Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume 01, Article ID 65768, 1 pages doi:10.50/01/65768 Research Article A Alterative Estimator for Estimatig the Fiite Populatio Mea Usig Auxiliary Iformatio i Sample Surveys Ramkrisha S. Solaki, Housila P. Sigh, ad Ajaa Rathour School of Studies i Statistics, Vikram Uiversity, Ujjai 56010, Idia Correspodece should be addressed to Ramkrisha S. Solaki, ramkssolaki@gmail.com Received 8 February 01; Accepted 9 March 01 Academic Editors: J. Hu ad J. López-Fidalgo Copyright q 01 Ramkrisha S. Solaki et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper suggests a class of estimators for estimatig the fiite populatio mea Y of the study variable y usig kow populatio mea X of the auxiliary variable x. Asymptotic expressios of bias ad variace of the suggested class of estimators have bee obtaied. Asymptotic optimum estimator AOE i the class is idetified alog with its variace formula. It has bee show that the proposed class of estimators is more efficiet tha usual ubiased, usual ratio, usual product, Bahl ad Tuteja 1991, ad Kadilar ad Cigi 003 estimators uder some realistic coditios. A empirical study is carried out to judge the merits of suggested estimator over other competitors practically. 1. Itroductio The literature o survey samplig describes a great variety of techiques for usig auxiliary iformatio to obtai more efficiet estimators. Ratio, product, ad regressio methods of estimatio are good examples i this cotext see Sigh 1. If the correlatio betwee the study variable y ad the auxiliary variable x is positive high, the ratio method of estimatio is quite effective. O the other had, if this correlatio is egative high the product method of estimatio evisaged by Robso ad rediscovered by Murthy 3 ca be employed quite effectively. The classical ratio ad product estimators are cosidered to be the most practicable i may situatios, but they have the limitatio of havig at most the same efficiecy as that of the liear regressio estimator. I the situatio where the relatio betwee the study variable y ad the auxiliary variable x is a straight lie ad passig through the origi, the ratio ad product estimators have efficiecies equal to the usual regressio estimator. But i may practical situatios, the lie does ot pass through the origi, ad i such
ISRN Probability ad Statistics circumstaces the ratio ad product estimators do ot perform equally well to the regressio estimator. Keepig this fact i view, a large umber of authors have paid their attetio toward the formulatio of modified ratio ad product estimators, for istace, see Sigh, 5, Srivastava 6 8, Reddy 9, 10, Gupta 11, Sahai 1, Sahai ad Ray 13, Ray ad Sahai 1, Srivekataramaa ad Tracy 15, 16, Badyopadhyay 17,Vos 18, Sisodia ad Dwivedi 19, Adhvaryu ad Gupta 0, Chaubey et al. 1, Mohaty ad Sahoo, Sigh ad Shukla 3, Naik ad Gupta,BahladTuteja 5, Mohaty ad Sahoo 6, Upadhyaya ad Sigh 7, Sigh ad Tailor 8, Kadilar ad Cigi 9, Sigh et al. 30, ad others. I this paper, we have evisaged a ew class of estimators for populatio mea of study variable y usig iformatio o a auxiliary variable x which is highly correlated with the study variable ad have show that the suggested class of estimators is more efficiet tha some existig estimators. Cosider a fiite populatio U u 1,u,...,u N of size N from which a sample of size is draw accordig to simple radom samplig without replacemet SRSWOR. Let y ad x be the sample mea estimators of the populatio meas Y ad X, respectively, of the study variable y ad the auxiliary variable x. LetC y S y /Y, C x S x /X, ρ S yx /S x S y correlatio coefficiet betwee the variables y ad x ad k ρc y /C x, where S y N 1 1 N (, y i Y S x N 1 1 N (, x i X i 1 S xy N 1 1 N ( ( x i X y i Y. i 1 i 1 1.1 The remaiig part of the paper is orgaized as follows. Sectio gives the brief review of some estimators for the populatio mea of study variable y with its properties. I Sectio 3, a ew class of estimators for the populatio mea is described, ad the expressios for the asymptotic bias ad variace are obtaied. Asymptotic optimum estimator AOE i the suggested class is obtaied with its variace formula. Sectio addresses the problem of efficiecy comparisos, while i Sectio 5 a empirical study is carried out to evaluate the performace of differet estimators.. Reviewig Estimators It is very well kow that the sample mea y is a ubiased estimator of populatio mea Y,aduder SRSWOR, its variace is give by Var ( y ( ( 1 f 1 f S y Y Cy,.1 where f /N.
ISRN Probability ad Statistics 3 The usual ratio ad product estimators of populatio mea Y of the study variable y are, respectively, defied as y R y X x, y P y x X.. Bahl ad Tuteja 5 suggested expoetial ratio-type ad product-type estimators for populatio mea Y, respectively, as ( X x y Re y exp, X x ( x X y Pe y exp. x X.3 as Kadilar ad Cigi 31 suggested a chai ratio-type estimator for populatio mea Y y CR y X x.. Followig the procedure adopted by Kadilar ad Cigi 31, oe may defie a chai product-type estimator for populatio mea Y as ( x y CP y..5 X To the first degree of approximatio, the biases ad variaces of estimators y R, y p, y Re, y Pe, y CR,ady CP are, respectively, give by B ( ( y R YC x 1 k, B ( ( y P YC xk, B ( ( ( Y y Re Cx 3 k, 8.6.7.8
ISRN Probability ad Statistics B ( ( ( Y y Pe Cx k 1,.9 8 B ( y CR YC x 3 k,.10 B ( ( y CP YC x 1 k,.11 Var ( ( y R Y [ Cy C x 1 k,.1 Var ( ( y P Y [ Cy C x 1 k,.13 Var ( ( [ y Re Y Cy C x 1 k,.1 Var ( ( [ y Pe Y Cy C x 1 k,.15 Var ( y CR Var ( y CP Y [ Cy Cx 1 k,.16 Y [ Cy Cx 1 k..17 From.1 ad.1.17, we have made some efficiecy comparisos betwee the estimators y, y R, y p, y Re, y Pe, y CR,ady CP, as show i Table 1. 3. Suggested Class of Estimators We defie the followig class of estimators for the populatio mea Y as ( ( x α δ x X t α,δ y exp (, 3.1 X x X where α, δ are suitable chose scalars. It is to be metioed that for δ 0, the class of estimators t α,δ reduces to the followig class of estimators which is due to Sahai ad Ray 13. [ t α,0 y ( x X α, 3.
ISRN Probability ad Statistics 5 Table 1: Efficiecy comparisos betwee differet estimators. Estimator More efficiet tha estimator If y y R k 1/ y y P k 1/ y R y k > 1/ y P y k< 1/ y y Re k 1/ y y Pe k 1/ y Re y k > 1/ y Pe y k< 1/ y y CR k 1 y y CP k 1 y CR y k > 1 y CP y k< 1 y CR y R k>3/ y CP y P k< 3/ While for α 0, it reduces to the ew trasformed class of estimators defied as ( δ x X t 0,δ y exp (. 3.3 x X To obtai the bias ad variace of suggested class of estimators t α,δ, we write yc Y 1 e 0, xc X 1 e 1, 3. such that E e 0 e 1 E e 0 E e 1 0, ( ( E e 0 C y, ( E e 1 ρc y C x C x, kc x. 3.5
6 ISRN Probability ad Statistics Expressig 3.1 i terms of e s, we have [ { t α,δ Y 1 e 0 1 e 1 α δe1 exp e 1 [ Y 1 e 0 1 e 1 α exp { δe1 ( } 1 e 1 1 }. 3.6 We assume that e 1 < 1, so that 1 e 1 α, exp{ δe 1 / 1 e 1 / 1 },ad 1 e 1 / 1 are expadable. Now expadig the right-had side of 3.6, we have [ { t α,δ Y 1 e 0 1 αe 1 { 1 δe ( 1 α α 1 e 1 } 1 e 1 [ { Y 1 e 0 1 αe 1 { 1 δe ( 1 [ { Y 1 e 0 1 [{ Y 1 e 0 1 δ e ( 1 8 1 e 1 α α 1 e 1 } } 1 e 1 e 1 δ e 1 8 1 e 1 α δ e 1 α δ e 1 e 0 e 1 } α δ δ δ e 1 } 8 α δ δ δ ( e 1 8 e 0e 1 }. 3.7 Neglectig the terms of e s havig greater tha two i 3.7, we have t α,δ Y [ 1 e 0 { α δ e 1 e 0 e 1 } δ δ e 1 3.8 or ( t α,δ Y Y [e 0 { α δ e 1 e 0 e 1 } δ δ e 1. 3.9 Takig expectatio of both sides of 3.9, we get the bias of the class of estimators t α,δ to the first degree of approximatio as B ( t α,δ YC x [ { } α δ δ δ k. 3.10 Squarig both sides of 3.9 ad eglectig terms e s havig power greater tha two, we have ( t α,δ Y Y [e 0 α δ { δ δ e 1 e 0e 1 }. 3.11
ISRN Probability ad Statistics 7 Takig expectatio of both sides of 3.11, we get the variace of t α,δ to the first degree of approximatio as which is miimum whe Var ( t α,δ Y [ C y α δ C x{ α δ k}, 3.1 α δ k. 3.13 Thus, the resultig miimum variace of t α,δ is obtaied as Var mi ( t α,δ S y ( 1 ρ. 3.1 The miimum variace of t α,δ equals to the approximate variace of the usual liear regressio estimator defied as y lr y β ( X x, 3.15 where β s xy /s x, s xy 1 1 i 1 x i x y i y ad s x 1 1 i 1 x i x. It is iterestig to ote that if we set δ 0 ad α 0 i 3.1, we get the variaces of the classes of estimators t α,0 ad t 0,δ, respectively, as Var ( ( t α,0 Var ( ( t 0,δ Y [ Cy Y [ Cy αcx α k, ( δ C x δ k. 3.16 If we assume the value of α is specified by α o say, the the variaces of t α,0 ad t α,δ are, respectively, give by Var ( ( t α αo,0 Var ( ( t α αo,δ Y [ Cy α o Cx α o k, Y [ Cy α o δ C x{ α o δ k}. 3.17 If the value of δ is specified δ o, the the variaces of the estimators t 0,δ ad t α,δ are, respectively, give by Var ( ( t 0,δ δo Y [ Cy Var ( ( t α,δ δo ( δo C x δ o k, Y [ Cy α o δ C x{ α δ o k}. 3.18
8 ISRN Probability ad Statistics To illustrate our geeral results, we cosider a particular case of the proposed class of estimators t α,δ with its properties. If we set α, δ 1, 1 i 3.1, we get a estimator of populatio mea Y as [ ( ( x x X t 1,1 y exp. 3.19 X x X Puttig α, δ 1, 1 i 3.10 ad 3.1, we get the bias ad variace of t 1,1,tothefirst degree of approximatio, respectively, as B ( 3 t 1,1 YC 8 x 1 K, 3.0 Var ( ( [ t 1,1 Y Cy 3C x 3 k. 3.1 Expressio 3.0 clearly idicates that the proposed estimator t 1,1 is better tha covetioal ubiased estimator y if k> 3/, a coditio which is usually met i survey situatios.. Efficiecy Comparisos From 3.17, we have Var ( ( t α αo,δ Var t α αo,0 ( { α o δ k} α o α o k [ Y Cx αo δ [ α o δ α o δ k α o 8α o k < 0, i.e., if δ δ α o k < 0, i.e., 0 <δ< k α o if or k α o <δ<0,.1 or equivaletly mi {0, k α o } <δ<max {0, k α o }..
ISRN Probability ad Statistics 9 From 3.18, we have Var ( ( t α,δ δo Var t 0,δo ( [ 1 f Y α Cx δo i.e., α δ o k δ o δ ok [ α αδ o αk < 0, 0 <α< k δ o if or k δ o <α<0, }.3 or equivaletly mi {0, k δ o } <α<max {0, k δ o }.. From.1,.1.17,ad 3.1, we have i Var ( ( t α,δ Var y ( Y [ Cx α δ k α δ 0 < α δ <k or k< α δ < 0,.5 or equivaletly mi 0,k < α δ < max 0,k..6 ii Var ( ( t α,δ Var yr ( [ 1 f Y α δ Cx { α δ k} 1 k < α δ < k 1 or k 1 < α δ <,.7 or equivaletly mi {, k 1 } < α δ < max {, k 1 }..8
10 ISRN Probability ad Statistics iii Var ( t α,δ Var ( y p ( [ 1 f Y α δ Cx { α δ k} 1 k <θ< k 1 or k 1 <θ<,.9 or equivaletly mi {, k 1 } < α δ < max {, k 1 }..10 iv Var ( ( t α,δ Var yre ( Y Cx α δ { α δ k} 1 k 1 < α δ < k 1 or k 1 < α δ < 1,.11 or equivaletly mi {1, k 1 } < α δ < max {1, k 1 }..1 v Var ( ( t α,δ Var ype ( Y Cx α δ { α δ k} 1 k 1 < α δ < k 1 or k 1 < α δ < 1,.13 or equivaletly mi { 1, k 1 } < α δ < max { 1, k 1 }..1
ISRN Probability ad Statistics 11 vi Var ( t α,δ Var ( ycr Y C x [ α δ { α δ k} 1 k < α δ < k 1 or k 1 < α δ <,.15 or equivaletly mi {, k 1 } < α δ < max {, k 1 }..16 vii Var ( t α,δ Var ( ycp ( [ 1 f Y α δ Cx { α δ k} 1 k < α δ < k 1 or k 1 < α δ <,.17 or equivaletly mi {, k 1 } < α δ < max {, k 1 }..18 From.1,.1,.1,.16, ad 3.1, we have made some efficiecy comparisos betwee the estimators t 1,1, y, y R, y Re,ady CR, as show i Table. It is clearly idicated from Table that the estimator t 1,1 is more efficiet tha estimators y, y R, y Re,ady CR if 5 <k<7 3..19 5. Empirical Study To judge the merits of the suggested estimator t 1,1 over usual ubiased estimator y, usual ratio estimator y R,BahladTuteja 5 expoetial ratio-type estimator y Re ad Kadilar ad Cigi 31 chai ratio-type estimator y CR we have cosidered three populatios. Descriptios of the populatios are give below.
1 ISRN Probability ad Statistics Table : Efficiecy comparisos betwee differet estimators. Estimator More efficiet tha estimator If t 1,1 y k > 3/ t 1,1 y R k>5/ t 1,1 y Re k>1 t 1,1 y CR k<7/3 Table 3: PREs of differet estimators of populatio mea y. Estimator PRE, y for populatio I II III y sample mea 100.00 100.00 100.00 y R ratio estimator 173.6 6.77 51.5 y Re bahl ad Tuteja 5 133.16 151.03 199.15 y CR kadilar ad Cigi 31 1.83 86.8 30.1 t 1,1 proposed estimator.6 97.05 631.59 Populatio I (source: Srivastava [7, page 06 y : fiber per plat i jute fiber crops, x : height, Cy 0.0568, Cx 0.0086, ρ 0.718, k 1.91. Populatio II (source: Gupta ad Shabbir [3 y : the level of apple productio amout 1uit 100 toes, x : the umber of apple trees 1uit 100 trees, Cy 17.7, Cx.080, ρ 0.8, k 1.6968. Populatio III (Source: Kadilar ad Cigi [33 y : the level of apple productio amout, x : the umber of apple trees, Cy 3.1056, Cx 1.85, ρ 0.9, k 1.3955. We have computed the percet relative efficiecies PREs of differet estimators y, y R, y Re, y CR,adt 1,1 of the populatio mea Y with respect to y by usig the followig formula: PRE (, y Var( y Var 100, 5.1 ad results are summarized i Table 3. Table 3 exhibits that the proposed estimator t 1,1 is more efficiet tha usual ubiased estimator y, usual ratio estimator y R,BahladTuteja 5 estimator y Re, ad Kadilar ad
ISRN Probability ad Statistics 13 Cigi 31 estimator y CR i the sese of havig the largest PRE i all three populatios. I populatios II ad III, the proposed estimator t 1,1 has the largest gai i efficiecy over all the estimators, while i populatio I, there is margial gai i efficiecy as compared to y CR. It is further oted that the coditio 5/ <k<7/ has bee satisfied i populatio I 1.5 < 1.91 <.33, II 1.5 < 1.6968 <.33, ad III 1.5 < 1.3955 <.33. Thus, we recommed the proposed estimator t 1,1 for its use i practice wherever the coditio 5/ <k<7/ is satisfied. Refereces 1 S. Sigh, Advaced Samplig Theory with Applicatios. Vol. I, II, Kluwer Academic, Dordrecht, The Netherlads, 003. D. S. Robso, Applicatios of multivariate polykays to the theory of ubiased ratio-type estimatio, Joural of the America Statistical Associatio, vol. 5, pp. 511 5, 1957. 3 M. N. Murthy, Product method of estimatio, Sakhya, vol. 6, pp. 69 7, 196. M. P. Sigh, O the estimatio of ratio ad product of the populatio parameters, Sakhya, vol. 7, pp. 31 38, 1965. 5 M. P. Sigh, Ratio cum product method of estimatio, Metrika, vol. 1, pp. 3, 1967. 6 S. K. Srivastava, A estimator usig auxiliary iformatio i sample surveys, Bulleti/Calcutta Statistical Associatio Bulleti, vol. 16, pp. 11 13, 1967. 7 S. K. Srivastava, A geeralized estimator for the mea of a fiite populatio usig multi auxiliary iformatio, Joural of the America Statistical Associatio, vol. 66, pp. 0 07, 1971. 8 S. K. Srivastava, A class of estimators usig auxiliary iformatio i sample surveys, The Caadia Joural of Statistics, vol. 8, o., pp. 53 5, 1980. 9 V. N. Reddy, O ratio ad product methods of estimatio, Sakhya B, vol. 35, o. 3, pp. 307 316, 1973. 10 V. N. Reddy, O trasformed ratio method of estimatio, Sakhya C, vol. 36, pp. 59 70, 197. 11 P. C. Gupta, O some quadratic ad higher degree ratio ad product estimators, Joural of the Idia Society of Agricultural Statistics, vol. 30, o., pp. 71 80, 1978. 1 A. Sahai, A efficiet variat of the product ad ratio estimators, Statistica Neerladica, vol. 33, o. 1, pp. 7 35, 1979. 13 A. Sahai ad S. K. Ray, A efficiet estimator usig auxiliary iformatio, Metrika, vol.7,o., pp. 71 75, 1980. 1 S. K. Ray ad A. Sahai, Efficiet families of ratio ad product-type estimators, Biometrika, vol. 67, o. 1, pp. 11 15, 1980. 15 T. Srivekataramaa ad D. S. Tracy, A alterative to ratio method i sample surveys, Aals of the Istitute of Statistical Mathematics, vol. 3, o. 1, pp. 111 10, 1980. 16 T. Srivekataramaa ad D. S. Tracy, Extedig product method of estimatio to positive correlatio case i surveys, The Australia Joural of Statistics, vol. 3, o. 1, pp. 93 100, 1981. 17 S. Badyopadhyay, Improved ratio ad product estimators, Sakhya, vol., o. 1-, pp. 5 9, 1980. 18 J. W. E. Vos, Mixig of direct, ratio, ad product method estimators, Statistica Neerladica, vol. 3, o., pp. 09 18, 1980. 19 B. V. S. Sisodia ad V. K. Dwivedi, A modified ratio estimator usig coefficiet of a auxiliary variable, Joural of the Idia Society of Agricultural Statistics, vol. 33, o., pp. 13 18, 1981. 0 D. Adhvaryu ad P. C. Gupta, O some alterative samplig strategies usig auxiliary iformatio, Metrika, vol. 30, o., pp. 17 6, 1983. 1 Y. P. Chaubey, T. D. Dwivedi, ad M. Sigh, A efficiecy compariso of product ad ratio estimator, Commuicatios i Statistics. A. Theory ad Methods, vol. 13, o. 6, pp. 699 709, 198. S. Mohaty ad L. N. Sahoo, A class of estimators based o mea per uit ad ratio estimators, Statistica, vol. 7, o. 3, pp. 73 77, 1987. 3 V. K. Sigh ad D. Shukla, Oe-parameter family of factor type ratio estimators, Metro, vol. 5, o. 1-, pp. 73 83, 1987. V. D. Naik ad P. C. Gupta, A geeral class of estimators for estimatig populatio mea usig auxiliary iformatio, Metrika, vol. 38, o. 1, pp. 11 17, 1991.
1 ISRN Probability ad Statistics 5 S. Bahl ad R. K. Tuteja, Ratio ad product type expoetial estimators, Joural of Iformatio & Optimizatio Scieces, vol. 1, o. 1, pp. 159 16, 1991. 6 S. Mohaty ad J. Sahoo, A ote o improvig the ratio method of estimatio through liear trasformatio usig certai kow populatio parameters, Sakhya, vol. 57, o. 1, pp. 93 10, 1995. 7 L. N. Upadhyaya ad H. P. Sigh, Use of trasformed auxiliary variable i estimatig the fiite populatio mea, Biometrical Joural, vol. 1, o. 5, pp. 67 636, 1999. 8 H. P. Sigh ad R. Tailor, Use of kow correlatio coefficiet i estimatig the fiite populatio mea, Statist Trasactio, vol. 6, o., pp. 555 560, 003. 9 C. Kadilar ad H. Cigi, A improvemet i estimatig the populatio mea by usig the correlatio coefficiet, Hacettepe Joural of Mathematics ad Statistics, vol. 35, o. 1, pp. 103 109, 006. 30 H. P. Sigh, R. Tailor, ad R. Tailor, O ratio ad product methods with certai kow populatio parameters of auxiliary variable i sample surveys, Statistics ad Operatios Research Trasactios, vol. 3, o., pp. 157 180, 010. 31 C. Kadilar ad H. Cigi, A study o the chai ratio-type estimator, Hacettepe Joural of Mathematics ad Statistics, vol. 3, pp. 105 108, 003. 3 S. Gupta ad J. Shabbir, Variace estimatio i simple radom samplig usig auxiliary iformatio, Hacettepe Joural of Mathematics ad Statistics, vol. 37, o. 1, pp. 57 67, 008. 33 C. Kadilar ad H. Cigi, Ratio estimators i stratified radom samplig, Biometrical Joural, vol. 5, o., pp. 18 5, 003.
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