Geometric Distribution as a Randomization Device: Implemented to the Kuk s Model

Transcription

1 It. J. Cotem. Math. Sceces, Vol. 8, 03, o. 5, HIKARI Ltd, Geometrc Dstrbuto as a Radomzato Devce: Imlemeted to the Kuk s Model Sarjder Sgh Deartmet of Mathematcs Texas A&M Uversty-Kgsvlle Kgsvlle, TX 78363, USA sarjder@yahoo.com Iderjt Sgh Grewal Deartmet of Mathematcs, Statstcs ad Physcs Pujab Agrcultural Uversty Pujab, 4004, Ida Isg969@au.edu Coyrght 03 Sarjder Sgh ad Iderjt Sgh Grewal. Ths s a oe access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the orgal work s roerly cted. Abstract I ths aer, a terestg mrovemet the Kuk 990 radomzed resose model has bee suggested. Recetly, Guerrero ad Sadr 007 have show that the famly of estmators roosed by Kuk 990 fars better tha the Smmos famly terms of effcecy ad rotecto. The roosed mroved method s show to have more rotecto ad effcecy tha the Kuk 990 model whle dog real surveys ractce. Mathematcs Subject Classfcato: 6D05 Keywords: Estmato of roorto; Radomzed resose samlg, Resodets rotecto.. INTRODUCTION The roblem of estmato of roorto of a sestve character usg a radomzato devce survey samlg s well kow sce Warer 965. A detaled revew ad

2 44 Sarjder Sgh ad Iderjt Sgh Grewal alcatos of such techques ca be had from Fox ad Tracy 986. Kuk 990 troduced a geous radomzed resose model whch f a resodet belogs to a sestve grou A, the he/she s structed to use a deck of cards havg roorto of cards wth the statemet, I belog to grou A ad, ad f the resodet belogs to osestve grou A the the resodet s requested to use a dfferet deck of cards havg c roorto of cards wth the statemet, I do ot belog to grou A. Assume be the true roorto of ersos belogg to the sestve grou A. Obvously, the robablty of yes aswer the Kuk 990 model s gve by: kuk. Further assume a smle radom wth relacemet samle of resodets s selected from the oulato, ad be the umber of observed yes aswers. The umber of eole that aswer "yes" s bomally dstrbuted wth arameters kuk ad. For the Kuk 990 model, a ubased estmator of the oulato roorto s gve by: ˆ kuk,. wth varace, gve by: V ˆ kuk.3 kuk kuk A varety of radomzed resose models such as the Warer 965, Magat 994, Magat ad Sgh 990 are secal cases of Kuk 990 model. Magat 994 model s further mroved by Gestvag ad Sgh 006. Guerrero ad Sadr 007 have show that the famly of radomzed resose models roosed by Kuk 990 fars better tha the Smmos famly terms of effcecy ad rotecto. I the ext secto, we suggest a terestg mrovemet the Kuk 990 model. Chaudhur 0 has gve a decet revew o radomzed resose samlg.. PROPOSED RANDOMIZED RESPONSE MODEL I the roosed radomzed resose model, each resodet selected the samle s rovded wth two decks of cards the same way as Kuk 990 model. I the frst deck of cards, let be the roorto of cards wth the statemet, I belog to grou A ad be the roorto of cards wth the statemet, I do ot belog to grou A. I the secod deck of cards, let be the roorto of cards wth the statemet, I do ot belog to grou A ad be the roorto of cards wth the statemet, I belog to grou A. U to here, t s same as the Kuk 990 radomzed resose model. Now the roosed model, f a resodet belogs to grou A, he/she s structed to draw cards, oe-by-oe usg wth relacemet, from the frst deck of cards utl he/she gets the frst card bearg the

3 Geometrc dstrbuto as a radomzato devce 45 statemet of hs/her ow status, ad requested to reort the total umber of cards, say X, draw by hm/her to obta the frst card of hs/her ow status. If a resodet belogs to grou c A, he/she s structed to draw cards, oe-by-oe usg wth relacemet, from the secod deck of cards utl he/she gets the frst card bearg the statemet of hs/her ow status, ad requested to reort the total umber of cards, say Y, draw by hm/her to obta the frst card of hs/her ow status. Obvously G X ad G Y, that s, X ad Y follow Geometrc dstrbutos wth arameters ad resectvely, because cards are draw based o wth relacemet samlg. Let Z be the umber of cards reorted by the th resodet, the we have the followg theorem. Theorem.. A ubased estmator of the oulato roorto s gve by: ˆ Z,. Proof. Gve G X ad G Y, therefore, we have: Z E. By takg the exected value o both sdes of. ad usg. we have: ˆ Z E Z E E whch roves the theorem. Theorem.. The varace of the roosed estmator ˆ s gve by: ˆ V.3 Proof. Sce the resoses are deedet, thus we have:

4 46 Sarjder Sgh ad Iderjt Sgh Grewal Z ˆ V Z V V V Now we have:.4 V Z E Z E Z.5 Gve X G ad Y G, therefore, we have E Z.6 Usg. ad.6.5, ad the usg t.4 we have.3. Hece the theorem. I the ext secto, we make rules about the choce of ad the roosed radomzed resose devce by keeg the effcecy of the roosed estmator ad the rotecto of resodets as the major ssues to be cosdered real surveys. 3. EFFICIENCY COMPARISONS The ercet relatve effcecy RE of the roosed model over the Kuk 990 model s defed as: V ˆ kuk RE 00% 3. V ˆ Kee md that f, the the Kuk 990 model s same as the Warer 965 model, ad f ad 0 < <, the the Kuk 990 model s same at the Magat 994 model. Now the Kuk 990 model, the choce of ad s such that the dfferece should be maxmum to reduce the varace of the estmator ˆ kuk. by keeg the resodets cooerato md. The RE value s free from the samle sze. I a vestgato, we decded to kee 0. 7 ad 0. whch could be cosdered as a good choce of radomzato devce arameters the Kuk s model by keeg the resodets cooerato md ad maxmum dfferece betwee ad. We searched for choce of ad for dfferet values of such that the ercet relatve effcecy value remas hgher tha 00%. The results so obtaed are reseted Table 3..

5 Geometrc dstrbuto as a radomzato devce 47 Table 3.. The values of, RE ad RE values for 0. 7, 0. ad RE RE RE

6 48 Sarjder Sgh ad Iderjt Sgh Grewal Dscusso of Results: Assume the vestgator exects the value of close to 0., the stead of choosg 0. 7 ad 0., f the vestgator decdes to do survey wth 0. ad 0. 3, the relatve effcecy remas 66.7%; ad the relatve effcecy may be creased to 365.3% by choosg Note that the tervewer does ot kow whch deck of cards has bee used by the tervewee thus t wll be hard to guess from whch deck the umber of reorted cards s draw. The choce 0. 3 ad 0. 7 may gve better cooerato wth the resodets, but the ercet relatve effcecy remas 64.4%. It s very terestg to ote that the choce 0. 7 ad 0. gves less effcet results. Aga assume the vestgator exects the value of close to 0., the stead of choosg 0.7 ad 0., f the vestgator decdes to do survey wth 0. ad 0. 3, the relatve effcecy remas 9.%; ad the relatve effcecy may be creased to 5.9% by choosg I the same way the rest of the results Table 3. ca be terreted. We coclude that t s always ossble to make the suggested geometrc dstrbuto radomzato devce estmator more effcet tha the Kuk s model. REFERENCES [ ] A. Chaudhur. Radomzed resose ad drect questog techques surveys. CRC Press, Lodo, 0. [ ] J. A. Fox. ad P.E. Tracy. Radomzed resose: A method for sestve surveys. SAGE Publcatos. 986 [ 3 ] C. R. Gjestvag ad S. Sgh. A ew radomzed resose model. Joural of the Royal Statstcal Socety, B, , [ 4 ] M. Guerrero ad M.F. Sadr. A ote o the comarso of some radomzed resose rocedures. Joural of Statstcal Plag ad Iferece,37007, [ 5 ] A.Y.C. Kuk. Askg sestve questos drectly. Bomerka, 77, 990, [ 6 ] N.S. Magat A mroved radomzed resose strategy. J. R. Statst. Soc., B, , [ 7 ] N.S. Magat ad R. Sgh A alteratve radomzed resose rocedure. Bometrka, [8 ] S.L. Warer. Radomzed resose: a survey techque for elmatg evasve aswer bas. J. Amer. Stat. Assoc., 60, 965, Receved: November, 0