J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) 1 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/030101 Improved Ratio Estimators of Populatio Mea I Adaptive Cluster Samplig Subhash Kumar Yadav 1, Sheela Misra, Sat Sara Mishra 1, ad ipapor Chutima 3 1 Departmet of Mathematics ad Statistics (A Cetre of Excellece o Advaced Computig)Dr. RML Avadh Uiversity, Faizabad- 4001, U.P., Idia Departmet of Statistics, Uiversity of Luckow, Luckow-6007, U.P., Idia 3 Departmet of Mathematics, Faculty of Sciece,Mahasarakham Uiversity, Maha Sarakham-44150, Thailad Received: 13 Aug. 015, Revised: 18 Sep. 015, Accepted: 3 Oct. 015 Published olie: 1 Ja. 016 Abstract: I this paper, we study the estimators of the populatio mea i adaptive cluster samplig by usig the iformatio of the auxiliary variable, that is, the populatio coefficiet of variatio, the coefficiet of skewess kurtosis of the auxiliary variable ad the correlatio coefficiet betwee mai variable ad the auxiliary variable. The large sample properties of the proposed estimators have bee studied up to the first order of approximatio. A umerical study is also carried out to judge the theoretical fidigs. The umerical example showed that if the populatio is rare ad hidde clustered populatio, all estimators i adaptive cluster samplig are more efficiet tha the estimators i simple radom samplig with the same coditio. Keywords: Adaptive Cluster Samplig, Auxiliary Variable, Mea squared error, Efficiecy 1 Itroductio Adaptive cluster samplig, proposed by Thompso(1990), is a efficiet method for samplig rare ad hidde clustered populatios. I adaptive cluster samplig, a iitial sample of uits is selected by simple radom samplig. If the value of the variable of iterest from a sampled uit satisfies a pre-specified coditio C,that is (i,y i > c) the the uit s eighborhood will also be added to the sample. If ay other uits that are adaptively added also satisfy the coditio C,the their eighborhoods are also added to the sample. This process is cotiued util o more uits that satisfy the coditio are foud. The set of all uits selected ad all eighborig uits that satisfy the coditio is called a etwork. The adaptive sample uits, which do ot satisfy the coditio, are called edge uits. A etwork ad its associated edge uits are called a cluster.if a uit is selected i the iitial sample ad does ot satisfy the coditio C, the there is oly oe uit i the etwork. It is well kow that the variable about which we have full iformatio is kow as auxiliary variable ad the iformatio is kow as auxiliary iformatio which is highly (positively or egatively) correlated with the variable uder study. Wheever auxiliary variable (iformatio) is kow, oe would like to use it at the desig or estimatio stage sice it is well kow ad established that the use of auxiliary iformatio i samplig theory ehaces the efficiecy of the estimators ad it is i use sice the use of samplig itself. I this paper, we will study the estimator of populatio mea i adaptive cluster samplig usig a auxiliary variable. Estimators Uder Simple Radom Samplig Let (x i,y i ),i = 1,,, be the pair of observatios for the auxiliary ad study variables, respectively for the populatio of size usig Simple Radom Samplig With Out Replacemet (SRSWOR). Let µ x ad µ y be the Correspodig author e-mail: sat x003@yahoo.co.i c 016 SP atural Scieces Publishig Cor.
S. K. Yadav et al.: Improved ratio estimators of populatio mea... populatio meas of auxiliary ad study variables respectively ad x ad ȳ be the respective sample meas. Ratio estimators are used whe the lie of regressio of y o x passes through origi ad the variables X ad Y are positively correlated to each other, while product estimators are used whe X ad Y are egatively correlated to each other, otherwise regressio estimators are used to estimate the populatio parameters uder cosideratio. Cochra (1940) proposed the classical ratio estimator for estimatig the populatio mea,µ y = 1 variable as follows: where ȳ= 1 y i ad x= 1 x i, assumig that µ x = 1 ȳ R = ȳ µx x y i of the study x i, the populatio mea of the auxiliary variable is kow. The expressios for the Mea Squared Error (MSE) of the estimator give i (1) up to the first order of approximatio are respectively, as follows: MSE(ȳ R )= where C y = S y µ y, C x = S x µ x, S y = 1 S xy = 1 1 (x i µ x )(y i µ y ) 1 µ [ y +Cx ρc ] yc x (y i µ y ), f =, S x = 1 1 (1) () (x i µ x ), ρ xy = S xy S y S x, Sisodia ad Dwivedi (1981) proposed the followig estimator usig the coefficiet of variatio of auxiliary variable as: µx +C x ȳ R1 = ȳ (3) x+c x The MSE of the estimator ȳ R1, up to the first order of approximatio respectively are: MSE(ȳ R1 ) µ [ y + θ1c x ] θ 1 ρ x x,where θ1 = µ x (4) µ x +C x Upadhyay ad Sigh (1999) proposed two ratio type estimators utilizig the coefficiet of variatio ad the coefficiet of kurtosis of auxiliary variable β (x) β(x) µ x +C x ȳ R = ȳ (5) β (x) x+c x Cx µ x + β (x) ȳ R3 = ȳ (6) C x x+β (x) The MSE of the estimators ȳ R ad ȳ R3, up to the first order of approximatio respectively are: MSE(ȳ R ) MSE(ȳ R3 ) µ [ y + θ C x θ ] ρ x x µ [ y + θ3c x ] θ 3 ρ x x where θ = β (x) µ x β (x) µ x +C x ad θ 3 = C xµ x C x µ x +β (x) Sigh ad Tailor (003) proposed a ratio type estimator usig correlatio coefficiet betwee auxiliary variable ad the variable uder study as: µx + ρ xy ȳ R4 = ȳ (9) x+ ρ xy The MSE of the estimator ȳ R4, up to the first order of approximatio respectively are: MSE(ȳ R4 ) µ [ y + θ4 C x θ ] 4ρ x x,where θ4 = µ x (10) µ x + ρ xy (7) (8) c 016 SP atural Scieces Publishig Cor.
J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) / www.aturalspublishig.com/jourals.asp 3 Kadilar ad Cigi (003) suggested the followig estimator utilizig the auxiliary variable as: ( µ ) ȳ R5 = ȳ x x (11) The MSE of the estimator ȳ R5, up to the first order of approximatio respectively are: MSE(ȳ R5 ) µ [ y + 4Cx 4ρ ] xyc x (1) Ya ad Tia (010) proposed two ratio type estimators usig coefficiets of skewess β 1(x) ad kurtosis β(x) of auxiliary variable as: β(x) µ x + β 1(x) ȳ R6 = ȳ (13) β (x) x+β 1(x) β1(x) µ x + β (x) ȳ R7 = ȳ (14) β 1(x) x+β (x) The MSE of the estimators ȳ R6 ad ȳ R7, up to the first order of approximatio respectively are: MSE(ȳ R6 ) MSE(ȳ R7 ) µ [ y + θ6 C x θ ] 6ρ x x µ [ y + θ7c x ] θ 7 ρ x x (15) (16) where θ 6 = β (x) µ x β (x) µ x +β 1(x) ad θ 7 = β 1(x) µ x β 1(x) µ x +β (x) 3 Estimators Uder Adaptive Cluster Samplig Let the populatio cosists of distict idetifiable uits labeled from 1,,...,. Let y i ad x i (i = 1,,...,) deote the observatio o the characteristic x ad y respectively, uder study for the i th uit of the populatio. Let deote the iitial sample size ad ν deote the fial sample size. Let Ψ i deote the etwork that icludes uit i ad m i be the umber of uits i that etwork. The iitial sample of uits is selected by simple radom samplig without replacemet. The estimator of the populatio mea for the variable of iterest uder adaptive cluster samplig based o Hase- Hurwitz estimator as, ȳ ac = 1 (w y ) i (17) where, (w y ) i is the average of the variable y uder study i the etwork that icludes uit i of the iitial sample, that is:(w y ) i = m 1 i (y j ) j Ψ i The variace of ȳ ac is give by, V(ȳ ac )= ( 1) [(w y ) i µ y ] (18) Dryver ad Chao (007) proposed the followig ratio estimator i adaptive cluster samplig as, ȳ acr = ȳac x ac µ x (19) c 016 SP atural Scieces Publishig Cor.
4 S. K. Yadav et al.: Improved ratio estimators of populatio mea... where x ac = 1 (w x ) i ad(w x ) i is the average of the auxiliary variable x i the etwork that icludes uit i of the iitial sample, that is, (w x ) i = m 1 i (x j ) j Ψ i The first order approximated MSE of ȳ acr is MSE(ȳ acr ) µ [ wy +Cwx ρ ] wx.wyc w wx Chutima (013) proposed the followig ratio type estimators of populatio mea usig parameters of the auxiliary iformatio based o Sisodia ad Dwivedi (1981) estimator ad Upadhyaya ad Sigh (1999) two estimators uder adaptive cluster samplig as, µy +C wx ȳ acr1 = ȳ ac (1) x ac +C wx µx β (wx )+C wx ȳ acr = ȳ ac () x ac β (wx )+C wx µx + β (wx ) ȳ acr3 = ȳ ac (3) x ac + β (wx ) where C wx is the populatio coefficiet of variatio of w x, β (wx ) is the populatio coefficiet of kurtosis of w x. Where S wy = 1 1 [(w y ) i µ y ], S wx = 1 1 [(w x ) i µ x ], S wx.wy = 1 1 [(w x ) i µ x ][(w y ) i µ y ]=ρ wx.wy S wx S wy The mea square errors of above estimators usig Taylor series method up to the first order of approximatios respectively are, MSE(ȳ acr1 ) MSE(ȳ acr ) MSE(ȳ acr3 ) µ [ wy + θw1 C wx θ ] w1ρ wx.w w wx µ [ wy + θw C wx θ ] wρ wx.w w wx µ [ wy + θw3 C wx θ ] w3ρ wx.w w wx where θ w1 = µ y µ y +C wx, θ w = µ xβ (wx) µ µ x β (wx) +C wx, θ w3 = x µ x +β ad R= µ x (wx) µ y (0) (4) (5) (6) 4 Proposed Estimators Motivated by Sigh ad Tailor (003), Kadilar ad Cigi (003) ad Ya ad Tia (010) estimators of populatio mea i simple radom samplig give above, we proposed the estimators based o these metioed estimators of populatio mea i adaptive cluster samplig as, µy + ρ wx.wy ȳ acr4 = ȳ ac (7) x ac + ρ wx.wy µ ȳ acr5 = ȳ x ac x (8) ac β(wx )µ x + β 1(wx ) ȳ acr6 = ȳ ac (9) β (wx ) x ac + β 1(wx ) ) ȳ acr7 = ȳ ac ( β1(wx )µ x + β (wx ) β 1(wx ) x ac + β (wx ) (30) c 016 SP atural Scieces Publishig Cor.
J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) / www.aturalspublishig.com/jourals.asp 5 Usig the method discussed bhutima (013), the mea square errors of above estimators up to the first order of approximatios respectively are, MSE(ȳ acr4 ) MSE(ȳ acr5 ) MSE(ȳ acr6 ) MSE(ȳ acr7 ) µ [ wy + θw4 C wx θ ] w4ρ wx.w w wx µ [ wy + 4Cwx ] 4ρ wx.w w wx µ [ wy + θw6 C wx θ ] w6ρ wx.w w wx µ [ wy + θw7c wx ] θ w7 ρ wx.w w wx (31) (3) (33) (34) where θ w4 = µ x µ x β µ x +ρ wx.wy, θ w6 = (wx) µ x β (wx) +β,θ w7 = 1(wx) µ x β 1(wx) µ x β 1(wx) +β (wx) 5 umerical Illustratio I this sectio, the simulatio x values ad y values from Chutima ad Kumpho (008) were studied. The data statistics of this populatios were show i Table 1.We take the sample size as =0 i Table, value of MSE which are computed usig equatios i Sectio 4, are give. Table 1: Data Statistics = 400 =0 µ y = 1.5 µ x = 0.5550 S y = 5.050 θ 1 = 0.114 S wy = 3.56 θ w1 = 0.137 S x =.400 θ = 0.876 S wx = 1.948 θ w = 0.9357 S xy = 11.037 θ 3 = 0.04 S wx.wy = 6.48 θ w3 = 0.006 ρ xy = 0.910 θ 4 = 0.379 ρ wx.wy = 0.96 θ w4 = 0.375 C y = 4.131 θ 6 = 0.817 C wy =.914 θ w6 = 0.864 C x = 4.35 θ 7 = 0.064 C wx = 3.510 θ w7 = 0.046 β 1(x) = 6.83 β (x) = 55.090 β 1(wx ) = 7.953 β (wx ) = 91.369 Table : MSE Values of Estimators MSE(ȳ R1 )=0.966 MSE(ȳ R )=0.407 MSE(ȳ R3 )=1.117 MSE(ȳ R4 )=0.57 MSE(ȳ R5 )=1.904 MSE(ȳ R6 )=0.311 MSE(ȳ R7 )=1.068 MSE(ȳ acr1 )=0.435 MSE(ȳ acr )=0.309 MSE(ȳ acr3 )=0.595 Proposed Estimators MSE(ȳ acr4 )=0.1 MSE(ȳ acr5 )=1.010 MSE(ȳ acr6 )=0.093 MSE(ȳ acr7 )=0.54 6 Coclusio I the preset mauscript, we developed ew ratio type estimators for the estimatio of populatio mea by usig auxiliary iformatio i adaptive cluster samplig scheme. The bias ad the mea squared error of proposed estimators have bee obtaied up to the first order of approximatio. A empirical study is carried out ad from the estimated MSE of the estimators i Table-, it is clear that if the populatio is rare ad hidde clustered populatio, all estimators i adaptive cluster samplig are more efficiet tha the estimators of populatio mea i simple radom samplig, give the same coditio. Further amog the all metioed estimators of populatio mea alog with all proposed estimators i adaptive cluster samplig, the proposed estimator, ȳ acr6 has the smallest estimated mea square error. Therefore, it should preferably be adopted for the estimatio of populatio mea i adaptive cluster samplig scheme. c 016 SP atural Scieces Publishig Cor.
6 S. K. Yadav et al.: Improved ratio estimators of populatio mea... Ackowledgemet The authors are very much thakful to the editor i chief ad the ukow leared referee for critically examiig the mauscript ad givig valuable suggestios to improve it. Refereces [1]. Chutima, Adaptive cluster samplig usig auxiliary variable,j.math.stat., 9, 49-55 (013). []. Chutima ad B. Chutima, Ratio estimator usig auxiliary variable for adaptive cluster samplig, Joural of Thai Statistical Associatio, 6(),41-56 (008) [3] W.G.Cochra, The estimatio of the yields of the cereal experimets by samplig for the ratio of grai to total produce, Jour. Agri. Sci.,59,15-16 (1940). [4] A.L. Dryver ad C. Chao, Ratio estimators i adaptive cluster samplig, Evirometrics,18, 607-60 (1967). [5] C. Kadilar, ad H. Cigi, A study o the chai ratio-type estimator,hacettepe Joural of Mathematics ad Statistics,3,105-108 (003). [6] H. P. Sigh ad R. Tailor, Use of kow correlatio coefficiet i estimatig the fiite populatio mea, Statistics i Trasitio,6,555-560 (003). [7] B.V.S. Sisodia ad V.K. Dwivedi, A modified ratio estimator usig coefficiet of variatio of auxiliary variable, J. Idia Soc. Agric. Statist., 33, 13-18 (1981). [8] S.K. Thompso, Adaptive cluster samplig, J. Am. Statist. Assoc., 85, 1050-1059 (1990). [9] L.. Upadhyaya ad H.P. Sigh, Use of trasformed auxiliary variable i estimatig the fiite populatio mea,biometri. J., 41, 67-636(1999). [10] Z. Ya ad B. Tia, Ratio Method to the Mea Estimatio Usig Co-efficiet of Skewess of Auxiliary Variable,, ICICA 010, Part II,CCIS 106, 103-110(010). Subhash Kumar Yadav is workig as assistat professor i the Departmet of Mathematics ad Statistics, Dr Ram Maohar Lohia Avadh Uiversity, Faizabad. He has got published may research papers i the field samplig of Statistics. Sheela Misra, the supervisor of Dr Subhash Kumar Yadav is workig as Professor i the Departmet of Statistics, Uiversity of Luckow, Luckow. She is a very good academicia as well as the admiistrator. She has successfully orgaized may iteratioal cofereces i Statistics. She has doe a lot of work i differet field of Statistics. Sat Shara Mishra is Associate Professor at Departmet of Mathematics ad Statistics, Dr Ram Maohar Lohia Avadh Uiversity, Faizabad. He is a seior faculty i the Departmet. He has supervised may Ph.D. i differet fields of Mathematics ad Statistics. He has got published a lot of research papers i differet jourals of repute. ipapor Chutima is workig as seior Faulty i the Departmet of Mathematics, Mahasarakham Uiversity, Maha Sarakham, Thailad. She is doig very good work i the field samplig of Statistics. She has got published may research papers i differet reputed jourals of Statistics. c 016 SP atural Scieces Publishig Cor.