Regression-cum-Exponential Ratio Estimators in Adaptive Cluster Sampling. Muhammad Shahzad Chaudhry 1 and Muhammad Hanif 2

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1 ISSN Journal of Saisics Volume, 015. pp Absrac Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling Muhammad Shahzad Chaudhry 1 and Muhammad Hanif In his paper, Regression-cum-Exponenial Raio Esimas have been proposed f esimaing he populaion mean using wo auxiliary variables in Adapive Cluser Sampling. The expressions f he Mean Square Err and Bias of he proposed Esimas have been derived. A simulaion sudy has been carried ou o demonsrae and compare he efficiencies and precisions of he Esimas. The proposed Esimas have been compared wih Convenional Raio, Regression, Exponenial Raio Esimas boh under Simple Random Sampling Wih-Ou Replacemen (SRSWOR) and Adapive Cluser Sampling. Keywds Transfmed populaion, Simulaed populaion, Expeced final sample size, Comparaive percenage relaive efficiency, Unlikely assumpion 1. Inroducion The Adapive Cluser Sampling (ACS) is suiable and efficien f he rare and clusered populaion. In Adapive Cluser Sampling,he iniial sample is seleced by a Convenional sampling design such as Simple Random Sampling hen he neighbhood of each uni seleced is considered if he value of he sudy variable from he sampled uni mee a pre-defined condiion C usually y > 0. The neighbing uni is added and examined if he condiion is saisfied and he process coninues unil he new uni mees he condiion. 1 Governmen College Universiy, Lahe, Pakisan. almoeed@homail.com Naional College of Business Adminisraion and Economics, Lahe, Pakisan. drmianhanif@gmail.com

2 58 Muhammad Shahzad Chaudhry and Muhammad Hanif The final sample comprises all he unis sudied and he iniial sample. A newk consiss of hose unis ha mee he predefined condiion. The unis ha do no mee he specified condiion are known as edge unis. A cluser is a combinaion of newk and edge unis. The neighbhood can be defined by social and insiuional relaionships beween unis. The firs-der neighbhood consiss of he sampling uni iself and four adjacen unis denoed as eas (above), wes (below), nh (righ), and souh (lef). The second-der neighbhood consiss of firs-der neighbing unis and he unis including nheas, nhwes, souheas, and souhwes unis i.e. diagonal quadras. Consider he following example o undersand he Adapive Cluser Sampling process using firs-der neighbhoods and he condiion C ha y i > 0. Figure 1 show ha he sudy region is divided ino 50 quadras and ha he populaion is divided ino hree clusers accding o he condiion. There are 35 newks of size one as any uni ha does no mee he condiion is a newk of size one. Each cluser can be divided ino a newk (ha saisfies C) and individual newks of size 1(ha do no saisfy C) i.e. edge unis. Thus, firs newk has 6 unis wih 8 edge unis. Newk has 5 unis wih 10 edge unis. Newk 3 has 4 unis wih 7 edge unis Figure 1: The hree newks are shaded accding o condiion Thompson (1990) firs inroduced he idea of he Adapive Cluser Sampling o esimae he rare and clusered populaion and proposed four unbiased Esimas in Adapive Cluser Sampling. Smih e al. (1995) sudied he efficiency of Adapive Cluser f esimaing densiy of winering waer fowl and found ha he efficiency is highes as compare o Simple Random Sampling design when he wihin newk variance is close o populaion variance. Dryver (003) found ha ACS perfms well in a Univariae seing. The simulaion on real daa of bluewinged and red-winged resuls shows ha Hviz-Thompson Type Esima (195) was he mos efficien Esima using he condiion of one ype of duck o esimae ha ype of duck. F highly crelaed variables he ACS perfms well f he parameers of ineres. Chao (004) proposed he Raio Esima in

3 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling Adapive Cluser Sampling and showed ha i produces beer esimaion resuls han he iginal Esima of Adapive Cluser Sampling. Dryver and Chao (007) suggesed he Classical Raio Esima in Adapive Cluser Sampling (ACS) and proposed wo new Raio Esimas under ACS. Chuiman and Kumphon (008) proposed a Raio Esima in Adapive Cluser Sampling using wo auxiliary variables. Chuiman (013) proposed Raio Esimas using populaion coefficien of variaion and coefficien of Kurosis, Regression and difference Esimas by using single auxiliary variable.. Some Esimas in Simple Random Sampling Le, N be he oal number of unis in he populaion. A random sample of size n is seleced by using Simple Random Sampling Wih-Ou Replacemen. The sudy variable and auxiliary variable are denoed by y and x wih heir populaion means Y and X, populaion sandard deviaion S y and S x, coefficien of variaion C y and C x respecively. Also ρ xy represen populaion crelaion coefficien beween X and Y. Cochran (1940) and Cochran (194) proposed he Classical Raio and Regression Esimas are given by: X 1 y (.1) x y X x yx (.) The Mean Square Err (MSE) of he Esimas of equaion (.1) and (.) are as follows: MSE( 1) Y Cy Cx xycxc (.3) y MSE Y C y 1 xy (.4) respecively. 1 1 where n N Bahl and Tueja (1991) proposed he Exponenial Raio Esima o esimae he populaion mean is given by: 3 y exp X x (.5) X x The Mean Square Err and Bias of he Exponenial Raio Esima 3 are given by: 59

4 60 Muhammad Shahzad Chaudhry and Muhammad Hanif ) C x Y Cy xycxcy 4 3 xycc x y Bias( 3) Y Cx 8 MSE( 3 3. Some Esimas in Adapive Cluser Sampling (.6) (.7) Suppose, a finie populaion of size N is labeled as 1,,3,,N and an iniial sample of n unis is seleced wih a Simple Random Sample Wih-Ou Replacemen. Le, w, w and w denoes he average y-value, average x- yi xi zi value and average z-value in he newk which includes uni i such ha, 1 wyi y, 1 j i x and 1 j wzi zj m m m, respecively. i j A i i j A i i j A Adapive Cluser Sampling can be considered as Simple Random Sample Wih- Ou Replacemen when he averages of newks are considered (Dryver and Chao, 007 and Thompson, 00). Consider he noaions w, w and w are he sample means of he sudy and auxiliary variables in he ransfmed populaion respecively, such ha, 1 w n w, 1 w n w and w z 1 n n i 1 w zi y n i n i yi y x x n i 1 Consider C wy, C and C wz represens populaion coefficien of variaions of he sudy and auxiliary variables respecively. Le ρ wy and ρ wzwy represen populaion crelaion coefficiens beween w x and w y, and, w z and w y respecively. Le us define, wy Y w ewy, x X w e, and z X ewz (3.1) Y X X where e, wy e and ewz are he sampling errs of he sudy and auxiliary variables respecively, such ha E e E e E e (3.) wy wz 0 wy and wy wy wy C, E wy e C E ewz E e e C C E e E e e C C (3.3) and wz wy wzwy wz wy Cwz (3.4) xi z

5 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling 61 Thompson (1990) developed an unbiased Esima f populaion mean in ACS based on a modificaion of he Hansen-Hurwiz (1943) Esima which can be used when sampling is wih replacemen wihou replacemen. In erms of he n newks (which may no be unique) inerseced by he iniial sample 4 1 n yi y n i 1 w w where 1 wyi m yj i j A i N Var( 4) wyi Y N 1 (3.5) i1 Dryver and Chao (007) proposed a Modified Raio Esima f he populaion mean keeping in view Adapive Cluser Sampling. wyi (3.6) is0 ˆ 5 X RX w xi is0 The Mean Square Err of 5 is, N MSE( 5) wyi Ri N 1 (3.7) i1 where R is he populaion raio beween w and xi w in he ransfmed populaion. yi Chuiman (013) proposed a Modified Regression Esima f he populaion mean of he sudy variable in Adapive Cluser Sampling as follows: 6 wy w( X wy) (3.8) where Swy wysswy Y wycwy w (3.9) S S X C The approximae Mean Squared Err of 6 is given by: MSE( ) S (1 ) YC (1 ) (3.10) 6 wy wy wy wy No and Hanif (014) proposed a Modified Exponenial Raio Esima f he populaion mean in ACS, following he Bahl and Tueja (1991) as given by: y

6 6 Muhammad Shahzad Chaudhry and Muhammad Hanif X 7 wy exp (3.11) X The Bias and Mean Square Err of he Esima are as follows: 3C wyccwy Bias( 7 ) Y 8 MSE( 7 ) = C Y Cwy wyccwy 4 4. Proposed Esimas in Adapive Cluser Sampling (3.1) (3.13) Following he Bahl and Tueja (1991), Chuiman (013) and No and Hanif (014) he proposed Modified Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling wih wo auxiliary variables are given by: 8 ( ) X wy Z wz exp (4.1) X X 9 wy ( Z wz) exp X (4.) 4.1 Bias and Mean Square Err of Esima 8 : The Esima equaion (4.1) may be wrien as: X X (1 e ) 8 Y 1 ewy Z Z 1 ewz exp (4.1.1) X X (1 e) e 8 Y 1 ewy Zewzexp (4.1.) e e 8 Y Yewy Zewz exp (4.1.3) e 1

7 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling 1 e e 8 Y Yewy Zewz exp 1 (4.1.4) Expanding he series up o he firs der, rewriing equaion (4.1.4) as: e e 8 Y Yewy Zewz exp 1 (4.1.5) e e 8 Y Yewy Zewz exp (4.1.6) 4 Expanding he exponenial erm up-o he second degree, we ge equaion (4.1.6) as: e e 1 e e 8 Y Yewy Zewz 1 (4.1.7) 4 4 e e e 8 Y Yewy Zewz 1 (4.1.8) 4 8 e 3e e 3e e 3e 8 Y 1 Yewy 1 Zewz Simplifying, igning he erms wih degree hree greaer 8 Y Ye 3Ye Yewye Zewze Yewy Zewz 8 (4.1.9) (4.1.10) Applying expecaions on boh sides of equaion (4.1.10), and using noaions of equaion ( ), we ge: E( 8 Y 3YC Y wycc yx ) ZwzCC wz (4.1.11) 8 In der o derive Mean Square Err of equaion (4.1), we have equaion (4.1.6) by igning he erm degree greaer as: e 8 Y Yewy Zewz exp (4.1.1) 63

8 64 Muhammad Shahzad Chaudhry and Muhammad Hanif e 8 Y Yewy Zewz 1 (4.1.13) Ye Yewye Zewze 8 Y Yewy Zewz (4.1.14) ( 8 Y Ye Yewye ) Zewze Y Yewy Zewz (4.1.15) Taking square and expecaions on he boh sides of equaion (4.1.15), and using noaions of equaion ( ) MSE( 8 ) Y C Y Cwy Z Cwz Y wyccwy 4 YZ wzwycwzcwy YZwz CCwz (4.1.16) 4. Bias and Mean Square Err of Esima 9 : The Esima of equaion (4.) may be wrien as follows: X X (1 e) 9 Y 1 ewy Z Z 1 ewz exp (4..1) X 9 Y 1 ewy Zewz expe (4..) Expanding he Exponenial erm up-o he second degree, we ge equaion (4..) as: e 9 Y Yewy Zewz 1 e ( ) (4..3) e e e 9 Y 1 e ( ) Yewy 1 e ( ) Zewz 1 e ( ) (4..4) Simplifying, igning he erms wih degree hree greaer

9 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling e 9 Y Yewy Ye Y ( ) Yewye Zewz Zewze (4..5) Applying expecaions on boh sides of equaion (4..5), and using noaions of equaion ( ), we ge: C E( 9 Y ) Y Y wyccyx ZwzCCwz (4..6) In der o derive Mean Square Err of equaion (4.), we have equaion (4..3) by igning he erm degree greaer as: Y Ye Ze 1 e 9 wy wz (4..7) 9 Y Yewy Zewz Ye Yewye Zewze (4..8) Y Ye Ze Ye Ye e Ze e (4..9) 9 wy wz wy wz Taking square and expecaions on he boh sides of equaion (4..9), and using noaions of equaion ( ), we ge: Y Cwy Y C Z Cwz YZ wzwycwzc wy MSE( 9 ) (4..10) Y wyccwy YZ wzccwz 5. Conclusion To compare he efficiency of proposed Esimas wih he oher Esimas, a simulaed populaion is used and perfmed simulaions f he sudy. The condiion C f added unis in he sample is y > 0. The y-values are obained and averaged f keeping he sample newk accding o he condiion and f each sample newk x-values are obained and averaged. F he simulaion sudy en housands ieraion was run f each Esima o ge accuracy esimaes wih he simple random sampling wihou replacemen and he iniial sample sizes of 10,0,30,40 and 50. In he ACS he expeced final sample size varies from sample o sample. Le, E(v) denoes he expeced final sample size in ACS, is sum of he probabiliies of inclusion of all quadras. In he ACS he expeced final sample size varies from 65

10 66 Muhammad Shahzad Chaudhry and Muhammad Hanif sample o sample. F he comparison, he sample mean from a SRSWOR based on E(v) has variance using he usual fmula: ( N E ( ) ( v Var y )) (5.1) NE() v The esimaed Mean Squared Err of he esimaed mean is given by: ^ 1 r MSE ( Y ) (5.) * * r i 1 Where * is he value f he relevan Esima f sample i, and r is he number of ieraions. The Percenage Relaive Efficiency is given by: PRE Var( y) ^ 100 (5.3) MSE * 5.1. Populaion: In his populaion, he wo rarely clusered populaions from he Thompson (199) have been considered as he auxiliary variables in Table 1 and Table as x and z, respecively. Dryver and Chao (007) generaed he values f he variable of ineres using he following wo models. y i 4x i i where i ~ N0, x i (5.1.1) y 4w where i ~ N w (5.1.) i xi i 0, xi The variabiliy of he sudy variable is propional o he auxiliary variable iself in model (5.1.1) whereas i is propional o he wihin-newk mean level of he auxiliary variable in model (5.1.). Consequenly, he wihin newk variances of he sudy variable in he wo newks consising of me han one unis are much larger in he populaion generaed by model (5.1.1). To evaluae he perfmance of proposed Esimas, we simulae he values f variable of ineres using he model (5.1.3), variable x and z have been used as auxiliary variables. Le yi, w xi and w zi denoe he i h value f he variable of ineresy, averages of newks f he auxiliary variables w and w respecively. In he given model, he averages of he newks (ransfmed populaion; Thompson, 00) f variable x and z have been used as auxiliary variables showing he srong crelaion a he newk (region) level. yi 4i 4wzi i where i ~ N w w (5.1.3) x 0, xi zi z

11 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling In simulaed populaion from model (5.1.3), he variance of sudy variable is propional o he sum of he newk means level of auxiliary variables. The simulaed populaion using model (5.1.3) is given in Table 3. Chao e al. (010) and Dryver and Chao (007) have proved ha Convenional Esimas in SRSWOR perfm beer han Adapive Cluser Sampling Esimas f srong crelaion a uni level. In his populaion, we evaluae he Esimas when having he crelaion a newk level no on uni level. The condiion C f added unis in he sample is y > 0. The y-values are obained and averaged f keeping he sample newk accding o he condiion and f each sample newk x-values and z-values are obained and averaged. The crelaion coefficien beween variable of ineres and auxiliary variables x and z is and , respecively. The crelaion coefficien in he ransfmed populaion beween variable of ineres and auxiliary variables x and z is and , respecively. Thus, crelaion is high a newk (region) level han he sampling uni level. Dryver and Chao (007) showed ha usual Esimas in SRSWOR perfm beer han Adapive Cluser Sampling Esimas f srong crelaion a uni level bu perfms wse when having he srong crelaion a newk level. The Adapive Esimas will be me efficien if here is a sufficienly high crelaion in he ransfmed populaion. Thompson (00) invesigaed ha Adapive Cluser Sampling is preferable han he comparable Convenional sampling mehods if he wihin newk variance is sufficienly large as compared o overall variance of he sudy variable and presened he condiion when he Modified Hansen-Hurwiz Esima f Adapive Cluser Sampling have lower variance han variance of he mean per uni f a simple random sampling wihou replacemen of size Ev () if and only if, 1 1 N n K Sy ( yi wi ) n E( v), (5.1.4) Nn( N 1) k 1iAi E( v)[ N n] Sy S. (5.1.5) wy D N[ E( v) n] where wihin newk variance is defined by: 1 K Swy ( y ) D i w. (5.1.6) i ( N 1) k 1iA i 67

12 68 Muhammad Shahzad Chaudhry and Muhammad Hanif Using equaion (5.1.6) he wihin newk variance of he sudy variable is found o be Le, f he iniial sample size n = 10, expeced sample size will be Ev () = 34.73wih he condiion of ineres y > 0, using he condiion (5.1.5), we have: 34.73[400 10] ( ). (5.1.7) 400[ ] > (5.1.8) Thus, he overall variance is found o be very high as compare o he wihin newk variance f he sudy variable. Adapive Cluser Sampling is preferable han he Convenional sampling if he wihin newk variance is large enough as compare o overall variance (Thompson 00). So, Adapive Esimas are expeced o perfm wse han he Esimas in Simple Random Sampling. The wihin newk variances of he sudy variable f he newk (38, 31, 37, 30, 3, 30, 9, 30, 9, 30, 31, 3, 33) is , f he newk (38, 33, 35, 31, 9, 31, 33, 38, 36, 33, 35, 36) is 8, f newk (, 4, 0, 6, 19, 5) is , f he newk (37, 4, 43, 40, 38, 40, 36, 31, 4, 31, 41) is , f he newk (45, 5, 44, 49) is , and f he newk (47, 50, 46, 45, 49, 53, 49, 5, 53) is The wihin newk variances do no accouns a large pion of overall variance. Thus, Adapive Esimas are expeced o perfm wse han he comparable usual Esimas. The Convenional Esimas are me efficien han he Adapive Esimas if wihin-newk variances do no accoun f a large pion of he overall variance (Dryver and Chao 007). Simulaion experimen has been conduced f all he Adapive Esimas and Simple Random Sampling Esimas. The esimaed relaive Bias (Table 4) of all he Esimas decreases by increasing he sample size. The Bias decreases by increasing he sample size as recommended ha Bias decreases f large sample sizes (Lohr, 1999). The Percenage Relaive Efficiency (Table 5) of all he ACS Esimas remains much lower han he SRS Esimas excep he proposed Esimas 8 and 9. The proposed Regression-cum-Exponenial Raio Esima 8 has he maximum Percenage Relaive Efficiency. The proposed regressioncum-exponenial raio Esima 9 also has greaer percenage relaive efficiency han all oher Esimas. The regression Esima 6 in ACS did no perfm well in erm of percenage relaive efficiency. Thus, he proposed Regression-cum- Exponenial Raio Type Esimas are me robus f rare Clusered populaion

13 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling even when he efficiency condiions are no fulfilled. Dryver and Chao (007) assumed 0/0 as zero f he Raio Esima. The usual Raio Esima and Raio Esima in ACS did no perfm and reurn no value (*) f he all he sample sizes. In his simulaion sudy, 0/0 is no assumed 0. The use of Regression-cum- Exponenial Raio Esimas is beer in Adapive Cluser Sampling han assuming an unlikely assumpion f he Raio Esima in Adapive Cluser Sampling. We did he simulaion sudy of logarihmic esimas in ACS and found good. We recommend he deail sudy of ACS design wih logarihmic esimas as fuure research. Acknowledgemens The auhs are graeful o Yves G. Berger, Universiy of Souhampon, UK f he programming advice o generae populaion, simulaion sudies and valuable suggesions regarding he improvemen of his paper. 69 Table 1: Auxiliary variable x (Thompson, 199) f populaion

14 70 Muhammad Shahzad Chaudhry and Muhammad Hanif Table : Auxiliary variable z (Thompson, 199) f populaion

15 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling Table 3: Simulaed values of y under model (5.1.3) using Wx and Wy

16 7 Muhammad Shahzad Chaudhry and Muhammad Hanif Table 4: Esimaed Relaive Bias f he Esimas n * * * * * * * * * * Table 5: Comparaive Percenage Relaive Efficiencies f he Esimas Simple Random Sampling Adapive Cluser Sampling Ev () y * * * * * * * * * * Table 6: Descripive measures of he populaions X x Z z Y y w z w y Cx xz Cz xy Cy wz wy C wz C wz yz wy C wy wywz

17 Regression-cum-Exponenial Raio Esimas in Adapive Cluser Sampling 73 References 1. Bahl, S. and Tueja, R. K. (1991). Raio and Produc Type Exponenial Esima. Infmaion and Opimizaion Sciences, 1, Chao, C. T. (004). Raio esimaion on Adapive Cluser Sampling. Journal of Chinese Saisical Associaion, 4(3), Chuiman, N. (013). Adapive Cluser Sampling using auxiliary variable. Journal of Mahemaics and Saisics, 9(3), Chuiman, N. and Kumphon, B. (008). Raio Esima using wo auxiliary variables f Adapive Cluser Sampling. Thailand Saisician, 6(), Cochran, W. G. (1940). The esimaion of he yields of cereal experimens by sampling f he raio of grain o oal produce. Journal of Agriculural Sciences, 30, Cochran, W. G. (194). Sampling hey when he sampling unis are of unequal sizes. Journal of he American Saisical Associaion, 37, Cochran, W.G. (1977). Sampling Techniques. John Wiley, New Yk. 8. Dryver, A. L. (003). Perfmance of Adapive Cluser Sampling Esimas in a mulivariae seing. Environmenal and Ecological Saisics, 10, Dryver, A. L. and Chao, C. T. (007). Raio Esimas in Adapive Cluser Sampling. Environmerics, 18, Lohr, S. L. (1999). Sampling Design and Analysis Pacific Grove. Duxbury Press, CA. 11. No-ul-Amin and Hanif, M. (013). Exponenial Esimas in Survey Sampling. Unpublished Thesis. Submied o Naional College of Business Adminisraion and Economics, Lahe. 1. Hansen, M. M. and Hurwiz, W. N. (1943). On he hey of sampling from finie populaion. The Annals of Mahemaical Saisics, 14, Hviz, D. G. and Thompson, D. J. (195). A generalizaion of sampling wihou replacemen from a finie universe. Journal of American Saisical Associaion, 47, Smih, D. R., Conroy, M. J. and Brakhage, D. H. (1995). Efficiency of Adapive Cluser Sampling f esimaing densiy of winering waerfowl. Biomerics, 51, Thompson, S. K. (1990). Adapive Cluser Sampling. Journal of American Saisical Associaion, 85, Thompson, S. K. (199). Sampling. John Wiley and Sons, New Yk. 17. Thompson, S. K. (00). Sampling. John Wiley and Sons, New Yk.