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1 esearch Joural o athematcal ad tatstcal ceces I 0 60 stmato o ato o two Populato meas a lass o ato-cum egresso tpe estmators usg ular character wth double amplg the presece o o-respose bstract KPatel ad aredra Kumar Departmet o tatstcs, Uda Pratap utoomous ollege, Varaas, UP, IDI valable ole at: wwwsca, wwwscame eceved 0 th Jue 0, revsed th Jul 0, accepted 0 th ugust 0 I ths paper, we represet estmato o rato o two populato meas ato-cum regresso tpe estmators usg aular character wth double samplg the presece o o-respose ad also stud ad dscuss ther propertes ter comparg the proposed estmator wth relevat estmators, we coclude that the proposed estmators are more ecet o the bass o ther mea square error For umercal support, the emprcal stud s gve Keword: ato estmator, regresso estmator, stud varable, aular varable, double samplg, o-respose Itroducto he ecec o ato, product ad regresso tpe estmators or estmato o rato o two populato meas o stud characters deped o the ormato o aular character Whe pror ormato closel related to stud varable s avalable, rato estmator s used ad case o lear relato shp betwee stud ad aular varables ests, regresso estmator s more useul egresso estmator s more sutable tha rato estmator regresso le does t pass through org stmato o rato o two populato meas s wdel used calculato o growth rate medcal scece, crop producto rate agrculture, lterac rate educato ad the eld o soco-ecoomcs o-respose errors ma arse mal due to ault samplg rame, based method o selecto o uts the selected sample etc or the estmato o populato mea the presece o o-respose, Hasa ad Hurvtz has gve a techque o sub samplg rom the o-respodg uts Further l-adr suggested a method to sedg several waves o questoare b mal surves to reduce the eect o orespose gh, rpath,, Upadhaa ad gh 6, rvastava et al, gh et al 8,9 ad gh ad gh 0 have proposed deret tpes o estmators or estmato o rato o two populato meas usg aular varable Whe all the ormato regardg aular varate are ot ow the we use double samplg rvastava, Kregera, have suggested deret rato, regresso ad product tpe estmators or estmato o populato mea usg double samplg Usg Hase ad Hurwtz techque, case o ow ad uow populato meas o aular characters have bee doe b ochra, ao,6, Khare ad umar ad Khare ad ehma 8 have proposed covetoal ad alteratve estmators or estmato o populato mea the presece o o-respose he estmato o rato o two populato meas usg aular varables the presece o o-respose have bee proposed b Khare ad pade 9, Khare ad ha 0,,,, Khare et al,, Kumar ad Patel 6 I recet paper, we preset a class o rato-cum-regresso tpe estmators or estmato o rato o two populato meas usg aular character double samplg presece o o-respose lso relatve bas ad mea square error o the suggested estmator are derved or ecec comparso to the relevat estmators ad studed propertes For perormace o the suggested estmator, percetage relatve ececes are also calculated mathematcall he estmators Let Y =,, ad p =,,, be the o- egatve value o p-th ut o the populato o the stud characters =, ad the aular character wth ther populato meas Y =,, ad Let us suppose the populato o uts ca be dvded to respose class o uts ad o-respose class o uts such that + = havg meas Y ad Y respectvel wth stud varates =, ad ad respectvel wth aular varate I Hase ad Hurwtz techque, a sample o < uts s selected out o populato uts usg smple radom Iteratoal cece ogress ssocato 9

2 esearch Joural o athematcal ad tatstcal ceces I 0 60 samplg wthout replacemet method ad s made a surve schedule each oe o the sample It s observed that sample o sze <, ol uts respodg ad = uts ot respodg wth meas ad regardg stud varates =, ad ad regardg aular varates hus, a sub sample o r= uts, tae rom orespodg uts, s selected radoml wth mea r r regardg stud varates =, ad regardg aular varates ad ormato rom each r uts s collected b persoal tervew Hece, the Hase Hurwtz ubased estmator or populato mea Y o stud characters =, ad o aular character based o + r uts are gve as r ad r are the ubased estmator o Y ad Y respectvel but based estmator o, characters r Y o = ad r are the ubased estmator o c ad respectvel but based estmator o o character d also the varace o Hase- Hurwtz ubased estmator o stud characters =, ad aular character are gve b V ad V ad W W =, tratum weght o o- respose class o uts ; the mea square o p Y p Y characters =, based o uts o the populato d, ; the mea square o Y p Y p characters =, based o o-respodg uts o the populato lso, p ; the mea square o p characters based o uts o the populato d, p ; the mea square o p characters based o o-respodg uts o the populato Let Y Y characters =, ; the rato o two populato meas o stud he, order to estmate, Dee a covetoal estmator ; the rato o two Hase-Hurvtz ubased estmators or populato meas, Y o stud characters = I ths case, whe populato mea o aular character s ot ow he order to estmate, a prelmar sample o sze > s selected rom populato o uts usg WO techque havg mea regardg aular varate the a sub sample o prelmar sample o sze s selected usg WO techque havg mea regardg aular varate We dee a covetoal estmator,whe ormato o both stud characters =, ad aular character are complete, ad a alteratve estmator, whe both stud characters =, have complete but aular character has complete ormato, or estmato o rato o two populato meas double samplg presece o orespose as ollow ad p p ad p p Usg above stuato, Khare ad ha have suggested a mproved classes o covetoal ad alteratve estmators or estmato o rato o two populato meas double samplg presece o o-respose as ollow ad Iteratoal cece ogress ssocato 0

3 esearch Joural o athematcal ad tatstcal ceces I 0 60 he proposed class o estmators: O the bass o above ormato ad codto, we propose a class o rato-cum regresso tpe estmators or estmato o rato o two populato meas usg aular character wth double samplg presece o o-respose s gve as a' b ' a b ad,, a ad b are ether real umber or uctos o ow parameters o obta elatve as ad ea quare rror: For ths purpose,let us assume, Y, Y, ' ad uch that- = = = = = 0 Usg above substtuto, the equato epress as terms o ad eglectg rd ad hgher terms, we have ' s { 6 ad he optmum value o δ s gve as opt Y Y, ', ' }, ' ad a a b,, ' lso,,,,, ', ',, ',,, ' d,,,, Y, Y, =,, ad For optmum value o ad, we have to mmze epresso wrt φ ad θ respectvel, we have m ad m Puttg the optmum value o θ ad φ, we get m { able- presets some members o the proposed class o estmators or deret values o,, a ad b equato able- ember o stmators a b ' ' ' 0 0 ' ' ' ' 6 a' b ' a b a' b ' a b 8 a' b ' a b } a b ρ ρ a b = -, Y, Y, = +, ad a ad =, he relatve bas ad mea square error o members o the proposed estmator, meto table, ca be epressed b Iteratoal cece ogress ssocato

4 esearch Joural o athematcal ad tatstcal ceces I 0 60 Iteratoal cece ogress ssocato b sutabl puttg the values o,, a ad b, equato 6 & respectvel, whch are gve as ad he optmum value o δ s gve as opt, stad or β ρ, β ρ lso the epresso o relatve bas ad mea square error o estmators, meto,, ad, have bee derved ad gve as } { m m cec comparso heoretcal comparso o proposed estmator over other estmators omparso wth 0 whe θ > 0 the θ < -λ whe θ < 0 the θ > -λ ad, 0 whe φ > 0 the φ< -λ whe φ < 0 the φ > -λ omparso wth 0 whe θ > the θ- < -λ whe θ < the θ- > -λ ad, 0 whe φ > the φ-< -λ whe φ< the φ- > -λ omparso wth 0 whe θ > the θ- < -λ whe θ < the θ- > -λ ad, 0 whe φ > 0 the φ< -λ whe φ < 0 the φ > -λ omparso wth We observe that, these two estmators have almost same sgcace or optmum values o costats omparso wth λ > 0 ad

5 esearch Joural o athematcal ad tatstcal ceces I 0 60 mathematcal comparso For perormace o our proposed classes o estmator, we have to calculate mea square error mathematcall ad compare wth relevat estmators usg the data whch has bee used b Khare ad ha ad belog to the data o phscal growth o upper soco ecoomc group o 9 schools gog chldre o Varaas uder a I stud, Departmet o pedatrcs, HU, durg 98-8 I ths data Let ad =,,the aular ad stud characters, are deed as Y : he heght o chldre cm Y : he weght o chldre g : he chest crcumerece o chldre cm the values o parameters related to the stud characters Y =, ad aular character, whe rst % e chldre uts has bee cosdered as o-respose uts, are gve as : = 9 = = Y = 96 Y = 9968 = 86 = 006 = 06 = = 000 = 00 = 060 = 086 = 00 = 09 = 000 = 0 = 068 = = = = 0009 = = 96 able : the mea square error ad relatve ecec o the estmators or deret values o ocluso ter aalzg o above table-, we coclude that the proposed class o estmators s more ecet tha relevat estmators, ad as well as ts members, 6, ad 8 but t s equall precse to estmator or deret values o ad ed value o ad lso t s observed that the ad percet o the proposed class o estmators crease wth crease o the value o able- estmators % % % m m m Iteratoal cece ogress ssocato

6 esearch Joural o athematcal ad tatstcal ceces I 0 60 eereces Hase H ad Hurwtz W, he problem o orespose sample surves, Joural o the merca tatstcal ssocato,, L-adr, samplg procedure or maled questoares Joural o the merca tatstcal ssocato,, gh P, O the estmato o rato ad product o populato parameters, aha,,, rpath P, cotrbuto o the samplg theor usg multvarate ormato, PhD thess submtted to Pujab Uverst, Patala, Ida, 90 rpath P, geeral class o estmators or populato rato, aha,,-, Upadhaa L ad gh HP, class o estmators usg aular ormato or estmatg rato o two te populato meas, Gujarat tat ev,, rvastava, a rvastava, ad Khare, O geeralzed cha estmator or rato ad product o two populato meas usg aular characters,,, gh VK, gh, Har P, gh, Housla P ad hula D, geeral class o ha estmators or rato ad product o two meas o populato, ommu tstst heor-ethod,, - 99a 9 gh VK, gh, Har P ad gh Housla P, stmato or rato ad product o two populato meas two- phase samplg, Jour tstst Pla, Ierece,, 6-99b 0 gh VP ad gh HP, ha estmators or populato rato double samplg, lgarh Jour tatat, ad 8, rvastava K, two phase estmator samplg surve, ust Jour tatst,, -, 90 Kregera, cha rato tpe estmator te populato double samplg usg two aular varables etra,, Kregera, egresso-tpe estmators usg two aular varables ad model o double samplg rom te populatos, etra,, ochra WG, amplg techques, hrd edto Joh wle ad sos, ew or 9 ao, P, ato ad regresso stmates wth subsamplg the orespodets, Paper ssocato eetg preseted at a specal cotrbuted sesso o the Iteratoal tatstcal, eptember, -6, oo, Japa 98 6 ao, P egresso estmators wth sub samplg o orespodets I: Gumer Leps ad V Uppulur, arcel Deer, eds Data qualt cotrol theor ad Pragmatcs, ew Yor, Khare, Pade K ad rvastava U, mproved class o two phase samplg estmators or populato mea usg aular character presece o orespose, J c es HU,, Khare ad Habb ur ehma, Geeralzed rato cum regresso tpe estmator or populato mea usg aular varable wth double samplg the presece o o respose, Joural o socal ad ecoomc statstcs,, Khare ad Pade K, class o estmators or rato o two populato meas usg aular character presece o o-respose, J c es HU, 0, Khare ad ha, stmato o the rato o two populato meas usg aular character wth uow populato mea presece o o-respose, Prog o aths HU, 6,, Khare ad ha, stmato o te populato rato usg two phase samplg presece o o-respose, lgarh J tat,, Khare ad ha, stmato o the rato o two populatos meas usg mult-aular characters the presece o o-respose Publshed tatstcal echques Le estg, elablt, amplg heor ad Qualt otrol edted b Pade, arosa Publshg House, ew Delh, 6-00 Khare ad ha, Improved class o estmators or rato o two populato meas wth double samplg the o-respodets, tatsta, heor eth, 9, Khare, rvastava U ad Kumar K, ha tpe estmators or rato o two populato meas usg aular characters the presece o o-respose, J c es, HU, 6, Khare, rvastava U ad Kumar K, Improved classes o ha tpe estmators or rato o two populato meas usg two aular characters the presece o o-respose, Iteratoal Joural o dvace tatstcs ad Probablt,,-60 6 Kumar ad Patel K, stmato o rato o two populato meas usg regresso estmator presece o o-respose, esearch Joural o athematcal ad tatstcal ceces,, -, 0 Iteratoal cece ogress ssocato