Ratio-Type Estimators in Stratified Random Sampling using Auxiliary Attribute
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1 roceedg of te Iteratoal Multoferece of Egeer ad omuter cett 0 Vol I IME 0 Marc - 0 Hog Kog Rato-ye Etmator tratfed Radom amlg ug Auxlary Attrbute R V K g A Amed Member IAEG Abtract ome rato-tye etmator ave bee rooed tratfed radom amlg ug auxlary attrbute e exreo for te ba ad mea quare error of te rooed etmator ave bee derved u to frt order of aroxmato omaro ave bee made t tradtoal combed rato etmator ad t o tat te rooed etmator are more effcet ta combed rato etmator uder certa codto For llutrato a emrcal tudy a bee carred out Keyord: roorto Ba ME Rato-tye etmator tratfed radom amlg I IRODUIO e rato etmator a oberved to be more rece ta te uual amle mea etmator uder dfferet codto for etmatg te oulato mea of te tudy caracter everal reearcer dverted ter atteto te drecto of ug ror value of certa oulato arameter to fd te etmate tat are more rece earl () ued coeffcet of varato of tudy caracter at etmato tage I ractce coeffcet of varato eldom o Motvated by earl () or varou autor cludg e () eodya ad Deved () g et al () ad Uadyaya ad g () ued te o coeffcet of varato of auxlary caracter for etmatg oulato mea of te tudy caracter rato metod of etmato g et al () frt made te ue of ror value of coeffcet of urto etmatg te oulato varace of tudy caracter Later ued by earl ad Iteraac (0) Recetly g ad alor (00) rooed a modfed rato etmator by ug te o value of correlato coeffcet ag to coderato Maucrt receved Aug 0; reved Ja5 0 RVKg eor lecturer ad Head Deartmet of Matematc Kebb tate Uverty of cece ad ecology Alerogera (e-mal: grv@gmalcom) AAamed Atat lecturer Umau dafodyo uverty ooto gera (e-mal:amedgborom@yaoocom ) te ot beral correlato coeffcet betee auxlary attrbute ad tudy varable Jajj et al (00) ad g et al (00) defed rato etmator of oulato mea e te ror formato o auxlary varable oeg ome attrbute avalable oder a radom amle of ze to be tae from a oulato of ze tratfed to trata Let a amle of ze be dra by mle radom amlg tout relacemet from a tratum of ze Let y ad deote te obervato o a radom varable y ad reectvely for t ut uoe tere a comlete dcotomy te oulato t reect to te reece or abece of a attrbute ay ad t aumed tat attrbute tae oly to value 0 ad a f I ut of te oulato oee attrbute 0 otere e e ave te follog defto; A - deote te total umber of ut te oulato oeg attrbute A - deote te total umber of ut te tratum oeg attrbute a - deote te total umber of ut te amle dra from tratum oeg attrbute IB: I: 0-05 (rt); I: 0-0 (Ole) IME 0
2 roceedg of te Iteratoal Multoferece of Egeer ad omuter cett 0 Vol I IME 0 Marc - 0 Hog Kog A - deote te roorto of ut te oulato oeg attrbute A - deote te roorto of ut te tratum oeg attrbute a - deote te roorto of ut te amle dra from tratum oeg attrbute I tratfed radom amlg te tradtoal combed rato etmator for te oulato mea ug auxlary attrbute defed a; yt R () t Were y y y t y t e Ba ad Mea quare Error (ME) of te tradtoal etmator are gve by equato () ad () reectvely a; y Ba R () ME y R R () y Were te egt of tratum R te oulato rato te umber of ut amle from tratum te oulato ze of tratum y te oulato varace tratum te oulato varace of auxlary attrbute tratum ad y te oulato covarace betee auxlary attrbute ad varable of teret tratum II ROOED EIMAOR e rooed etmator ca be rtte te form belo; ( y b ( )) ( ) ( m m ) m m ( 0) () Were m ( 0) ad m are eter real umber or te fucto of te o arameter of te attrbute uc a coeffcet of varato coeffcet of urto B ( ) ad ot beral correlato coeffcet b b y y y ad Remar : e e ut b 0 m ad m 0 te rooed etmator reduce to tradtoal etmator e follog ceme reet te etmator of te oulato mea c ca be obta by utable coce of cotat m ad m EIMAOR y b y b y b y b b y b 5 y b b VALUE OF m m 0 b IB: I: 0-05 (rt); I: 0-0 (Ole) IME 0
3 roceedg of te Iteratoal Multoferece of Egeer ad omuter cett 0 Vol I IME 0 Marc - 0 Hog Kog y b b y b b b y b b y b 0 b b b b b b b b III EFFIIE OMARIO We comare te tradtoal etmator t te rooed etmator ( 0) ad te codto for c te rooed etmator ll ave te leat mea quare error ere obtaed a follo; ME ME R y by y R Ry b y y R R R Let A B b y e Ba ad Mea quare Error (ME) u to frt order aroxmato of te rooed etmator are gve by equato () ad () reectvely a; Ba ( 0) () Were 5 b b b b b y b ME R 0 () Were R R R b R R R5 b R R b 0 b b R b R IB: I: 0-05 (rt); I: 0-0 (Ole) b e e ave y R AB R A R R A R A B R 0 A R R BR 0 Were tere are to codto a follo; () We R R 0 B R A 0 R R B R A R R () We R R 0 B R A 0 R R B R A R R We eter of tee codto atfed te rooed etmator 0 te tradtoal etmator ll be more effcet ta IME 0
4 roceedg of te Iteratoal Multoferece of Egeer ad omuter cett 0 Vol I IME 0 Marc - 0 Hog Kog IV EMIRIAL UD e formato o 500 tudet tae from tudet re- Medcal Regtrato Umau Dafodyo Uverty ooto (0/0 eo) a ued a data for emrcal tudy e egt of te tudet te varable of teret ad ter geder a ued a auxlary attrbute (Male= ad Female=0) e tratfcato baed o te faculte By ug eyma allocato e ave y () y We ave comuted amle ze eac tratum e ummary formato o te emrcal data are gve belo; y Hegt of te tudet Geder of te tudet b E t ma tor Ba 5 able II RELAIVE BIAE AD EFFIIEIE Rel Ba M E Relat ve effc ecy o Rat o ( R ) B R X R R od for effce cy A X or A X Udef ot ed atfed x0-5 atfed 005 x0-5 atfed 00 5 x0-5 atfed x0-5 atfed x0-5 atfed x0-5 atfed x0-5 atfed x0-5 atfed x0-5 atfed 0 able I DAA AII tratu m o Faculty tratum ze aml e ze tratum arameter Agrc y 0 y 5 b 5 Vetary 00 y Medce 0 y b Educato 5 5 Art ad Ilamc tude y 0 y 5 b y 05 0 y 5 b 5 5 La y 5 0 y 5 b 5 olof 5 y 0 Healt c 0 y b 5 ocal 0 y cece 0 0 y b 05 cece 5 Maagem et c y 0 y b y y b IB: I: 0-05 (rt); I: 0-0 (Ole) IME 0
5 roceedg of te Iteratoal Multoferece of Egeer ad omuter cett 0 Vol I IME 0 Marc - 0 Hog Kog From able II e oberved tat te rooed etmator tat ue ome o value of oulato roorto erform better ad le ba ta te tradtoal etmator V OLUIO I te reet aer e ave develoed ome rato-tye etmator tratfed radom amlg for etmatg oulato mea by ug formato o auxlary attrbute By comaro t foud tat te rooed etmator are more effcet ad le ba ta te tradtoal combed rato etmator ee teoretcal codto are alo atfed by te reult of a alcato t orgal data For ractcal uroe te coce of te etmator deed uo te avalablty of te oulato arameter REFEREE [] A R e Etmato of te oulato mea e te coeffcet of varato o ommu tatt eory Met A () 5- [} B V odya ad V K Deved modfed rato etmator ug coeffcet of varato of auxlary varable Jour Id oc Agr tattc - [] D earl e utlzato of o coeffcet of varato te etmato rocedure Joural of Amerca tattcal Aocato 5 5- [] D earl ad Itaraac A ote o a etmator for te varace tat utlze te urto e Amerca tattca 5-0 [5] H g ad Ralor Ue of o correlato coeffcet etmatg te fte oulato mea tattc rato [] H Jajj M K arma ad L K Grover A famly of etmator of oulato mea ug formato o auxlary attrbute a J tatt () [] Jg B adey ad KHrao O te utlzato of a o coeffcet of urto te etmato rocedure of varace A It tat Mat [] L Uadayaya ad H g O te etmato of te oulato mea t o coeffcet of varato Bometrcal Joural () 5- [] Rg aua aa ad Fmaradace Rato etmator mle radom amlg ug formato o auxlary attrbute a J tat OerRe () -500 [0] V K g R V Kg ad G g Effcecy comaro of modfed cla of rato tye etmator ug coeffcet of varato of auxlary varable roc Mat oc B H U Vol 05- IB: I: 0-05 (rt); I: 0-0 (Ole) IME 0
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