Development of Efficient Estimation Technique for Population Mean in Two Phase Sampling Using Fuzzy Tools

Transcription

1 JAMSI, 13 (017), No. 5 Develomen of Efficien Esimaion Technique for Poulaion Mean in Two Phase Samling Using Fuzzy Tools P. PARICHHA, K. BASU, A. BANDYOPADHYAY AND P. MUKHOPADHYAY Absrac The resen invesigaion deals wih he roblem of esimaion of oulaion mean in wo-hase (double) samling. Uilizing informaion on wo auxiliary variables, one chain exonenial raio and regression ye esimaor has been roosed and is roeries are sudied under wo differen srucures of wohase samling. To make he esimaor racicable, unbiased version of he roosed sraegy has also been develoed. The dominance of he suggesed esimaor over some conemorary esimaors of oulaion mean has been esablished hrough numerical illusraions carried over he daa se of some naural oulaion and arificially generaed oulaion. Caegorizaion of he dominance ranges of he roosed esimaion sraegies are deloyed hrough defuzzificaion ools, which are followed by suiable recommendaions. Mahemaics Subjec Classificaion 000: 6D05 Keywords: Double samling, sudy variable, auxiliary variable, chain-ye, regression, bias, fuzzy alicaions, variance, efficiency 1. INTRODUCTION Informaion on variables correlaed wih he sudy variable is oularly known as auxiliary informaion. The use of sulemenary informaion on auxiliary variable for esimaing he finie oulaion mean of he variable under sudy has layed an eminen role in samling heory and racices. Auxiliary informaion may be ruhfully uilized a lanning, design and esimaion sages o develo imroved esimaion rocedures in samle surveys. Use of auxiliary informaion a esimaion sage was inroduced during he 1940 s wih a comrehensive heory rovided by Cochran. Someimes, informaion on auxiliary variable may be readily available for all he unis of oulaion; for examle, onnage (or sea caaciy) of each vehicle or shi is known in survey samling of ransoraion and number of beds available in differen hosials may be known well in advance in healh care surveys. If such /jamsi Universiy of SS. Cyril and Mehodius in Trnava

2 6 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay informaion lacks, i is someimes, relaively chea o ake a large reliminary samle where auxiliary variable alone is measured, such racice is alicable in wo-hase (or double) samling. Two-hase samling haens o be a owerful and cos effecive (economical) echnique for obaining he reliable esimae in firshase (reliminary) samle for he unknown arameers of he auxiliary variables. For examle, Sukhame (196) menioned ha in a survey o esimae he roducion of lime cro based on orchards as samling unis, a comaraively larger samle is drawn o deermine he acreage under he cro while he yield rae is deermined from a sub samle of he orchards seleced for deermining acreage. In order o consruc an efficien esimaor of he oulaion mean of he auxiliary variable in firs-hase (reliminary) samle, Chand (1975) inroduced a echnique of chaining anoher auxiliary variable wih he firs auxiliary variable by using he raio esimaor in he firs hase samle. The esimaor is known as chainye raio esimaor. This work was furher exended by Kiregyera (1980, 1984), Sahoo e al. (1994), Tracy e al. (1996), Singh and Esejo (007), Gua and Shabbir (007), Shukla e al. (01), Choudhury and Singh (01) and among ohers where hey roosed various chain-ye raio and regression esimaors. I may be noed ha he mos of hese esimaion rocedures of oulaion mean in wo-hase samling are biased which become a serious drawback for heir racical alicaions. Aar from he bias esimaes, i may also be observed ha dominance ranges of he recen develoed ones over he convenional ones are no clearly menioned. Simulaion sudy carried over he daa se of naural and arificially generaed oulaion have been uilized o obain he rend of efficacy of hese recen develoed sraegies. The dominance condiions of he newly develoed esimaors are very essenial for heir recommendaions o he real life roblems. Moivaed wih his argumen, only Chaerjee e al. (015) has given a new direcion o find he dominance range of he esimaion sraegy hrough fuzzy ools in successive samling and i may be noed ha no aem has been made ye o find he dominance ranges of he esimaors in wo hase samling scheme for he esimaion of oulaion mean in samle surveys.

3 JAMSI, 13 (017), No. 7 Encouraged and fascinaed wih he work discussed earlier, we have roosed chain exonenial raio and regression ye esimaor of oulaion mean and sudied is roeries under wo differen srucures of wo-hase samling. Considering he realisic siuaions, we have develoed unbiased version of he roosed esimaors. Performances of he roosed esimaor have been examined hrough heoreical and numerical illusraions, which resens he effeciveness of he roosed sraegies. To caegorize he dominance ranges of he roosed esimaion sraegic, fuzzificaion and defuzzificaion rule are emloyed. Recommendaions of he roosed esimaion sraegy have been u forward o he survey saisicians.. FORMULATION OF THE CLASS OF ESTIMATOR.1. Samle Srucure and Some Exising Esimaion Procedures Le y, k x and k z be he values of he sudy variable y, firs auxiliary variable k x and second auxiliary variable z resecively associaed wih he k h uni of he finie oulaion U = (U 1, U, U 3,..., U N). We wish o esimae he oulaion mean Y of he sudy variable y in resence of auxiliary variables x and z, when he oulaion mean X of x is unknown bu informaion on z is readily available for all he unis of oulaion. Thus, o esimae Y, a firs hase samle S S U of size n is drawn by simle random samling wihou relacemen scheme (SRSWOR) from he enire oulaion U and observed for he auxiliary variables x and z o furnish he esimae of X. Again a second-hase samle S of size m m hase samle SRSWOR scheme o observe he sudy variable y. Hence onwards, we use he following noaions: Z : Poulaion mean of he auxiliary variable z. n is drawn from he firs x, z: Samle means of he resecive variables based on he firs hase samle of size n.

4 8 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay x, y, z : Samle means of he resecive variables based on he second hase samle of size m. b yx (n), b (n): Samle regression coefficiens beween he variables shown in subscris and based on he samle sizes indicaed in he braces. s yx (n), s (n), s yz (n): Samle covariance beween he variables shown in subscris s x and based on he second hase samle of size n. : Samle mean square of he variable x based on he second hase samle of size n. β yx, β, β yz: Poulaion regression coefficiens beween he variables shown in subscris. To esimae he oulaion mean Y, he classical raio esimaor is resened as y y r = X. x where y and x are he samle means of variables y and x resecively based on he second hase samle S. If X is unknown, we esimae Y under wo-hase samling se u as (1) y 1= x x () where x is he samle mean of he auxiliary variable x based on he firs-hase (reliminary) samle S. Srivasava (1970) generalized he raio mehod of esimaion and is srucure in wohase samling is given by α x = y, x where α is a real scalar, which can be suiably deermined by minimizing he mean square error (M. S. E.) of he esimaor. The way in which he esimae of Y is imroved using he auxiliary informaion on x can also be exended o imrove he esimae of X in he firs-hase samle, if (3)

5 JAMSI, 13 (017), No. 9 anoher auxiliary variable z closely relaed o x bu remoely relaed o y is used. Thus, assuming ha he oulaion mean of he auxiliary variable z is known, Chand (1975) roosed a chain-ye raio esimaor as y = c xrd x (4) x where x rd = Z, z and Z are he samle mean based on he firs hase samle of z size n and oulaion mean of he auxiliary variable z resecively. Singh and Esejo (007) considered a raio - roduc ye esimaor in double samling as x 3 = y + 1- x x x where is a real scalar which may be suiably deermined o minimize he mean square error of he esimaor 4. Singh and Vishwakarma (007) consruced exonenial raio and roduc ye esimaor of oulaion mean in wo hase samling as (5) and resecively. x- x 4= y ex x + x x - x 5 = y ex x + x (6) (7).. Proosed Class of Esimaor Moivaed wih he earlier work, we have defined a class of chain exonenial raio and regression ye esimaors as x x ld - x = y 1-k + k ex (8) x x ld + x where k is a real consan which can be suiably deermined by minimizing he M. S. E. of he class of esimaors and id x = x + b (n) Z - z.

6 10 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay 3. FORMULATION OF THE CLASS OF ESTIMATOR BIAS AND MEAN SQUARE ERRORS OF THE PROPOSED CLASS OF ESTIMATOR I can be easily noed ha he roosed class of esimaors defined in equaions (9) is chain exonenial raio and regression ye esimaor. Therefore, i is biased for oulaion mean Y. Hence, we roceed o obain heir biases and mean square errors under large samle aroximaions using he following ransformaions: y =Y(1+e ), x =X(1+e ), x = X(1+ e ), z =Z(1+ e ), s =S (1 + e ), s = S (1+ e ) 5 z z 6 where E(e i) = 0 for ( i = 1,,..., 6), e i for ( i = 1,,..., 6) are relaive error erm. Under above ransformaions he class of esimaors can be reresened as Z (e3-e )- β (e 4 +e4e5-e4e 6) 1 k X = Y(1 + e 1) (1- k) 1+e 3 1 e (9) + ex -1 e +e3 Z 1+ - β (e 4 +e4e5-e4e 6) X We have derived he exressions for bias and mean square error of he roosed class of esimaors and resened hem below. We have he following execaions of he samle saisics under wo - hase samling se u as

7 JAMSI, 13 (017), No y 1 x 3 x 4 z E(e )= f C, E(e ) = f C, E(e ) = f C, E(e ) = f C E(e e )= f ρ C C, E(e e ) = f ρ C C, 1 1 yx y x 1 3 yx y x 3 x x z E(e e )= f C, E(e e ) = E(e e )= f ρ C C, μ μ E(e e ) = f, E(e e ) = f, ZS ZSz μ01 μ10 E(ee5) = f, E(ee 6) = f, XS XS E(e e )= f ρ C C. 1 4 yz y z z (10) where f 1= -, f = -, f 3= -, n N m N n m N 1 q r qr i i i N i=1 μ = (x - X) (y - Y) (z - Z) ; (, q, r 0) Exanding binomially, exonenially, using resuls from equaion (9) and reaining he erms u o firs order of samle size, we have derived he exressions of bias B(.) and mean square error M(.) of he class of esimaor as S μ10 μ S 003 yz S S x yx B( )= E-Y = Y kf - - +f XSz XSz XSz YX X YX 1 y M =E -Y =Y f C + (k /4)a+kb +c (1) where a = f + f ρ C and b = -f ρ ρ C C + f ρ C C f C, c = f C - f ρ C C 3 x yz y x 3 yx y x 3 x 3 x 3 yx y x (11) 4. BIAS REDUCTION FOR THE PROPOSED CLASS OF ESTIMATOR In some siuaions of racical imorance, bias becomes a serious drawback. Therefore, unbiased versions of he roosed classes of esimaors are more desirable. Moivaed wih his argumen and influenced by he bias correcion echniques of Tracy e al. (1996) and Bandyoadhyay and Singh (014) we roceed o derive he unbiased version of our roosed class of esimaor.

8 1 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay From equaion (1), we observe ha he exression of bias of he esimaor conains he oulaion arameers such as μ 003, μ, 10 S, yx yz x y and S. Since z S is known while z μ 003, μ, 10 S, yx yz x y unknown, relacing μ 003, μ, 10 S, yx yz x y S, S, S, Y, X and S, S, S, Y, X, S, S, S, Y, X and S yz Syz are S yz by heir resecive samle esimaor (based on he second hase samle of size m) m 003, m10, s yz, s x, s y, y, x and yz s, we ge an esimaor of B as s m10 m s 003 yz s s x yx b( )= y k f f x sz xsz xsz yx x yx (13) m q r. i m i m i m m i = 1 where m = x - x y - y z - z qr 1 Now moivaing wih he bias reducion echniques of Tracy e al. (1996) and Bandyoadhyay and Singh (014), we have derived he unbiased version of he roosed class of esimaor o he firs order of aroximaions as which becomes ' = - b( ) x x - x s m m s s s (14) ld yz x yx = y 1-k + k ex y k f f x x ld + x x sz xsz xsz yx x yx Thus, he variance of o he firs order of aroximaion are obained as 1 y V =M =Y f C + (k /4)a+kb +c (15) Thus, from equaions and (1) and (15) i is o be noed ha he class of esimaors is referable over he class of esimaor as is unbiased (u o firs order of samle size) class of esimaors of Y while he class of esimaor is biased.

9 JAMSI, 13 (017), No MINIMUM VARIANCE OF PROPOSED CLASS OF ESTIMATOR I is obvious from he equaion (15) ha he variance of he roosed class of esimaor deends on he value of he consan. Therefore, we desire o minimize heir variances and discussed hem below. The oimaliy condiion under which roosed class of esimaors have minimum variance is obained as where k k = -b/a (16) Subsiuing he oimum value of he consan k in equaion (15), we have he minimum variance of he class of esimaor as b Min. V( ) = Y f1c y - + C a (17) REMARK 5.1: I is o menioned ha he oimum value of k deends on unknown oulaion arameers such as C x, C y, C z, ρ yx and ρ. Thus, o make he class of esimaors racicable, hese unknown oulaion arameers may be esimaed wih heir resecive samle esimaes or from as daa or guessed from exerience gahered over ime. Such roblems are also considered by Reddy (1978), Tracy e al. (1996) and Singh e al. (007). 6. EFFICIENCY COMPARISONS OF THE PROPOSED CLASS OF ESTIMATOR To examine he erformances of he roosed class of esimaors under wo differen cases of wo - hase samling se u as suggesed in his aer, we have comared heir efficiencies wih some exising esimaors of oulaion mean such as y (samle mean esimaor) and i i = 1,,..., 5. The Variance/ M. S. E.s/ minimum M. S. E.s of he exising esimaors i are obained u o he firs order of aroximaions under he Cases of he wo hase - samling se u and resened below.

10 14 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay M 1 = Y f1c y+fcx -f ρyxcyc x Min. M = S f - f ρ y 1 3 yx Min. M = f 1- ρ + f ρ S 3 1 yx yx y f3 M 4 = Y f1c y + C x - f3ρyxcycx 4 f3 M 5 = Y f1c y + C x + f3ρyxcycx 4 We have demonsraed he suerioriy of he suggesed esimaor over he esimaor i (i = 1,,..., 5) hrough numerical illusraion and grahical inerreaions Emirical Invesigaions hrough Naural Poulaion We have chosen four naural oulaions o illusrae he efficacious erformance of our roosed classes of esimaors. The source of he oulaions, he naure of he variables y, x, z and he values of he various arameers are as follows. Poulaion I - Source: Cochran (1977) y: Number of lacebo children. N = 34, n = 15, m =10, Y = 4.9, C = 1.013, C = 1.318, C = 1.070, ρ = 36, ρ = 430 and ρ = 837. yx yz y x z Poulaion II - Source: Shukla (1966) y: Measuremen of weigh of children. N = 50, n = 15, m = 8, Y =.584, C = 0.943, C = , C = , ρ = 8, ρ = 0.37 and ρ = 3. z yx yz y x Poulaion III - Source: Handique e al. (011) y: Fores imber volume in cubic meer (Cum) in 0.1 ha samle lo. N = 500, n = 700, m = 80, Y = 4.63, C = 5, C = 8, C = 4, ρ = 9, ρ = and ρ = 6. yx yz y x z Poulaion IV - Source: Sukhame and Sukhame (1970) y: Area (acres) under whea in 1937.

11 JAMSI, 13 (017), No. 15 N = 34, n = 10, m = 7, Y = 01.41, C = 4, C = 6, C = 1, ρ = 3, ρ = and ρ = 3. yx yz y x z To have a angible idea abou he erformance of he roosed class of esimaor we have comued ercen relaive efficiencies (PREs) of hem and he exising esimaors i (i = 1,,..., 5) under similar realisic siuaions and he findings are V y dislayed in Table 1 where PREs are designaed as PRE = 100 and V T denoe variance/minimum M. S. E. of an esimaor T. Table 1: PRE of various esimaors Esimaor Poulaion -I Poulaion -II Poulaion -III Poulaion -IV MT 6.. Emirical Invesigaions hrough Arificially Generaed Poulaion An imoran asec of simulaion is ha one builds a simulaion model o relicae he acual sysem. Simulaion allows comarison of analyical echniques and hels in concluding wheher a newly develoed echnique is beer han he exising ones. Moivaed by Singh and Deo (003) and Singh e al.(001) who have been adoed he arificial oulaion generaion echniques, we have generaed five ses of indeenden random samles of size N (N = 100) namely x, y, x, y and z (k =1,, 3,..., N) from a sandard normal disribuion 1k 1k k k k wih he hel of R-sofware. By varying he correlaion coefficiens ρ yx and ρ, we

12 16 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay have generaed he following ransformed variables of he oulaion U wih he values of σ y = 50, μ y= 40, σ x = 5, μ x = 50, σ = 9 andμ = 30 as z y 1 =μ k y+σ y ρxyx ρ k yx y 1, x k 1 =μ k x +σxx 1 and z k k =μ z+σ z ρx ρ k z k. Thus, we have derived following efficiency comarisons of our roosed sraegy wih he recen relevan ones wih he above arificially generaed oulaion echniques as: Table : PRE of various esimaors z Esimaors PRE ρ yx =, ρ ρ yx =, ρ ρ yx =, ρ ANALYSIS OF EMPIRICAL STUDY THROUGH FUZZY TOOLS From he above emirical sudy, i is o be noed ha he dominance of he roosed sraegy over he exising ones have been esablished hrough daa se of real life oulaion and arificially generaed oulaion. However, he dominance condiions of he roosed sraegy over he exising ones have no ye been obained clearly. Therefore, we desire o derive he condiions where roosed esimaors erforms exremely well or dominae mildly he samle mean esimaor y. This invesigaion hels us in choosing he suiable oulaion where our

13 JAMSI, 13 (017), No. 17 roosed work may be alied effecively which is very essenial for he recommendaions of our roosed work for heir racical alicaion. Moivaed wih his argumen, we roceed o build u a decision making machinery hrough fuzzy ools which will enable us o measure he degree of efficiency of he esimaor for differen choices of correlaions ρ yx, ρ and ρ yz. Thus, we have comued he PRE of he roosed class esimaors (under is resecive oimum condiion as discussed in secion 5) wih resec o he samle mean esimaor y and resened hem in Table 3. REMARK: 7.1. Here we have considered he sabiliy naure of he coefficien of variaions of he sudy variable and auxiliary variables (Reddy 1978) and hus we have aken he coefficien of variaions of y, x and z o be aroximaely equal.

14 18 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay Table 3: PRE of he roosed class of esimaors ρ = 0. (fix.) yx ρ ρ yz PRE ρ = 0.3 (fix.) yx ρ ρ yz PRE ρ = (fix.) yx ρ ρ yz PRE ρ = (fix.) yx ρ ρ yz PRE

15 JAMSI, 13 (017), No. 19 Table 3 coninued... ρ = (fix.) yx ρ ρ yz PRE ρ = (fix.) yx ρ ρ yz PRE ρ = (fix.) yx ρ ρ yz PRE ρ = (fix.) ρ ρ yx yz PRE Develomen of he Fuzzy Logic Conroller (FLC) is accomlished by sudying he emirical daa furnished in Table 3 where he FLC roduces he degree of efficiency for a given range of. xy, and. They are aken as o be he hree inu fuzzy yz variables having 8, 6 and 7 linguisics resecively (lised in Tables 4.a, 4.b & 4.c).

16 0 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay Enire range of [0. xy ] is divided ino 8 equal ars and ha for xy [ ] ino 6 equal ars and for [0.3 yz ] ino 7 equal ars. To yz each ar a linguisic is assigned suiably for all hree cases. Also he PRE is aken as he ouu fuzzy variable having he se of 16 linguisics (lised in Table 5) in he descending degree of efficiency. The range [101 PRE 76], as i is furnished in Table 5, is divided ino 16 equal ars as shown in he able. Table 4.a Table 4.b Ling Range A (mid.) Table 4.c C (half widh) Ling Range Ling Range A(mid.) C(half widh) vwc wc c c c c vsc A (mid..) C (half widh) vwc Mc wc c c c c Sc c Hc c Vhc sc hsc

17 JAMSI, 13 (017), No. 1 Table 5 Ling Range A(mid.) C(half widh) T T T T T T T T T T T T T T T T The Mamdani Inference Model is adoed here as i is he mos commonly used fuzzy mehodology and was one among he firs few conrol sysems buil using fuzzy se heory. Ebrahim Mamdani (1975) roosed and used i o conrol a seam engine and boiler combinaion by synhesizing a se of linguisic conrol rules obained from exerienced human oeraors. Mamdani s effor originaed from Lofi Zadeh s aer on fuzzy algorihms for comlex sysems and decision rocesses (1973).

18 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay The following sandard oeraor se is used in his model: Oeraors Tye Defaul Funcion AND BINARY MIN(a,b) OR BINARY MAX(a,b) IMPLICATION BINARY MIN(a,b) ALSO BINARY MAX(a,b) NOT UNARY 1-a STRONGLY UNARY a^ MODERATELY UNARY a^(1/) SLIGHTLY UNARY 4.a.(1-a) DEFUZZIFICATION DEFUZZIFICATION Cenre Of Area A hree-arameer (a, b, c) bell shaed coninuous membershi grade funcion is chosen for each linguisic of boh inu and ouu variables (This is a direc generalizaion of Cauchy Disribuion) so ha membershi funcions can be fine grained according o he necessiy. The arameers a, b, c denoe resecively he middle oin of bell shaed curve (where he grade is max), he degree of eakedness (resembling he Kurosis in Normal disribuion) which aken o be 1 and half widh of he membershi funcion. c is ke consan=0.05 for, xy and. C akes values 5 for PRE and b is ke consan=1 hroughou. yz The funcion is defined by- f ( x; a, b, c) 1 1 x a c The se of values for he arameers are comued from he se of daa generaed in Table 3. b

19 JAMSI, 13 (017), No. 3 Table 6.a Sl. No. Lef end Of xy Righ end Of xy Mid Poin (a) Linguisics Inerreaion Half Widh Vwc Very weekly correlaed osiive Wc weekly osiively correlaed c1 moderaely correlaed c moderaely correlaed c3 moderaely correlaed c4 moderaely correlaed Sc srongly correlaed Hsc Highly srongly correlaed 0.05 Table 6.b Sl. No. Lef end Of Righ end Of Mid Poin (a) Linguisics Inerreaion Half Widh Mc mildly correlaed c1 moderaely correlaed c moderaely correlaed Sc srongly correlaed Hc highly correlaed Vhc Very highly correlaed 0.05 A 8x6x7 Fuzzy Associaion Marix (FAM) is consruced which is he basis of FLC engine and he Cenre of Area mehod (which resembles he execed value comuaion in robabiliy disribuion) is adoed for defuzzificaion which is mos widely used mehod and defined by- COA z = zμ(z)dz μ(z)dz A A

20 4 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay All comuaions are done wih he hel of sandard fuzzy sofware named XFuzzyVs3.0from IMSE-CNM which is available on inerne 7.1. Caegorizaion of Efficacy of Proosed Work The above analysis of emirical sudy using fuzzy ools gives he advanage o find ou he secific ranges of, xy and where our suggesed esimaor yz dominaes i. exremely ii. iii. mildly and equally he samle mean esimaor y. To elucidae hese aricular regions of, xy &, he same grah of PRE in differen views are resened below: yz Fig. 1. Horizonal Fron View ρ = 5 Fig.. Horizonal Rear View ρ = 5

21 JAMSI, 13 (017), No. 5 Fig. 3. Horizonal Fron Rear View ρ = Fig. 4. Horizonal Fron View ρ = Surface lo of PRE of he roosed class of esimaors (under is resecive oimum condiion as discussed in secion 5) agains ρyx, ρ yz, ρ from differen angles. (I may be noed from he Figs. 1-4 ha RYX = ρ yx, RXZ = ρ and RYZ = ρ yz.) 8. CONCLUSION The following inerreaions can be read ou from he resen sudy: The following conclusions may be read-ou from he resen sudy. (a) Table 1 exhibis ha for high osiive values of he correlaion coefficiens, he roosed class of esimaors yield imressive gains in efficiency over he exising esimaors i(i=1,,..., 5). This aern indicaes ha roosed class of esimaors are more efficien han he exising ones which enhances heir recommendaions o survey saisician for heir usage in real life roblem. (b) From able, i is observed ha for fixed values of ρ, he ercen relaive efficiencies of he class of esimaors are increasing wih he increasing values of ρ yx. This behavior indicaes ha our roosed classes of esimaors erforms saisfacorily if highly correlaed auxiliary variable resen in oulaion.

22 6 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay (c) From he above grahical reresenaions of figures 1-4, i may be seen ha PRE agains ρ yx and ρ gives a clear idea abou he efficiency of our roosed class yz of esimaor over he samle mean esimaor y under is resecive oimal condiion. I iself describes he secific ranges of ρ yx and ρ a where our yz esimaor dominaes (exremely, mildly or equally) y. From he differen views of he grah (aken from o and differen sides), i is clear ha he orion of he grah which is almos horizonal denoes ha he roosed class of esimaors is equally efficien wih y. Whereas he urising orions denoe he mildly efficien range and he eaks of he grah along wih heir neighbourhoods denoe he exremely efficien range of he class of esimaors. The following conclusions may be drawn abou he erformance of wih resec o he samle mean esimaor y. i. Fig. 1 and Fig. indicaes ha he class of esimaors is mildly efficien han y when 0.15 ρyx 7, 0.39 ρyz 7 and ρ = 5. ii. I is cleared from Fig. 1, Fig. ha he class of esimaors is moderaely efficien y when 7 ρyx 9, 7 ρyz 1 and ρ = 5. iii. I can be observed from Fig.1,Fig. ha he class of esimaors is exremely efficien when 9 ρyx 5, 1 ρyz 5 and ρ = 5. iv. Also from Fig. 3 and Fig. 4 he class of esimaors is moderaely efficien y when 0.31 ρyx 7, 0.5 ρyz 7 and ρ =. v. I can be observed from Fig. 3, Fig. 4 ha he class of esimaors is exremely efficien when 7 ρyx 5, 7 ρyz 5 and ρ =.

23 JAMSI, 13 (017), No. 7 Thus i is ereced ha he use of an auxiliary characer is highly rewarding in erms of he roosed class of esimaors. Moreover, he roosal of he class of esimaors in he resen sudy is jusified as i unifies several desirable resuls including roducing unbiased esimaes (u o firs order of samle size) and finding he dominance range of he roosed sraegy. Looking a he nice behaviour of he roosed sraegy, hey are recommended o he survey saisicians for heir alicaions in real life roblems. REFERENCES [1] ANDERSON, T. W., 1958, An Inroducion o Mulivariae Saisical Analysis. John Wiley & Sons, Inc. New York. [] BANDYOPADHYAY, A. AND SINGH, G. N., 014, Predicive esimaion of oulaion mean in wo-hase samling. Communicaions in Saisics- Theory and Mehods. DOI: / , (Aricle in Press). [3] CHAND, L., 1975, Some raio ye esimaors based on wo or more auxiliary variables.unublished Ph. D. hesis. Iowa Sae Universiy, Ames, Iowa (USA). [4] CHOUDHURY, S. AND SINGH, B. K., 01, A class of chain raio roduc ye esimaors wih wo auxiliary variables under double samling scheme. Journal of he Korean Saisical Sociey, 41, [5] CHATTERJEE, A., SINGH, G. N., BANDYOPADHYAY, A. AND MUKHOPADHYAY, P. 015, A General Procedure for Esimaing Poulaion Variance in Successive Samling using Fuzzy Tools. Haceee Journal of Mahemaics and Saisics. ISSN: (S. C. I. E. indexed ) (Aricle in Press) DOI: /HJMS [6] COCHRAN, W. G., 1940, The esimaion of he yields of cereal exerimens by samling for he raio gain o oal roduce. J. Agric. Soc. 30, [7] COCHRAN, W. G., 1977, Samling Techniques. Wiley Easern Limied, New Delhi, III, Ediion. [8] GUPTA, S. AND SHABBIR, J., 007, On he use of ransformed auxiliary variables in esimaing oulaion mean by using wo auxiliary variables. Journal of Saisical Planning and Inference, 137, [9] HANDIQUE, B. K., DAS, G., AND KALITA, M. C., 011, Comaraive sudies on fores samling echniques wih saellie remoe sensing inus. Projec Reor, NESAC, [10] KIREGYERA, B., 1980, A chain raio ye esimaors in finie oulaion double samling using wo auxiliary variables. Merika. 17, [11] KIREGYERA, B., 1984, Regression ye esimaors using wo auxiliary variables and he model of double samling from finie oulaions. Merika. 31, [1] REDDY, V. N., 1978, A sudy on he use of rior knowledge on cerain oulaion arameers in esimaion. Sankhya, Series C. 40, [13] SHUKLA, D., PATHAK, SAND THAKUR, N. S., 01, Esimaion of oulaion mean using wo auxiliary sources in samle surveys. Saisics in Transiion, 13(1), [14] SINGH, S., JOARDER, A. H. AND TRACY, D. S., 001, Median esimaion using double samling. Ausralian & New Zealand Journal of Saisics, 43(1),

24 8 P. Parichha, K. Basu, A. Bandyoadhyay and P. Mukhoadhyay [15] SINGH, S. AND DEO, B., 003, Imuaion by ower ransformaion. Saisical Paers, 4, [16] SINGH, H. P. AND ESPEJO, M. R., 007, Double samling raio-roduc esimaor of a finie oulaion mean in samling surveys. Journal of Alied Saisics, 34(1), [17] SINGH, H. P. AND VISHWAKARMA, G. K., 007, Modified Exonenial Raio and Produc Esimaors for Finie Poulaion Mean in Double Samling. Ausrian journal of saisics, 36 (3), 17 5 [18] SINGH, H. P., CHANDRA, P., JOARDER, A. H. AND SINGH, S., 007, Family of esimaors of mean, raio and roduc of a finie oulaion using random nonresonse. Tes, 16, [19] SRIVASTAVA, S. K., 1970, A wo-hase samling esimaor in samle surveys. Ausralian Journal of Saisics. 1(1), 3-7. [0] SUKHATME, B., 196, Some raio-ye esimaors in wo - hase samling. Journals of he American Saisics Associaions. 57, [1] SUKHATME, P. V. AND SUKHATME, B. V., 1970, Samling heory of surveys wih alicaions. Asia Publishing House, India. [] TRACY, D. S., SINGH, H. P., AND SINGH, R., 1996, An alernaive o he raio-cum-roduc esimaor in samle surveys. Journal of Saisical Planning and Inference. 53, P. Parichha Dearmen of Mahemaics, Asansol Engineering College, Asansol , India. arhaarichha1989@gmail.com K. Basu Dearmen of Mahemaics, Naional Insiue of Technology, Durgaur 71309, India kajla.basu@gmail.com A. Bandyoadhyay Dearmen of Mahemaics, Asansol Engineering College, Asansol , India. arnabbandyoadhyay4@gmail.com P. Mukhoadhyay Dearmen of Mahemaics, Asansol Engineering College, Asansol , India. arhamukhoadhyay1967@gmail.com