40(0), Kadilar, C. and Cingi, H. (005). A new eimator in ratified random sampling. Communication in Statiics-Teory and Metods, 34, Kadilar, C., Candan, M. and Cingi, H. (007). Ratio eimators using robu regression. Hacettepe Journal of Matematics and Statiics, 36(), Kadilar, C. and Cingi, H. (003). Ratio eimator in ratified sampling. Biometrical Journal, 45, Koyuncu, N. and Kadilar, C. (00). On te family of eimators of population mean in ratified random sampling. Pakian Journal of Statiics, 6(), Noor-ul-Amin, M. and Hanif, M. (0). Some exponential eimators in survey sampling. Pakian Journal of Statiics, 8(3), Rasid, R., Noor-ul-Amin, M. and Hanif, M. (05). Exponential eimators for population mean using te transformed auxiliary variables. Applied Matematics Information and Science, 9(4), Saoo,., Misra, G., and Nayak, S. (00). On two different classes of eimators in two-pase sampling using multi-auxiliary variables. Model Assied Statiics and Applications, 5(), Sanaulla, A., Ali, H.A., Noor-ul-Amin, M. and Hanif, M. (04). Generalized Exponential cain ratio eimators under ratified two-pase random sampling. Applied Matematics and Computation, 6, Sabbir, J. and Gupta, S. (007). On improvement in variance eimation using auxiliary information. Commun. in Stati.-Teory and Metods, 36(), Sabbir, J. and Gupta, S. (00). Eimation of te finite population mean in twopase sampling wen auxiliary variables are attribute. Hacettepe Journal of Matematics and Statiics, 39(), -9.. Sing, H.P. and Viswakarma, G.K. (008). A family of eimators of population mean using auxiliary information in ratified sampling. Communication in Statiics- Teory and Metods, 37(7), Smilily, M. and Storm, H. (00). ICT usage in enterprises Euroat Publications, Issue.

9 Nadia, Noor-ul-Amin and Hanif Sousa, R., Sabbir, J., Real, P.C. and Gupta, S. (00). Ratio Eimation of te Mean of a Sensitive Variable in te Presence of Auxiliary Information. Journal of Statiical Teory and Practice, 4(3), Sousa, R., Gupta, S. Sabbir, J. and Real, P.C. (04). Improved Mean Eimation of a Sensitive Variable Using Auxiliary Information in Stratified Sampling. Journal of Statiics & Management Syem, 7(5&6), Sukatme, R.V. and Sukatme, B.V. (970). Sampling Teory of Surveys wit Applications, Iowa State University Press. 7. Warner, S.. (965). Randomized response: A survey tecnique for eliminating evasive answer bias. Journal of te American Statiical Association, 60, Yadav, S.K., Kadilar, C., Sabbir, J. and Gupta, S. (05). Improved family of eimators of population variance in simple random sampling. Journal of Statiical Teory and Practice, 9(), 99-6.

10 40 Eimation of a Population Mean of a Sensitive Variable Table Empirical and Teoretical MSE, PRE for te Eimators Relative to RRT Mean Eimator in Stratified Two Pase Random Sampling for Population MSE Eimation N N Y n n Eimation Empirical Teoretical PRE 000 N 550 Y tyd N 450 Y trd tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d

11 Nadia, Noor-ul-Amin and Hanif 403 Table Empirical and Teoretical MSE, PRE for te Eimators Relative to RRT Mean Eimator in Stratified Two Pase Random Sampling for Population MSE Eimation N N Y n n Eimation Empirical Teoretical PRE 000 N 550 Y tyd N 450 Y trd tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d

12 404 Eimation of a Population Mean of a Sensitive Variable Table 3 Empirical and Teoretical MSE, PRE for te Eimators Relative to RRT Mean Eimator in Stratified Two Pase Random Sampling for Real Data Set MSE Eimation N N Y n n Eimation Empirical Teoretical PRE 698 N 979 Y tyd N 36 Y trd N3 357 Y tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d tyd trd tregd ts0d tsd ts d ts3d ts 4d ts5d