Extension of Mangat Randomized Response Model

Transcription

1 Iteratioal Joural of Busiess ad Social Sciece Vol. 2 No. 8; May 2011 Etesio of Magat Radomized Respose Model Zawar Hussai Departmet of Statistics, Quaid-i-Azam Uiversity 45320, Islamabad 44000, Pakista zhlagah@yahoo.com Salma Arif Cheema Departmet of Statistics, Virgiia Tech (VA), Blacksburg 26040, USA. Sidra Zafar Faculty of Mathematical Scieces, Fatima Jiah Wome Uiversity, Rawalpidi, Pakista Abstract I this study, a ia estimatio of populatio proportio of a stigmatized attribute has bee cosidered ad a estimator is developed whe the iformatio from the survey respodets is collected usig the radomized respose techique (RRT) by Magat ad Sigh (1990). Domiace picture of estimator has bee portrayed for a wide rage of values of populatio proportio by usig simple Beta distributio as Prior iformatio. It is observed estimator performed better tha the Maimum Likelihood Estimator (), i particular, for small ad moderate samples. Key Words: Sesitive attribute, ia estimatio, Simple radom samplig ad Mea squared error. 1. Itroductio Utruthful resposes might be received from respodets i a direct questioig approach regardig sesitive attributes. For may reasos, it might be ecessary to acquire iformatio about icidece of stigmatized attribute(s) i the populatio. Warer (1965) was the first researcher who proposed a method of survey to collect iformatio regardig stigmatized/ sesitive attributes by esurig privacy of the respodet. Now a large umber of improvemets ad variats of Warer s radomized respose model (RRM) have bee suggested i several studies, for eample, Greeberg et al. (1969), Chaudhuri ad Mukeree (1988), Magat ad Sigh (1990), Magat et al. (1995), Tracy ad Magat (1996), Mahmood et al. (1998), Bhargava ad Sigh (2000), Sigh et al. (2003), Christofides (2003), Kim ad Warde (2004), Zaizai ( ), Hussai et al. (2007), Perri (2008), Diaa ad Perri (2009, 2010) Hussai ad Shabbir(2011), Hussai et al. (2011a, b), Abid et al. (2011), However, i situatios where prior iformatio is available, ia approach of estimatio ca be adopted i order to estimate the ukow parameter by combiig sample ad prior iformatio. Wikler ad Frakli (1979), Pitz (1980), Spurrier ad Padgett (1980), O Haga (1987), Oh (1994), Migo ad Tachibaa (1997), Uikrisha ad Kute (1999), Bar-Lev ad Bobovich (2003), Barabesi ad Marcheselli (2006), ad Kim et al. (2006) are some of the researchers who made efforts i ia aalysis of radomized respose models. Usig the Magat ad Sigh (1990) RRM ad applyig the ia estimatio we ited to suggest a estimator of populatio proportio. Before movig to the formal developmet of estimator we preset the Magat Sigh (1990) RRMs i the followig sectio. The developmet of estimator is preseted i Sectio 2 ad a comparative study is preseted i Sectio Magat ad Sigh RRM The basic ratioale i Magat ad Sigh (1990) RRM is to establish a radom relatioship betwee the sesitive questios ad idividual s resposes. I this model, two radomizatio devices R 1 ad R 2 are used. The device R1 cosists of the two statemets: (i) do you belog to sesitive group?, ad (ii) go to R 2, preseted with probabilities T ad 1T respectively. The radomizatio device R2 cosists of the two statemets: (i) do you belog to sesitive group?,ad (ii) do you ot belog to sesitive group?, preseted with probabilities P ad 1 P. A respodet is selected ad the respose is recorded as yes if the respodets actual status matches with the selected questio ad o else wise. For a particular respodet, the probability for a yes aswer is give by P yes 2 T P PT 1 1 P 1 T, (1) r 261

2 Cetre for Promotig Ideas, USA The of is give by ˆ 1 P1 T ˆ ML 2 T P PT 1, (2) where, ˆ ad is the umber of yes resposes i the sample of respodets. Moreover, 1 1T 1 P1 1 T 1 P var ˆ ML. (3) 2P 1 2T 1 P 2 3. ia Estimatio of π usig Magat ad Sigh (1990) RRM For the ia estimatio of populatio proportio for this radomized respose model, we assume Beta distributio as the prior distributio for populatio proportio with parameters a ad b, which is as follows: 1 a1 b f 1 1,0 1, a, b 0. (4) ab, Let X i1 i,where X represets the total umber of yes resposes i a sample of size draw from the populatio with simple radom samplig with replacemet. Here i 1 with probability ad i 0 with probability 1, where is as defied i (1). The the coditioal distributio of X give is:! fx / X / 1!! Lettig f or! 2T 2P 2PT 1 PY P T 1!! T P PT PY P T P ad h 2 T P TP 1 T T P TP 2 1 TP T P PT T P T P, we have! f X / X / 2T 2P 2TP 1 f 1 h!! f X /!!! 2T 2P 2TP 1 X /!! i i f h 1, (5) i0 0!! i! i! where, 0,1,2,...,. The oit distributio of X ad is give by: 1 a1 b1! f X, 1 2T 2P 2TP 1 a, b!! Or i0 0! f!! h i i 1 262

3 Iteratioal Joural of Busiess ad Social Sciece Vol. 2 No. 8; May ! f X, 2T 2P 2TP 1 a, b!!!! i ai1 b1 f h 1. (6) i0 0!! i! i! Now the margial distributio of X ca be obtaied by itegratig the oit distributio of X ad. Thus the margial distributio of X is give by: 1! f X 2T 2P 2TP 1 a, b!!!! i f h a i, b. (8) i0 0!! i! i! Now, the posterior distributio of give X is defied as: f, X f / X / X. (9) f X Thus the posterior distributio of give X may be obtaied as:!! i ai1 f h 1 i0 0!! i! i! f / X / X!! i f h a i, b!! i! i! i0 0 The estimator, uder the squared error loss fuctio, is give by!! i f h a i 1, b i0 0!! i! i! ˆ!! i f h a i, b!! i! i! i0 0 b1. (10). (11) The Mea Squared Errors (MSEs) of both the classical () ad ia estimators are defied for a fied value of π ad are writte as ˆ ˆ 2 2 MSE ML E ML ˆ ML 1 (12) 0 Ad 2 2 MSE ˆ E ˆ ˆ 1. (13) 0 4. Compariso For differet values of desig parameters, usig (12) ad (13) we have calculated the MSEs of the both the estimators uder cosideratio. For this purpose R is used to write the codes. The results are displayed graphically i the Figures 1-4(see Appedi). To save the space we have preseted a limited umber of graphs but we have observed that for all values of the desig parameters P adt ad populatio proportio the estimator outperforms the over the complete rage of. We assumed hyperparameters a 1ad b 2 so that prior mea is 1. It is observed that estimator is bouded i the iterval 0,1 eve i the etreme realizatio of 3 yes resposes. For these etremes the may lie outside this iterval which is ot admissible. Thus, wheever it is easier to collect data through Magat ad Sigh RRM ia estimatio may be used to have more reliable estimates. 263

4 Cetre for Promotig Ideas, USA Refereces 1. Abid, M. Cheema, S. A. ad Hussai, Z. (2011). A Efficiecy Compariso of Certai Radomized Respose Strategies. Iteratioal Joural of Busiess ad Social Scieces, 2(7), Barabesi, L.; Marcheselli, M. (2006) A practical implemetatio ad ia estimatio i Frakli's radomized respose procedure. Commuicatio i Statistics- Computatio ad Simulatio. 35, Bar-Lev, S. K. Bobovich, E. ad. Boukai, B. (2003) A commo cougate prior structure for several radomized respose models. Test, 12 (1), Bhargava, M. ad Sigh, R. (2000). A modified radomizatio device for Warer s model, Statistica, 60, Chaudhuri, A. ad Mukeree, R. (1988). Radomized Respose: Theory ad Techiques. Marcel Dekker, New York. 6. Christofides, T. C. (2003). A geeralized radomized respose techique. Metrika 57, Diaa, G. ad Perri, P. F. (2009). Estimatig a sesitive proportio through radomized respose procedures based o auiliary iformatio, Statistical Papers, 50, Diaa, G. ad Perri, P. F. (2010). New scrambled respose models for estimatig the mea of a sesitive character, Joural of Applied Statistics, 37(11), Greeberg, B. G., Abul-Ela Abdel-Latif, A., Simmos, W. R. ad Horvitz, D. G. (1969). The urelated questio RR model: Theoretical framework, Joural of the America Statistical Associatio, 64, Hussai, Z., Shabbir, J. ad Gupta S. (2007). A alterative to Ryu et al. radomized respose model, Joural of Statistics & Maagemet Systems, 10(4), Hussai, Z., Shabbir, J. (2011). A Estimatio of Sesitive Proportio Utilizig Higher Order Momets of Auiliary Variable. Iteratioal Joural of Busiess ad Social Scieces, 2(2), Hussai, Z., Hamraz, M. ad Shabbir, J. (2011a). A improved Quatitative Radomized Respose Model. Iteratioal Joural of Busiess ad Social Scieces, 2(4), Hussai, Z., Aum, S. ad Shabbir, J. (2011b). Improved logit estimatio through Magat radomized respose model. Iteratioal Joural of Busiess ad Social Scieces, 2(5), Kim, J. M., Warde, D. W. (2004). A stratified Warer s radomized respose model. J. Statist. Pla. Iferece, Vol. 120/ Kim, J. M., Tebbs, J. M., A. S. W. (2006). Etesios of Magat s radomized respose model. J. Statist. Pla. Iferece, 36(4) Mahmood, M. Sigh, S. ad Hor, S. (1998). O the cofidetiality guarateed uder radomized respose samplig: a compariso with several ew techiques, Biometrical Joural, 40, Magat, N. S., Sigh, R. (1990). A alterative radomized respose procedure, Biometrika, 77, Magat, N. S., Sigh, S. ad Sigh, R. (1995). A ote o the iverse biomial radomized respose procedure, Joural of the Idia Society of Agricultural Statistics, 47, Migo, H., Tachibaa, V. (1997). ia approimatios i radomized respose models. Comput. Statist. Data Aal, O Haga, A. (1987). liear estimators for radomized respose models. J. Amer. Statist. Assoc, Oh, M. (1994). ia aalysis of radomized respose models: a Gibbs samplig approach. J. Korea. Statist. Soc, Pitz. G. (1980). ia aalysis of radomized respose models. J. Psychological Bull, Perri, P.F. (2008). Modified radomized devices for Simmos model, Model Assisted Statistics ad Applicatios, 3, Spurrier, J., Padgett, W. (1980). The applicatio of ia techiques i radomized respose. Sociological Methodol, Sigh, S., Hor, S., Sigh, R. ad Magat, N.S. (2003). O the use of modified radomizatio device for estimatig the prevalece of a sesitive attribute, Statistics i Trasitio, 6(4), Tracy, D. ad Magat, N. (1996). Some developmet i radomized respose samplig durig the last decade-a follow up of review by Chaudhuri ad Mukeree, Joural of Applied Statistical Scieces, 4,

5 MSE MSE Iteratioal Joural of Busiess ad Social Sciece Vol. 2 No. 8; May Uikrisha, N., Kute, S. (1999). ia aalysis for radomized respose models. Sakhya,Ser. B, Warer, S. L. (1965). Radomized respose: a survey techique for elimiatig evasive aswer bias, Joural of the America Statistical Associatio, 60, Wikler,R., Frakli,L. (1979). Warer s radomized respose model: A ia approach. J. Amer. Statist. Assoc, Zaizai, Y. (2005, 2006). Ratio method of estimatio of populatio proportio usig radomized respose techique, Model Assisted Statistics ad Applicatios, 1, Appedices Figure 1: Graph of MSEs of ˆML ad ˆ for 25, P 0.6 adt 0.7 Figure 2: Graph of MSEs of ˆML ad ˆ for 25, P 0.6 adt

6 MSE MSE Cetre for Promotig Ideas, USA Figure 3: Graph of MSEs of ˆML ad ˆ for 50, P 0.6 adt 0.7 Figure 4: Graph of MSEs of ˆML ad ˆ for 50, P 0.6 adt