Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 15 Issue 3 Versio 1.0 Year 2015 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA Olie ISSN: 2249-4626 & Prit ISSN: 0975-5896 A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept By F. B. Adebola & N. A. Adegoke Federal Uiversity of Techology Akure, Nigeria Abstract- This paper eamies a class of regressio estimator with cum-dual product estimator as itercept for estimatig the mea of the study variable Y usig auiliary variable X. We obtaied the bias ad the mea square error (MSE of the proposed estimator. We also obtaied MSE of its asymptotically optimum estimator (AOE. Theoretical ad umerical validatio of the proposed estimator was doe to show it s superiority over the usual simple radom samplig estimator ad ratio estimator, product estimator, cum-dual ratio ad product estimator. It was foud that the asymptotic optimal value of the proposed estimator performed better tha other competig estimators ad performed i eactly the same way as the regressio estimator, whe compared with the usual simple radom estimator for estimatig the average sleepig hours of udergraduate studets of the departmet of statistics, Federal Uiversity of Techology Akure, Nigeria. Keywords: differece estimator, auiliary variable, cum-dual product estimator, bias, mea square error, efficiecy, simple radom samplig. GJSFR-F Classificatio : FOR Code : MSC 2010: 62J07 AClassofRegressioEstimatorwithCumDualProductEstimatorAsItercept Strictly as per the compliace ad regulatios of : 2015. F. B. Adebola & N. A. Adegoke. This is a research/review paper, distributed uder the terms of the Creative Commos Attributio-Nocommercial 3.0 Uported Licese http://creativecommos.org/liceses/by-c/3.0/, permittig all o commercial use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.
A Class of Regressio Estimator with Cum- Dual Product Estimator As Itercept F. B. Adebola α & N. A. Adegoke σ Abstract- This paper eamies a class of regressio estimator with cum-dual product estimator as itercept for estimatig the mea of the study variable Y usig auiliary variable X. We obtaied the bias ad the mea square error (MSE of the proposed estimator. We also obtaied MSE of its asymptotically optimum estimator (AOE. Theoretical ad umerical validatio of the proposed estimator was doe to show it s superiority over the usual simple radom samplig estimator ad ratio estimator, product estimator, cum-dual ratio ad product estimator. It was foud that the asymptotic optimal value of the proposed estimator performed better tha other competig estimators ad performed i eactly the same way as the regressio estimator, whe compared with the usual simple radom estimator for estimatig the average sleepig hours of udergraduate studets of the departmet of statistics, Federal Uiversity of Techology Akure, Nigeria. Keywords: differece estimator, auiliary variable, cum-dual product estimator, bias, mea square error, efficiecy, simple radom samplig. I. Itroductio I estimatig the mea of the study variable Cochra (1940, used the auiliary iformatio X at the estimatio phase to icrease the efficiecy of the study variable. To estimate the ratio estimator of the populatio mea or total of the study variable Y, he used additioal kowledge o the auiliary variable X which was positively correlated with Y. Whe the relatioship betwee the study variable Y ad the auiliary variable X is liear through the origi ad Y proportioal to X, the ratio estimator will be more efficiet tha the ormal Simple Radom Samplig (SRS Sajib Choudhury et al (2012. Also, Robso (1957 proposed product estimator ad showed that whe the relatioship betwee the study variable Y ad the auiliary variable X is liear through the origi ad Y is iversely proportioal to X, the product estimator will be more efficiet tha the usual SRS. Murthy 1964 suggested the use of ratio ρc y estimator yy pp whe ad ubiased estimator yy whe 1 ρ C y, where CC yy, 2 > 1 2 1 2 C ad ρρ are coefficiets of variatio of y, ad correlatio betwee y ad respectively. Suppose that SRSWOR of uits is draw from a populatio of N uits to N estimate the populatio mea 1 Y y i of the study variable Y. All the sample uits N i 1 are observed for the variables Y ad X. Let y, where i 1,2,3,.., deote the set of the ( i i Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 49 Author α σ: Departmet of Statistics, Federal Uiversity of Techology Akure, Nigeria. e-mail: urudee.adegoke@yahoo.com 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 50 observatio for the study variable Y ad X. Let the sample meas the populatio meas of the auiliary variable X ad study variable Y based o the observatios. y The usual product estimator of is give as y p ad the usual regressio estimator is give as ˆ 1 1 ˆ y y reg y ( X, where y y i,, ad is the estimate i 2 s i 1 i 1 slope of regressio coefficiet of Y ad X. Cum-Dual Product estimator give as X y dp y, where is the u-sampled auiliary variable i X give as = NX N was obtaied by Badyopadhyay (1980. The use of auiliary iformatio i sample surveys was etesively discussed i well-kow classical tet books such as Cochra(1977, Sukhatme ad Sukhatme(1970, Sukhatme, Sukhatme, ad Asok (1984, Murthy(1967 ad Yates(1960 amog others. Recet developmets i ratio ad product methods of estimatio alog with their variety of modified forms are lucidly described i detail by Sigh (2003. II. The Proposed Class of Estimator The proposed a class of regressio estimator with dual product as the itercept for estimatig populatio mea Y, give as; + α(x Where, α is a suitable scalar. We obtaied the bias ad MSE of the proposed estimator. (1 up to the first order approimatio, this is obtaied by substitutig = NX ito equatio (1, We write The bias The MSE of e 0 = y Y ad Y e 1 = X E( is give as is give as Y = y X = y X (N NX X. Bias( = (1 f Y 2 = ( 1 f (S y 2 + 2gY ρs S y X Y + α( X N X (g2 Y 2 S 2 X + Y g S y 2 X Y N (, y (Y + αx + g 2 S 2 (Y + αx 2 X 2 MSE ( = ( 1 f (Y 2C 2 y + 2gρY C y (Y + αx + g 2 C 2 (Y + αx 2 s be ubiased of... (1.1 (2 The optimum value of the MSE ( is obtaied as follows, α MSE ( = ( 1 f Sy (2gY X Y (X + g 2 C 2 (2(X (Y + αx (3 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept Set equatio (3 to zero; we have 2gS y + 2g 2 2 X (Y + αx = 0 α = (R + β g The optimal MSE of Equatio (4 shows that the MSE ( Regressio Estimate of y o X. is the same as the MSE of Remark The bias of is the same as the bias of the dual product estimator ad whe α=0, the MSE ( boils dow to product cum estimator proposed by Badyopadhyay (1980. The bias ad MSE of are give as MSE(y p = ( 1 f.(5 ( Y 2C 2 y + 2gρY 2C C y + Y g 2 C 2 III. The Efficiecy Comparisos I this sectio, we compared the MSE of yy with the MSE of uder SRS scheme is give as MSE (y = ( 1 f Y 2C2 From equatio (2 ad (5, is better tha yy If MSE ( < MSE (y That is This holds if. MSE( opt = ( 1 f S y 2 (1 ρ 2 (1 Y + αx < 0 ad 2gρY C y + g 2 C 2 (Y + αx > 0 Or opt (2 Y + αx > 0 ad 2gρY C y + g 2 C 2 (Y + αx < 0 y p Bias(y p = ( 1 f (g2 Y 2C 2 + Y gρ C y (4. The MSE of yy ( 1 f ( Y 2C 2 y + 2gρY C y (Y + αx + g 2 C 2 (Y + αx 2 ( 1 f Y 2C2 The rage of α uder which (Y + αx ( 2gρY C y + g 2 2 (Y + αx < 0 is more efficiet tha usual SRS yy is Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 51 We also compared give as, mi { R, R (1 + 2ρC y }, ma { R, R (1 + 2ρC y }. with the usual ratio estimator yy RR. The MSE of the yy RR is 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 52 MSE It is foud that will be more efficiet tha the usual ratio estimator yy RR if < MSE (y R. That is, ( 1 f ( Y 2C 2 y + 2gρY C y (Y + αx + g 2 C 2 (Y + αx 2 ( 1 f ( Y 2C 2 y 2ρY 2 C y + Y 2C 2 This holds if the followig two coditios are satisfied (1. (g(y + αx + Y < 0 Ad 2ρY C y + C 2 (g(y + αx Y > 0. or (2. (g(y + αx + Y > 0 Ad 2ρY C y + C 2 (g(y + αx Y < 0. This coditio holds if α > R ( N ad α < R (N 2 2ρC y or α < R ( N ad α > R ( N 2 2ρC y We also compared is give as MSE( y R = ( 1 f ( Y 2C y 2 2ρY 2C C y + Y 2C 2 mi { R ( N 2, R (N 2ρC y }, ma { R ( N 2, R (N 2ρC y }. with the usual product estimator yy PP. The MSE of the yy PP MSE( y P = ( 1 f ( Y 2C 2 y + 2ρY 2C C y + Y 2C 2 It is foud that the proposed estimator y PD will be more efficiet tha the usual ratio estimator yy PP if MSE ( < MSE (y P. That is, ( 1 f ( Y 2C 2 y + 2gρY C y (Y + αx + g 2 C 2 (Y + αx 2 ( 1 f ( Y 2C 2 y + 2ρY 2 C y + Y 2C 2 This holds if the followig two coditios are satisfied (1. (g(y + αx Y < 0 Ad 2ρY C y + C 2 (g(y + αx + Y > 0.. Or (2. (g(y + αx Y > 0 Ad 2ρY C y + C 2 (g(y + αx + Y < 0. This coditio holds if α > R ( N 2 ad α < R ( N + 2ρC y or α < R ( N 2 ad α > R ( N + 2ρC y N 2 mi {R (, R ( N + 2ρC y }, ma {R ( N 2, R ( N + 2ρC y }. We compared the MSE of the proposed estimator with MSE of dual product estimator from equatio (2 ad (5 it is foud that the proposed estimator will be more efficiet tha that of Badyopadhyay (1980 estimator y p if MSE < MSE (y p That is ( 1 f ( Y 2C 2 y + 2gρY C y (Y + αx + g 2 C 2 (Y + αx 2 ( 1 f ( Y 2C 2 y + 2gρY 2C C y + Y 2g 2 C 2 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept This holds if 1. αx < 0 ad 2ρY C y + (2Y + αx > 0 Or 2. αx > 0 ad 2ρY C y + (2Y + αx < 0 The rage of α uder which the proposed estimator is more efficiet tha is y p Lastly, we compared MSE of with that of dual to ratio estimator proposed Srivekataramaa(1980, will be more efficiet tha if ( 1 f ( Y 2C 2 y 2gρY 2C C y + Y 2g 2 C 2 This holds if 1. 2Y + αx < 0 ad 2ρY C y + (αx > 0 Or 2. 2Y + αx > 0 ad 2ρY C y + (αx < 0 This coditio holds if 2R > α ad < α or 2R<α ad Therefore, the rage of α uder which the proposed estimator efficiet tha dual to ratio estimator is Compariso of AOE to mi {0, 2R ( ρc y + 1}, ma {0, 2R ( ρc y + 1} 2RρC y y R 2RρC y mi { 2R, 2R ( ρc y }, ma { 2R, 2R ( ρc y } OPT MSE ( < MSE (y R ( 1 f ( Y 2C y 2 + 2gρY C y (Y + αx + g 2 2 (Y + αx 2 is more OPT is more efficiet tha the other eistig estimators yy, the Ratio estimator yy RR, the product estimator yy pp, the dual to ratio estimator yy RR ad the dual to product estimator y p sice. MSE(y MSE( OPT = ( 1 f ( Y 2ρ 2 C 2 y > 0 MSE(y R MSE( OPT = ( 1 f ( Y 2C 2 (1 ρc 2 y > 0 MSE(y P MSE( OPT = ( 1 f ( Y 2C 2 (1 + ρc 2 y > 0 y R > α y R Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 53 MSE(y R MSE( OPT = ( 1 f ( Y 2C 2 ( ρc y 2 g > 0 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 54 Hece, we coclude that the proposed class of estimator tha other estimator i case of its optimality. IV. Numerical Validatio To illustrate the efficiecy of the proposed estimator over the other estimators yy,yy RR,yy pp,yy RR aadd y p. Data o the ages ad hours of sleepig by the udergraduate studets of the Departmet of Statistics Federal Uiversity of Techology Akure, Odo State, Nigeria were collected. A sample of 150 out of 461 studets of the departmet was obtaied usig simple radom samplig without replacemet. The iformatio o the age of the studets was used as auiliary iformatio to icrease the precisio of the estimate of the average sleepig hours. The estimate of the average hours of sleepig of the studets were obtaied ad also the 95% cofidece itervals of the average hours of sleepig were obtaied for the proposed estimator ad the other estimators. Table 1.0, gives the estimates of the average sleepig hours ad the 95% cofidece Iterval. As show i Table 1.0, the proposed estimator performed better tha the other estimators, also the width of the cofidece iterval of the proposed estimator is smallest tha the other competig estimators. Table 1.0. : Average Sleepig Hours ad 95% cofidece itervals for Differet Estimators for the udergraduate Studets of Departmet of Statistics, Federals Uiversity of Techology Akure. Nigeria Estimator Average Sleepig Hours LCL UCL WIDTH yy 6.08 5.930386531 6.229613469 0.299226939 y R y p y R y p y PD MSE(y p MSE( OPT = ( 1 f ( Y 2C 2 ( ρc y is more efficiet 6.210472103 6.042844235 6.378099971 0.335255737 5.952268908 5.778821411 6.125716404 0.346894993 6.141606636 5.988421023 6.294792249 0.306371226 6.01901342 5.862732122 6.175290562 0.31255844 6.072287 5.927857 6.216717 0.28886 The proposed estimator performed the same way as the regressio estimator whe compare with the usual estimator yy. The average Sleepig Hours ad 95% cofidece itervals for the proposed estimator ad the regressio estimator is give below, the two estimators have the same cofidece Iterval width. 2 + g > 0 Table 2.0. : Average Sleepig Hours ad 95% cofidece itervals for the proposed estimators ad regressio estimators for the udergraduate Studets of Departmet of Statistics, Federals Uiversity of Techology Akure. Nigeria Estimator y PD y REG Average Sleepig LCL UCL WIDTH Hours 6.072287 5.927857 6.2167177 0.28886 6.089652737 5.945222793 6.234082681 0.288859888 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept To eamie the gai i the efficiecy of the proposed estimator over the estimator yy,yy RR,yy pp, y R aadd y p, we obtaied the percetage relative efficiecy of differet estimator of Y with respect to the usual ubiased estimator yy i Table 2.0. The proposed estimator performed better tha the other estimators y, y R, y p, y R aadd y p ad performed eactly the same way as regressio estimator. Table 3.0. : The percetage relative efficiecy of differet estimator of YY with respect to the usual ubiased estimator y ESTIMATOR PERCENATGE RELATIVE FFICIENCY y 100 y R 79.66158486, y p 74.40554745 y R 95.39056726 y p 91.65136111 y REG 107.3067159 107.3067159 V. Coclusio We have proposed a class of regressio estimator with cum-dual product estimator as itercept for estimatig the mea of the study variable Y usig auiliary variable X as i equatio (1 ad obtaied AOE for the proposed estimator. Theoretically, we have demostrated that proposed estimator is always more efficiet tha other uder the effective rages of αα ad its optimum values. Table 1.0 shows that the proposed estimator performed better tha the other estimators as the width of the cofidece iterval of the proposed estimator is smallest tha the other competig estimators. The percetage relative efficiecy of differet estimator of Y with respect to the usual ubiased estimator y i Table 2.0 shows that the proposed estimator y PD performed better tha the other estimators y, y R, y p, y R aadd y p ad performed eactly the same way as regressio estimator whe compared to the usual estimator yy. Hece, it is preferred to use the proposed class of estimator i practice. Refereces Référeces Referecias 1. Badyopadhyay, S. (1980. Improved ratio ad product estimators. Sakhya Series C, 42(2, 45-49. 2. Cochra, W. G. (1940. The estimatio of the yields of the cereal eperimets by Samplig for the. 3. COCHRAN, W.G. (1977: Samplig Techiques.Joh Wiley, New York. 4. Murthy, M. N. (1964. Product method of estimatio. Sakhya A, 26, 69-74. 5. Murthy, M. N. (1967. Samplig Theory ad Methods, Statistical Publishig Society, Calcutta. 6. ROBSON, D.S.(1957: Applicatio of multivariate polykays to the theory of ubiased ratio type estimatio. J.Amer.stat. Assoc., 52, 411-422. 7. Sajib Choudhury ad B. K. Sigh (2012, A Efficiet Class of Dual to Product- Cum-Dual to Ratio Estimators of Fiite Populatio, Mea i Sample Surveys. Global Joural of Sciece Frotier Research. Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 55 2015 Global Jourals Ic. (US
A Class of Regressio Estimator with Cum-Dual Product Estimator As Itercept Global Joural of Sciece Frotier Research F Volume XV Issue III V ersio I Year 2015 56 8. SINGH,S. (2003: Advaced samplig theory with applicatios,vol. 1, Kluwer Academic Publishers, The Netherlads. 9. Srivekataramaa, T. (1980. A dual to ratio estimator i sample surveys. Biometrika, 67(1, 199-204. 10. SUKHATME, P.V., SUKHATME, B. V. ad A SOK, C. (1984: Samplig theory of surveys with applicatios. Id. Soc. Ag. Stat., New Delhi. 11. SUKHATME,P V. ad SUKHATME, B.V.(1970:Samplig theory of surveys with Applicatios. Asia Publishig House, New Delhi 12. YATES, F.(1960: Samplig methods i cesuses ad surveys. Charles Griffi ad Co., Lodo: 2015 Global Jourals Ic. (US