On Improved Estimation of Population Mean using Qualitative Auxiliary Information

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1 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No On Imroved Esimaion of Poulaion Mean using Qualiaive Auxiliary Informaion UBHAH KUMAR YADAV * A.A. ADEWARA. Dearmen of Mahemaics and aisics (A enre of Excellence Dr. RML Avadh Universiy Faizabad- 00 U.P. INDIA. Dearmen of aisics Universiy of Ilorin Ilorin. Kwara ae Nigeria * orresonding auhor: drskysas@gmail.com Absrac This aer deals wih he esimaion of oulaion mean of he variable under sudy by imroved raio-roduc ye exonenial esimaor using qualiaive auxiliary informaion. The exression for he bias and mean squared error (ME of he roosed esimaors has been derived o he firs order of aroximaion. A comaraive aroach has been aded o sudy he efficiency of roosed and revious esimaors. The resen esimaors rovide us significan imrovemen over revious esimaors leading o he beer ersecive of alicaion in various alied areas. The numerical demonsraion has been resened o elucidae he novely of aer. Keywords: Exonenial esimaor auxiliary aribue Proorion bias mean squared error efficiency. Mahemaics ubjec lassificaion 00: 6D05. Inroducion The use of sulemenary (auxiliary informaion has been widely discussed in samling heory. Auxiliary variables are in use in survey samling o obain imroved samling designs and o achieve higher recision in he esimaes of some oulaion arameers such as he mean or he variance of he variable under sudy. This informaion may be used a boh he sage of designing (leading for insance o sraificaion sysemaic or robabiliy roorional o size samling designs and esimaion sage. I is well esablished ha when he auxiliary informaion is o be used a he esimaion sage he raio roduc and regression mehods of esimaion are widely used in many siuaions. The esimaion of he oulaion mean is a burning issue in samling heory and many effors have been made o imrove he recision of he esimaes. In survey samling lieraure a grea variey of echniques for using auxiliary informaion by means of raio roduc and regression mehods has been used. Paricularly in he resence of muli-auxiliary variables a wide variey of esimaors have been roosed following differen ideas and linking ogeher raio roduc or regression esimaors each one exloiing he variables one a a ime. The firs aem was made by ochran (90 o invesigae he roblem of esimaion of oulaion mean when auxiliary variables are resen and he roosed he usual raio esimaor of oulaion mean. Robson (957 and Murhy (96 worked ou indeendenly on usual roduc esimaor of oulaion mean. Olkin (958 also used auxiliary variables o esimae oulaion mean of variable under sudy. He considered he linear combinaion of raio esimaors based on each auxiliary variable searaely making use of informaion relaed o he sulemenary characerisics having osiive correlaion wih he variable under consideraion. ingh (967a dwel uon a mulivariae exression of Murhy s (96 roduc esimaor. Furher he muli-auxiliary variables hrough a linear combinaion of single difference esimaors were aemed by Raj (965. In nex bid of invesigaion ingh (967b exended he raio-cum-roduc esimaors o mulisulemenary variables. An innovaive idea of weighed sum of single raio and roduc esimaors leading o mulivariae was develoed by Rao and Mudholkar (967. Much versaile effor was made by John (969 by considering a general raio-ye Esimaor ha in urn resened n unified class of esimaors obaing various aricular esimaors suggesed by revious auhors such as Olkin s (958 and ingh s (967a. rivasava (97 deal wih a general raio-ye esimaor unifying reviously develoed esimaors by eminen auhors engaged in his area of invesigaion. earls (96 and isodia & Dwivedi (98 used coefficien of variaion of sudy and auxiliary variables resecively o esimae oulaion mean of sudy variable. rivenkaaramana (980 firs roosed he dual o raio esimaor for esimaing oulaion mean. Kadilar and ingi ( analyzed combinaions of regression ye esimaors in he case of wo auxiliary variables. In he same siuaion Perri (005 roosed some new esimaors obained from ingh s ( b ones. ingh and Tailor (005 Tailor and harma (009 worked on raio-cum-roduc esimaors. harma and Tailor (00 roosed a raio-cum-dual o raio esimaor for he esimaion of finie oulaion mean of he sudy variable y. In he series of imrovemen Das and Triahi (978 rivasava and Jhajj (980 ingh e.al (988 Prasad and ingh ( Naik and Gua (99 eccon and Diana (996 Agarwal e al. (997 Uadhyay and ingh ( Abu-Dayyeh e al. (003 Javid habbir (006 Kadilar and ingi (006 ingh e.al (007 ingh e.al (009 Muhammad Hanif e.al (00 Yadav (0 Pandey e.al (0 hukla

2 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No e.al (0 Onyeka (0 ec have roosed many esimaors uilizing auxiliary informaion. In he resen sudy we sugges a new esimaor for esimaing oulaion mean of he variable under sudy. The use of auxiliary informaion may increase he recision of an esimaor when sudy variable Y is highly correlaed wih auxiliary variable X. When he variable under sudy Y is highly osiively correlaed wih he auxiliary variable X hen he raio ye esimaors are used o esimae he oulaion arameer and roduc esimaors are used when he variable under sudy Y is highly negaively correlaed wih he auxiliary variable X for imroved esimaion of arameers of variable under sudy. However here are siuaions when informaion on auxiliary variable is no available in quaniaive form bu in racice he informaion regarding he oulaion roorion ossessing cerain aribue is easily available (see Jhajj e.al. [7] which is highly associaed wih he sudy variable Y. For examle (i Y may be he use of drugs and may be he gender (ii Y may be he roducion of a cro and may be he aricular variey. (iii Y may be he amoun of milk roduced and a aricular breed of cow. (iv Y may be he yield of whea cro and a aricular variey of whea ec. (see habbir and Gua [5]. ( Le here be N unis in he oulaion. Le i i y i.. N be he corresonding observaion values of he i h uni of he oulaion of he sudy variable Y and he auxiliary variable resecively. Furher we assume ha i and i 0 i. N if i ossesses a aricular characerisic or does N n A a i i no ossess i. Le i and i denoe he oal number of unis in he oulaion and A a samle resecively ossessing he aribue P. Le N and n denoe he roorion of unis in he oulaion and samle resecively ossessing he aribue. Le a simle random samle of size n from his ( y oulaion is aken wihou relacemen having samle values i i i.. n. Naik and Gua [5] defined he following raio and roduc esimaors of oulaion mean when he rior informaion of oulaion roorion of unis ossessing he same aribue is available as P y( (. y( P (. The ME of esimaors and u o he firs order of aroximaion are f Y [ y + ( K ] (.3 ME ( f Y [ y + (+ K ] (. ingh e.al [7] defined he following raio roduc and raio & roduc esimaors resecively of oulaion mean using qualiaive auxiliary informaion as P 3 yex( P+ (.5 P yex( P (.6 P P 5 y α ex( + ( αex( P+ P (.7 Where α is a real consan o be deermined such ha he ME of 5 is minimum. Forα 5 reduces o he 3

3 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No. 03 esimaor 3 and forα 0 i reduces o he esimaor. The ME of he esimaors 3 and 5 u o he firs order of aroximaion are resecively as 3 f Y [ y + ( K ] ME ( f Y [ y + ( + K ] 5 f Y [ y + ( + α α + ρby( α] which is minimum for imum value of α as K + α α0 ( say and he minimum ME of 5 is ME f Y ( ρ M( min ( 5 y b 5 which is same as ha of radiional linear regression esimaor. where and ρ y y Y b i P N ( i P N y y K y y ρ N ( Yi N i is he oin biserial correlaion coefficien. b y y Y( P i N ( Yi Y N i f n N (.8 (.9 (.0. uggesed Esimaors Moivaed by Prasad [9] and Gandge e al. [6] we roose The exonenial raio ye esimaor as P kyex( P+ The exonenial roduc ye esimaor as P kyex( P The exonenial dual o raio ye esimaor as * P 3 y ex * + P * ( NP ni i * ( N n g P g i N where or i (+ i... n * g (+ g P g where N n. The exonenial raio and dual o raio ye esimaor as (. (. (.3 which usually gives

4 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No P + αex ( α ex P+ y * * P + P where α is a real consan o be deermined such ha he ME of he esimaor 3 and forα 0 i reduces o he esimaor 3. To obain he bias and mean squared error (ME of he esimaors le y Y( + e0 P( + e E( e 0 i 0 and such ha i and E ( e0 f y E ( e f E( e e f y fρby and 0 From (. by uing he values of y and we have P P(+ e ky(+ e0ex P+ P(+ e e ky(+ e0ex + e e ky( + e0ex is minimum. For (. α reduces o (.5 Exanding he righ hand side of (.5 and reaining erms u o second owers of e s and hen subracing Y from boh sides we have e e e e Y ky 0 + e + 0 Y 8 Taking execaions on boh sides of (.6 we ge he bias of he esimaor u o he firs order of aroximaion as B( kyf ( K + Y( k (.7 quaring boh sides of equaion (.6 gives ( Y k Y + e 0 e e + 8 e0e + Y ky + e 0 e e + 8 e0e (.6 and now aking execaion we ge he ME of he esimaor o he firs order of aroximaion as f Y [ k fy + (k k f( ρ by + ( k ] The minimum of is obained for he imal value of k which is A k B where A f( ρ by + B fy + f( ρ by + and Thus he minimum ME of he esimaor is obained as (.8 (.9 5

5 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No ME ( min A B Y (.0 Following he same rocedure as above we ge he bias and ME of he esimaor u o he firs order of aroximaion as B( kyf ( + K + Y( k (. f Y [ k fy + (k k f( + ρ by + ( k ] (. The minimum ME of he esimaor is obained for imal value of k as A MEmin( Y B (.3 where he imal value of k for he esimaor is A k B A f + ρ + ( b y where Exressing (.3 in erms of e s we have Y(+ e 3 0 ge ex and B fy + f( + ρby + Exanding he righ hand side of (. and reaining erms u o second owers of e s and hen subracing Y from boh sides we have 3 ge g e ge e Y Y 0 + e Y 8 (. (.5 Taking execaions on boh sides of (.5 we ge he bias of he esimaor 3 u o he firs order of aroximaion as g B ( 3 Yfg ( + K (.6 From equaion (.5 we have ge ( 3 Y Y( e0 + (.7 quaring boh sides of (.7 and hen aking execaions we ge ME of he esimaor 3 u o he firs order of aroximaion as g ME ( 3 f Y [ y + g( + K ] (.8 Exressing (. in erms of e s we have e ge Y( + e0 α ex + ( αex (.9 Exanding he righ hand side of (.9 and reaining erms u o second owers of e s and hen subracing Y from boh sides we have 6

6 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No α [ ( α] Where g + g e e e e Y Y 0 + e α α α Y 8 (.0 andα [ g + ( g α]. Taking execaions on boh sides of (.0 we ge he bias of he esimaor u o he firs order of aroximaion as From equaion (.0 we have α B( fy α + ρby 8 e Y Y( e0 + α ( quaring boh sides of equaion (. gives e ( Y Y e0 + α + αe0e and now aking execaion we ge he ME of he esimaor o he firs order of aroximaion as f Y [ y + α + αρ b y which is minimum for imum value of α as K+ g α α + g ( say and he minimum ME of is ME f Y y ( ρ M( min( b which is same as ha of radiional linear regression esimaor and also equal o 3. Efficiency comarisons Following are he condiions for which he roosed esimaor We know ha he variance of he samle mean y is V y f Y ( y ] ( 5. (. (. (.3 (. (.5 is beer han he and esimaors. To comare he efficiency of he roosed esimaor wih he exising and roosed esimaors from (3. and (.3(. (.8 (.9(.0 (.3 and (.8 we have (3. V( y M( ρb 0 M( ( ρby 0 M( ( + ρby 0 3 M( ( ρby M( ( + ρby 0 0 (3. (3.3 (3. (3.5 (3.6 7

7 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No M( < M( < M( < 3 if if if A f( ρ < ( B A f( ρ < ( B g 8 ρ < 0 + b y (3.7 (3.8 (3.9. Emirical udy To analyze he erformance of various esimaors of oulaion mean Y of sudy variable y we considered he following wo daa ses Daa. [ource: ukhame & ukhame [33] age 305] Y Area (in acres under whea cro in he circles and A circle consising more han five villages. N 89 n 3 Y P 0.36 b ρ y Daa. [ource: Mukhoadhyay [] age ] Y Household size and A household ha availed an agriculural loan from a bank. N 5 n 7 Y 9. P 0.00 b ρ - y The ercen relaive efficiency (PRE of he esimaors y roosed and menioned exising esimaors and ( wih resec o y usual unbiased esimaor have been comued and given in able. Table: PRE of various esimaors wih resec o y Esimaor PRE of. wih resec o y Poulaion I Poulaion II y ( ( onclusion From Table we see ha he roosed esimaor under imum condiion erforms beer han he usual samle mean esimaor y Naik and Gua esimaors ( and ingh e.al esimaors ( 3 and roosed esimaors ( and 3. The esimaors are beer han he corresonding esimaors and under imum condiions and 5 boh are equally efficien. References [] Abu-Dayyeh W. A. Ahmed M.. Ahmed R. A. and Mulak H. A. (003 ome esimaors of finie oulaion mean using auxiliary informaion Alied Mahemaics and omuaion [] Agarwal. K. harma U. K. and Kashya. (997 A new aroach o use mulivariae auxiliary informaion in samle surveys Journal of aisical Planning and Inference [3] Bahl. and Tueja R.K. (99 Raio and Produc ye exonenial esimaor Informaion and 8

8 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No Oimizaion sciences Vol.XII I [] Das A. K. and Triahi T. P. (98 A class of samling sraegies for oulaion mean using informaion on mean and variance of an auxiliary characer Proceedings of Indian aisical Insiue Golden Jubilee Inernaional onference of aisics: Alicaions and New Direcions 7 8. [5] Diana G. and Tommasi. (003 Oimal esimaion for finie oulaion mean in wo-hase samling aisical Mehods & Alicaions 8. [6] Gandge.N. Varghese T. and Prabhu-Ajgaonkar.G. (993 A noe on modified roduc esimaor Pakisan Journal of aisics 9 (3B [7] Jhajj H.. harma M. K. and Grover L. K. (006 A family of esimaors of oulaion mean using informaion on auxiliary aribue. Pak. J. ais. ( [8] John. (969 On mulivariae raio and roduc esimaors Biomerika [9] Kadilar. and ingi H. (00 Esimaor of a oulaion mean using wo auxiliary variables in simle random samling Inernaional Mahemaical Journal [0] Kadilar. and ingi H. (005 A new esimaor using wo auxiliary variables Alied Mahemaics and omuaion [] Muhammad Hanif e.al (00 ome New Regression Tyes Esimaors in Two Phase amling World Alied ciences Journal 8 (7: [] Mukhoadhyaya P. (000 Theory and mehods of survey samling Prenice Hall of India New Delhi India. [3] Murhy M. N. (96 Produc mehod of esimaion ankhya A [] Naik V. D. and Gua P.. (99 A general class of esimaors for esimaing oulaion mean using auxiliary informaion Merika [5] Naik V.D. and Gua P.. (996 A noe on esimaion of mean wih known oulaion roorion of an auxiliary characer Jour. Ind. oc. Agr. a. 8 ( [6] Olkin I. (958 Mulivariae raio esimaion for finie oulaions Biomerika [7] Onyeka A.. (0 Esimaion of oulaion mean in ossraified samling using known value of some oulaion arameer(s TATITI IN TRANITION-new series March 0 Vol. 3 No [8] Pandey H. Yadav.K. and hukla A.K. (0 An Imroved General lass of Esimaors Esimaing Poulaion Mean using Auxiliary Informaion Inernaional Journal of aisics and ysemsvolume 6 Number (0. 7. [9] Prasad P. (989 ome imroved raio ye esimaors of oulaion mean and raio in finie oulaion samle surveys ommunicaions in aisics: Theory and Mehods [0] Perri P. F. (00 Alcune considerazioni sull efficienza degli simaori raoro-cum-rodoo aisica & Alicazioni no [] Perri P. F. (005 ombining wo auxiliary variables in raio-cum-roduc ye esimaors Proceedings of Ialian aisical ociey Inermediae Meeing on aisics and Environmen Messina -3 eember [] Raj D. (965 On a mehod of using muli-auxiliary informaion in samle surveys Journal of he American aisical Associaion [3] Rao P.. R.. and Mudholkar G.. (967 Generalized mulivariae esimaor for he mean of finie oulaions Journal of he American aisical Associaion [] Robinson P. M. (99 A class of esimaors for he mean of a finie oulaion using auxiliary informaion ankhya B [5] habbir J. and Gua. On esimaing he finie oulaion mean wih known oulaion roorion of an auxiliary variable Pak. J. ais. 3( [6] ingh H.P. and Esejo M.R. On linear regression and raio-roduc esimaion of a finie oulaion mean The saisician [7] ingh R. hauhan P. awan N. and marandache F. Imrovemen in esimaing he oulaion mean using exonenial esimaor in simle random samlings Bullein of aisics & Economics [8] ingh M. P. (965 On he esimaion of raio and roduc of he oulaion arameers ankhyab [9] ingh M. P. (967a Mulivariae roduc mehod of esimaion for finie oulaions Journal of he Indian ociey of Agriculural aisics [30] ingh M. P. (967b Raio cum roduc mehod of esimaion Merika 3. [3] rivasava. K. (97 A generalized esimaor for he mean of a finie oulaion using muli- auxiliary informaion Journal of he American aisical Associaion [3] hukla D. Pahak. and Thakur N.. (0 Esimaion of oulaion mean using wo auxiliary sources in samle survey TATITI IN TRANITION-new series March 0 Vol. 3 No

9 Mahemaical Theory and Modeling IN -580 (Paer IN 5-05 (Online Vol.3 No [33] ukhame P.V. and ukhame B.V. amling heory of surveys wih alicaions Iowa ae Universiy Press Ames U..A 970. [3] Tracy D.. ingh H. P. and ingh R. (996 An alernaive o he raio-cum-roduc esimaor in samle surveys Journal of aisical Planning and Inference [35] Woler K. M. (985 Inroducion o Variance Esimaion inger-verlag New York. [36] Yadav.K. (0 Efficien Esimaors for Poulaion Variance using Auxiliary Informaion Global Journal of Mahemaical ciences: Theory and Pracical. Volume 3 Number (

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