Product and Exponential Product Estimators in Adaptive Cluster Sampling under Different Population Situations

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1 Proceedngs of the Pakstan Academy of Scences: A. Physcal and Computatonal Scences 53 (4): (06) Copyrght Pakstan Academy of Scences ISSN: (prnt), (onlne) Pakstan Academy of Scences Research Artcle Product and Exponental Product Estmators n Adaptve Cluster Samplng under Dfferent Populaton Stuatons Muhammad Shahzad Chaudhry *, and Muhammad Hanf Department of Statstcs, Government College Unversty, Lahore, Pakstan Natonal College of Busness Admnstraton & Economcs, Lahore, Pakstan Abstract: In ths paper, the product and exponental product estmators have been proposed for estmatng the populaton mean usng populaton mean of an auxlary varable, when there s negatve correlaton between the varables, under adaptve cluster samplng (ACS) desgn. The expressons for mean squared error (MSE) and bas of the proposed estmators have been derved. Two smulated populatons are used and smulaton studes have been conceded out to reveal and match the effcences of the estmators. The proposed estmators have been matched wth conventonal estmators and estmators n ACS. The smulaton results showed that the proposed product and exponental product estmators are more effcent as compare to conventonal as well as Hansen-Hurwtz and rato estmators n ACS. Keywords: Auxlary nformaton, smulated populaton, transformed populaton, bvarate normal dstrbuton, negatve correlaton, expected fnal sample sze, comparable varance, estmated relatve bas. INTRODUCTION Samplng selects a part of populaton of nterest to gan nformaton about the whole. Data are often produced by samplng a populaton of ndvduals or objects. The nferences made wll rely on the statstcs ganed from the data so these nferences can only be as good as the data. The samplng desgn refers to the method used to select the sample from the populaton. Deprved samplng desgn can produce deceptve conclusons. In conventonal samplng desgns the sample sze s determned pror to the survey. In these desgns the samplng unts are ndependent on observatons gathered throughout the survey. A man problem that arses n survey samplng s estmaton of the densty of rare and clustered populaton, such as plants, brds and anmals of scarce and dyng out speces, fsheres, patchy mnerals exploraton, toxc waste concentratons, drug addcted, and AIDS patents. The tradtonal random samplng desgns may be neffcent and frequently not succeed to offer samples wth useful nformaton for rare populaton, because t s possble that the majorty of the sampled elements gve no nformaton []. The Adaptve Cluster Samplng (ACS) desgn s approprate for the scarce and bunched populaton. In ACS, a frst sample s selected wth a usual samplng desgn then the vcnty of every element selected n frst sample s consdered. If the observed values of the study varable satsfy a specfc condton C, say y > C then the addtonal unts n the neghbourhood of the th unts s sampled. Each neghbourng element s ncluded and nvestgated f the predefned condton C s fulflled and the procedure keep on untl a new unt does not satsfy the condton. It s a type of network samplng, whch provdes mproved estmates as compared to conventonal desgns n case of clustered rare populaton. All the unts studed (ncludng the ntal sample) compose the fnal sample. The set consstng on those elements that met the condton are called a network. The elements that are fals to fulfl the condton are called edge unts. Clusters are groupng of networks and edge unts. Thompson [] frst proposed the dea of the ACS scheme and ntroduced modfed Hansen-Hurwtz [3] Receved, October 05; Accepted, November 06 *Correspondng author: Muhammad Shahzad Chaudhry; Emal: almoeed@hotmal.com

2 448 Muhammad Shahzad Chaudhry & Muhammad Hanf and Horvtz-Thompson [4] form estmators. Dryver [5] establshed that ACS performed extraordnary n unvarate stuaton but n the multvarate settng, the effcency of ACS depended on the relatonshp of the varables. The smulaton results for actual data of blue-wnged and red-wnged showed that Horvtz- Thompson [4] form of estmator was the manly effcent estmator n certan condtons. Dryver and Chao [6] proposed the conventonal rato estmator n ACS and also proposed two more rato estmators. Adaptve Cluster Samplng Process Consder a fxed populaton of N elements labelled,,3,,n are denoted as u = {u,u,,u N }. Consder a small ntal sample of sze n whch s selected from N by smple random sample wthout replacement (srswor). The frst sample s chosen by tradtonal samplng process n an ACS procedure and then the predefned neghbourng unts for all the unts of the frst sample are consdered for a partcular condton C. If any of the elements n the frst sample satsfy condton C there neghbourng elements are ncluded to the sample and observed. In general, f the characterstc of nterest s found at a partcular area then we contnue to locate around that area for more nformaton. Further, f any neghbourng unt satsfes the condton then ts neghbourhoods are also sampled and the process goes on. Ths teratve process stops when the new unt does not satsfy condton C. The vcnty can be decded n two ways. The samplng element and four neghbourng elements are known as the frst-order neghbourhood denoted by east, west, north, and south. The frst-order neghbourng elements and the elements northeast, northwest, southeast, and southwest are known as second-order neghbourhood. There are n total eght neghbourhood quadrats ncludng the frst-order neghbourhood and the second order neghbourhood. All the unts ncludng the ntal sample composed the fnal sample. A network conssts of elements that satsfy the specfed condton (usually y =). The networks of sze one are those elements whch fal to meet the predefned condton C n the frst sample. The edge unts are those whch do not satsfy the specfed crtera. A cluster s a mxture of network unts wth assocated edge unts. Clusters may have overlappng edge unts. The networks do not have common elements such that the unon of the networks becomes the populaton. Thus, t s possble to partton the populaton of all elements n a form of exclusve and entre networks. The networks related to clusters can be denoted by A, A, A 3,, A n or they can be shown wth darker lnes around the quadrats. These may be shaded as well. The edge unts can be denoted wth open crcles ( ). The entre regon s parttoned nto N rectangular or square unts of equal sze that can be set n a lattce system. The rectangular or square unts are called quadrats. The unts (u,u,,u N ) form a dsjont and comprehensve partton of the entre area so that unts labels (,,,N) categorse the poston of N quadrats. In ACS the populaton s measured n terms of quadrats only.. MATERIALS AND METHODS. Some Estmators n Smple Random Samplng Let N be the entre number of elements n the populaton. A random sample of sze n s selected by usng srswor. The study varable and auxlary varable are represented by y and x wth the populaton means YY and XX, populaton standard devaton S y and S x and coeffcent of varaton C y and C x respectvely. Also let ρ xy denote populaton correlaton coeffcent between X and Y. The sample means of the study and auxlary varables are denoted by y and x respectvely. Cochran [7] and Robson [8] proposed the classcal rato and classcal product estmators, respectvely, for estmatng the populaton mean stated as follows: and t X = y, () x x =. () t y X The mean squared error (MSE) of the estmators () and () are gven by:

3 Product Estmators n ACS 449 MSE ( t ) and MSE ( t ) θ Y Cy Cx xycc + ρ x y, (3) θ Y Cy Cx xycc + + ρ x y, (4) respectvely. Where θ= n N. The rato and product estmators are desgn based. Bahl and Tuteja [9] proposed the exponental rato and exponental product estmators to estmate the populaton mean are gven by: t 3 y exp X x =, (5) X + x t 4 y exp x X =. (6) X + x The MSE and bas of the exponental rato estmator t 3 are as follows: MSE( t 3 ) θ Y C + ρ CC 4 Cx y xy x y 3 ρ CC Bas( t3) θy Cx 8 xy x y, (7). (8) The MSE and bas of the exponental product estmator t 4 are gven by: MSE( t 4 ) Cx θ Y C + +ρ CC 4 y xy x y ρ CC Bas( t4) θy Cx + 8 xy x y. Some Estmators n Adaptve Cluster Samplng, (9). (0) Let a prelmnary sample of n elements s selected wth a srswor from a fnte populaton of sze N categorzed lke,,3,,n. The average y-value and average x-value n the network whch ncludes unt are wy = yj m and w x = xj j A m respectvely. ACS can be consdered as srswor when j A the averages of networks are consdered [, 6]. The averages of networks are consdered as transformed populaton. Transformed populaton s obtaned wth the replacement of the orgnal values of the networks wth the averages of the networks. In the case of transformed populaton s used, each sample of sze one selected wth srswor wll be representatve of the whole network f t ntersect to any unt of a network. In the transformed populaton, the sample means of the study and auxlary varables are w y n n = = w and y w x n n = = w x respectvely. Consder C wy and C represents populaton coeffcent of varatons of the study varable and auxlary varable n the transformed populaton respectvely and ρ wy represent populaton correlaton coeffcent between w x and w y n the ACS. Let us defne, e wy wy Y w = and x X e =. () Y X

4 450 Muhammad Shahzad Chaudhry & Muhammad Hanf Where e and wy e are relatve samplng errors of the study varable and auxlary varable respectvely, such that: ( wy ) = E( e ) = 0 and ( ) E e ( wy ) E e =θ and E( e ) Cwy E e e = θρ C C () wy wy wy =θ C (3) Thompson [] proposed an unbased modfed Hansen-Hurwtz [3] estmator for populaton mean n ACS and can be used as samplng done wth replacement or wthout replacement. Elements that do not meet the condton C are gnored f these elements are not selected n the prelmnary sample. In the form of n networks (whch possbly wll not be exclusve) overlapped by the prelmnary sample (because transformed populaton s used, ACS becomes srswor and each unt selected wll represent a whole network) s stated as follows: Where w y 5 n = y y n t = w = w. (4) = yj m s the average of the number of elements m n the network A. j A The varance of t 5 s gven by: θ N Var( t ( ) 5) = wy Y. (5) N = Dryver and Chao [6] proposed a modfed rato estmator to estmate the populaton mean n ACS s gven by: 0 t X RX 6 wy s = w = x s0 Where ˆR s the sample rato between The MSE of t 6 s gven by: ˆ N =. (6) w y and w x. θ N MSE( t ( ) 6) = wy R. (7) Where R s the populaton rato between w y and w x..3 Proposed Estmators n Adaptve Cluster Samplng The proposed modfed classcal product estmator n ACS wth one auxlary varable s stated as follows: t7 w X x = w. (8) y Followng the Bahl and Tuteja [9], the proposed exponental product estmator n ACS wth one auxlary varable s stated as follows: t X 8 = wy exp. (9) + X

5 Product Estmators n ACS Bas and Mean Square Error of Proposed Product Estmator t 7 Usng () the estmator (8) may be wrtten as follows: X( + e ) X t 7 = Y ( + ewy ), (0) so, t 7 = Y + ewy + e + eewy. Applyng expectatons on both sdes of (), and usng the notatons () we get as follows: Bas 7 t = ( ) 7 wy wy. () E t Y = Yθρ C C () In order to derve MSE of (8), we have (3) by gnorng the term degree or greater as follows: t 7 = Y + ewy + e, t Y = e + e 7 wy. Takng square and expectatons on the both sdes of (4), the obtaned as follows: MSE( ) 7 t = E( t Y) Y ( Cwy C wyccwy ) (3) (4) 7 θ + θ + θρ. (5).3. Bas and Mean Square Error of the Proposed Exponental Product Estmator t 8 Usng () the estmator (9) may be wrtten as follows: or 8 t X( + e) X 8 = Y ( + ewy ) exp, X( + e) + X e e 8 = Y + ewy exp ( + ), t ( ) t Y ( ewy ) e e e = + exp ( + ), 4 or t e 8 e = Y ( + ewy ) exp. 4 Expandng the exponental term up-to the second degree, we get (9) as follows: t e e e 8 Y ( + ewy ) (30) Smplfyng, gnorng the terms wth degree three or greater we get as follows: t 8 Y e e e e e Y ewy Applyng expectatons on both sdes of (3) as follows: Bas( ) 8 t = ( ) wy. C C E t Y Y ρ 4 8 C C wy wy 8 θ + +. (6) (7) (8) (9) (3) (3)

6 45 Muhammad Shahzad Chaudhry & Muhammad Hanf In order to derve MSE of (9), we have (3) as follows: e e + exp ( ). t 8 Y ( e ) wy Ignorng the terms wth power two or greater, we get (33) as follows: t 8 ( ) e Y + ewy exp (33). (34) Openng the exponental term, gnorng terms wth power two or more we get (34) as follows: or t 8 ( ) e Y + ewy +, (35) e e e t8 Y Y ewy + + wy. (36) Takng square and expectatons on the both sdes of (36), and usng notatons ( &3) we get as follows: MSE( ) 8 E t Y Y C C C 4 t = ( ) 3. RESULTS AND DISCUSSION 3. Smulaton Study C 8 θ wy + +ρwy wy. To evaluate and match the effcency of suggested estmators wth the already exstng estmators, two dfferent types of smulated populatons are used and executed smulatons for the comprehensve study. The condton C for ncluded elements n the sample s defned. To get the transformed populaton, the y- values are acqured and averaged for keepng the sample network wth respect to the condton and for every sample network parallel x-values are obtaned and averaged, then y-values and x-values are replaced wth ther networks averages, accordngly. For the smulaton study ten thousands teraton was executed to get MSE and bas for each estmator wth the srswor and the ntal sample szes of 5,0,5,0 and 5 for populatons and. In ACS, the ultmate sample sze s generally larger than the prelmnary sample sze. Let E(v) denote the expected fnal sample sze n ACS, ths s the sum of the probabltes of ncluson of all quadrats. The expected fnal sample sze fluctuates from one sample to another sample n ACS. For the comparson, the sample mean from a srswor based on E(v) has varance usng the formula stated as follows: The estmated relatve bas s defned as: (37) σ ( N Ev ( )) Var( y) = (38) NE() v ^ RBas t ( ) * r ( t* ) Y r = = Y Where t * s the value for the relevant estmator for sample, and r s the number of teratons. The estmated MSE of the estmated mean s gven by: ^ ( ) r MSE t = ( t Y ) * * r = (39) (40)

7 Product Estmators n ACS 453 The percentage relatve effcency s gven by: PRE = Var( y) ^ 00 MSE t 3. Populaton : Study Varable s Clustered and Auxlary Varable s Bnary ( ) * Dryver and Chao [6] used blue-wnged teal data (Fg. ) collected by Smth et al. [0] as an auxlary varable for proposed rato estmators under ACS and compared ther effcency wth conventonal rato estmator n srswor. Let y and x denote the th value for the varable of nterest y, auxlary varables x (say blue-wnged teal), Dryver and Chao [6] generated the values for the varable of nterest usng the followng two models: y y = +ε where 4x ε ~ ( ) (4) N 0, x (4) = 4w +ε where ε ~ N( w ) (43) x 0, x Fg.. Blue-Wnged Teal data [0] for populaton. The varablty of the varable of nterest y s proportonal to the auxlary varable tself n model (4) whle t s proportonal to the wthn-network mean level of the auxlary varable n model (43). Therefore, the wthn network varances of the varable of nterest n the networks are greatly bgger n the populaton produced wth model (4). In ths smulaton study the blue-wnged teal (BWT) s taken as the study varable y. To generate a data set for the auxlary varable x (Fg. ) that contrbutes hgh negatve correlaton (-0.999) correspond to the BWT data whch s the varable of nterest (y), the followng model s used: x = ( ) y +ε where ε ~ ( ) N 0, y (44) Fg.. Smulated x values, based on model (43) and usng BWT data for populaton. In model (44), the study varable y s treated as ndependent varable just to generate a data set for the auxlary varable x. Then 50 s added to and dvded by each value to avod the maxmum number of negatve values and the remanng negatve values are treated as zero. Dong so, the correlaton decreases

8 454 Muhammad Shahzad Chaudhry & Muhammad Hanf to Now changng the role of varables agan (.e., BWT s treated as study varable and not as an auxlary varable) the condton for the varable of nterest BWT s set as C 0 to added unt n the sample. There remanng only two networks wth ths condton but the correlaton found to be between the average values of the networks (Fg. 3, 4). Thus, there s a low correlaton at a unt level but hgh correlaton at the network level. Dryver and Chao [6] demonstrated that classcal estmators n srswor execute well than ACS estmators for hgh correlaton at unt level whle execute poorer when contan the hgh correlaton at network level Fg. 3. Transformed populaton- wth average values of the networks (Wy) Fg. 4. Transformed Populaton- wth average values of the networks (Wx). The varablty of the auxlary varable s proportonal to the varable of nterest n the model (44). The whole varance of the varable of nterest s whereas n the transformed populaton the varance decreased to The wthn network varance of the varable of nterest for the network (0, 03, 3639, 4,, 77) s An enormous part of whole varance s accounts by wthn network varance. Therefore, estmators n ACS are lkely to be more effcent than the equvalent estmators n srs. The estmators n srs are more effcent than the estmators n ACS f wthn-network varances do not report a great part of the overall varance [6]. The comparatve percentage relatve effcency (Table ) of the ACS estmators s much hgher than ther counterpart SRS estmators. The proposed modfed product t 7 and exponental modfed product estmator t 8 n ACS has maxmum percentage relatve effcency for the ntal sample sze and percentage relatve effcency starts ncreasng rapdly for comparable sample szes. Thus, the proposed product estmator n ACS perform much superor than the other usual estmators, the rato, and the Hansen-Hurwtz estmators n ACS when there s a hgh negatve correlaton among the study and the auxlary varables, under the gven condtons. Table. Comparatve percentage relatve effcences for populaton based on E(v). y t t t 3 t 4 t 5 t 6 t 7 t

9 Product Estmators n ACS 455 The estmated relatve bas (Table ) of the estmators decreases as sample szes ncreases n the ACS as well as n the srs. For ACS, as lke srswor, t s suggested a bgger sample sze for a small bas []. Table. Estmated relatve bas for populaton for dfferent sample szes. n E(v) t t t 3 t 4 t 5 t 6 t 7 t Populaton : Study Varable s Rare Clustered and Auxlary Varable s Abundant The populaton (Fg. 5 & Fg. 6) has been generated from a bvarate normal dstrbuton wth the mean vector µ and covarance matrx Σ, ths s Y ~ N( µσ, ). In partcular, we assumed µ= (0,0) and Σ= 5 X. The condton C to ncluded elements n the sample s y > 0 for populaton. The correlaton n the par of random varables of the populaton was found to be When negatve values are assumed zeroes n smulated populaton the correlaton reduces to The correlaton when the averages of the networks (Fg. 7, 8) assumed ncreases to Fg. 5. Smulated y values for populaton, based on bvarate normal dstrbuton Fg. 6. Smulated x values for populaton, based on bvarate normal dstrbuton.

10 456 Muhammad Shahzad Chaudhry & Muhammad Hanf Fg. 7. Transformed populaton- wth average values of the networks (Wy) Fg. 8. Transformed populaton- wth average values of the networks (Wx). The whole varance of the varable of nterest s whle ths varance reduced to n the transformed populaton. The wthn network varances of the study varable for the network (,,3) s 0.333, for the network (,,,,,,) s 0.38, and for the network (,,,,,) s The network varances do not accounts a huge part of whole varance. The overall varance s found to be hgh as compare to the wthn network varances for the study varable. Thus, adaptve estmators are expected to perform worse than the comparable usual estmators. The usual estmators wll be more effcent than the adaptve estmators f wthn-network varances do not account for a huge part of the overall varance [6]. The percentage relatve effcency (Table 3) of the ACS estmators s much lower than the SRS estmators, as expected. The usual product estmator has maxmum percentage relatve effcency, whle usual exponental product estmator has hgher percentage relatve effcency than the other conventonal and adaptve estmators n ACS. The bas of all the estmators decreases by ncreasng the sample sze. The estmated relatve bas s gven n Table 4. The bas decreases by ncreasng the sample sze as recommended that bas decreases for large sample szes []. Table 3. Comparatve percentage relatve effcences for populaton based on E(v). y t t t 3 t 4 t 5 t 6 t 7 t

11 Product Estmators n ACS 457 Table 4. Estmated relatve bas for populaton for dfferent sample szes. N E(v) t t t 3 t 4 t 5 t 6 t 7 t CONCLUSIONS The performance of proposed product estmator and exponental product s better than all the other estmators ncludng conventonal as well as Hansen-Hurwtz and rato estmators n ACS samplng for populaton. The proposed estmators become more effcent as ntal sample sze ncreases for the populaton, under the gven condtons. Thus, the proposed product estmator and proposed exponental product estmator should be employed for rare and clustered populaton when there s negatve correlaton between the study varable and the auxlary varable. The product and exponental product estmators may be studed for the populaton varance for negatvely correlated study and auxlary varables n ACS. Moreover, some logarthmc form of estmators may also be derved as a future research n ACS. 5. ACKNOWLEDGEMENTS The authors are ndebted to Yves G. Berger, Unversty of Southampton, UK for the programmng gudance to produce populatons, smulatons and precous suggestons about the upgradng of ths paper. 6. REFERENCES. Thompson, S. K. Samplng. Wley, New York (99).. Thompson, S.K. Adaptve cluster samplng. Journal of Amercan Statstcal Assocaton 85: (990). 3. Hansen, M.M. & W.N. Hurwtz. On the theory of samplng from fnte populaton. Annals of Mathematcal Statstcs 4: (943). 4. Horvtz, D.G. & D.J. Thompson. A generalzaton of samplng wthout replacement from a fnte unverse. Journal of Amercan Statstcal Assocaton 47: (95). 5. Dryver, A.L. Performance of adaptve cluster samplng estmators n a multvarate settng. Envronmental and Ecologcal Statstcs 0: 07 3 (003). 6. Dryver, A.L. & C.T. Chao. Rato estmators n adaptve cluster samplng. Envronmetrcs 8: (007). 7. Cochran, W.G. The estmaton of the yelds of cereal experments by samplng for the rato of gran to total produce. Journal of Agrcultural Scences 30: 6-75 (940). 8. Robson, D.S. Applcaton of multvarate polykays to the theory of unbased rato type estmators. Journal of Amercan Statstcal Assocaton 5: 5-5 (957). 9. Bahl, S. & R.K. Tuteja. Rato and product type exponental estmator. Informaton and Optmzaton Scences : (99). 0. Smth, D.R., M.J. Conroy, & D.H. Brakhage. Effcency of adaptve cluster samplng for estmatng densty of wnterng waterfowl. Bometrcs 5: (995).. Lohr, S.L. Samplng Desgn and Analyss. Pacfc Grove, Duxbury Press, CA (999).