Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA Olie ISSN: 2249-4626 & Prit ISSN: 0975-5896 Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig By Waikhom Warsee Chau & B. K. Sigh North Easter Regioal Istitute of Sciece ad Techology, Idia Abstract- I the preset study, we have proposed a class of ratio-cum-product estimators for estimatig fiite populatio mea of the study variable i two phase samplig. The bias ad mea square error of the proposed estimator have bee obtaied. The asymptotically optimum estimator (AOE i this class has also bee idetified alog with its approximate bias ad mea square error. Compariso of the proposed class of estimators with other estimators is also worked out theoretically to demostrate the superiority of the proposed estimator over the other estimators. Keywords: ratio-cum-product, mea, two phase samplig, asymptotically optimum estimator, bias, mea square error. GJSFR-F Classificatio : 62D05 ImprovedClassofRatioCumProductEstimatorsofFiitePopulatioMeaitwoPhaseSamplig Strictly as per the compliace ad regulatios of : 204. Waikhom Warsee Chau & B. K. Sigh. 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.
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Waikhom Warsee Chau & B. K. Sigh Abstract- I the preset study, we have proposed a class of ratio-cum-product estimators for estimatig fiite populatio mea Ȳ of the study variable y i two phase samplig. The bias ad mea square error of the proposed estimator have bee obtaied. The asymtotically optimum estimator (AOE i this class has also bee idetified alog with its approximate bias ad mea square error. Compariso of the proposed class of estimators with other estimators is also worked out theoretically to demostrate the superiority of the proposed estimator over the other estimators. Keywords: ratio-cum-product, mea, two phase samplig, asymptotically optimum estimator, bias, mea square error. I. Itroductio The literature o survey samplig describes a great variety of techiques for usig auxiliary iformatio i order to obtai improved estimators for estimatig some most commo populatio parameter such as populatio total, populatio mea, populatio proportio, populatio ratio etc. More ofte we are iterested i the estimatio of the mea of a certai characteristic of a fiite populatio o the basis of a sample take from the populatio followig a specified samplig procedure. Use of auxiliary iformatio has show its sigificace i improvig the efficiecy of estimators of ukow populatio parameters. Cochra (940 used auxiliary iformatio i the form of populatio mea of auxiliary variate at estimatio stage for the estimatio of populatio parameters whe study ad auxiliary variate are positively correlated. I case of egative correlatio betwee study variate ad auxiliary variate, Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 69 Robso (957 defied product estimator for the estimatio of populatio mea which was revisited by Murthy (967. Ratio estimator performs better tha simple mea estimator i case of positive correlatio betwee study variate ad auxiliary variate. Author : North Easter Regioal Istitute of Sciece ad Techology. e-mail: warc203@gmail.com 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig For further discussio o ratio cum product estimator, the reader is referred to Sigh (967, Shah ad Shah (978, Sigh ad Tailor (2005, Tailor ad Sharma (2009, Tailor ad Sharma (2009, Sharma ad Tailor (200, Choudhury ad Sigh (20 etc. Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 70 Whe the populatio mea X of the auxiliary variable x is ukow before start of the survey, it is estimated from a prelimiary large sample o which oly the auxiliary characteristic x is observed. The value of X i the estimator is the replaced by its estimate. After the a smaller secod-phase sample of the variate of iterest (study variate y is the take. samplig. This techique is kow as double samplig or two-phase Neyma (938 was the first to give the cocept of double samplig i coectio with collectig iformatio o the strata sizes i a stratified samplig Cosider a fiite populatio U =(u,u 2,u 3,..., u N of size N uits, y ad x are the study ad auxiliary variate respectively. Whe the populatio mea X of x is ot kow, a first phase sample of size is draw from the populatio o which oly the x characteristic is measured i order to furish a good estimate of X. After the a secod-phase sample of size ( < is draw o which both the variates y ad x are measured. The usual ratio ad product estimators i double samplig are: ad Y R d =ȳ x x Y P d =ȳ z z where x,ȳ ad z are the sample mea of x, y ad z respectively based o the sample of size out of the populatio N uits ad ȳ = i= y i, x = i= x i ad z = i= z i deote the sample mea of x ad z based o the first- phase sample of the size. Sigh (967 improved the ratio ad the product methods of estimatio by studyig the ratio cum product estimator for estimatig Ȳ as 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig ( X Ȳ =ȳ x z Z Motivated by Sigh (967, Choudhury ad Sigh (20 proposed a modified class of ratio cum product type of estimator for estimatig populatio mea Ȳ as I. II. Ȳ (α =ȳ ( X x α z Z Motivated by? ad as a extesio to the work of Choudhury ad Sigh (20, we have developed a improved class of ratio-cum-product estimators i double samplig to estimate the populatio mea proposed estimator. Ȳ theoretically ad studied the properties of the The Proposed Estimator The proposed improved class of ratio-cum-product estimators of populatio mea Ȳ i two-phase samplig is give as where α is a suitably chose costat. To obtai the bias ad MSE of Ȳ w(d ( Ȳ w(d x =ȳ x. z z α. ( to the first degree of approximatio, we write e 0 = ( ȳ Ȳ /Ȳ, e = ( x Ȳ / X, e 2 = ( x X / X, e 3 = ( z Z / Z, e 4 = ( z Z / Z. Expressig Y w(d i terms of e s ad eglectig higher power of e s, we have Y w(d = Ȳ ( + e 0 { ( + e 2 (+e ( + e 3 (+e 4 } α. Assumig the sample size to be large eough so that e <, e 4 < ad expadig ( + e,(+e 4 i powers of e, e 4, multiplyig out ad eglectig higher powers of e s,wehave Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 7 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Y w(d =Ȳ ( + e 0 [ { e e 2 e 3 + e 4 e 2 e 2 4 + e e 2 e e 4 Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 72 Ȳ w(d Ȳ =Ȳ +e 2 e 4 + e e 3 e 2 e 3 + e 3 e 4 }] α [ e 0 α e 0 e 2 + e 0 e 4 e 0 e 3 ( e e 2 e 3 + e 4 e2 2 + e2 2 2 + e2 3 2 e2 4 2 + e 0e + α 2 ( e 2 2 + e2 2 2 + e2 3 2 + e2 4 2 e e 2 e e 3 + e e 4 + e 2 e 3 e 2 e 4 e 3 e 4 ] The followig two cases will be cosidered separately. Case I: Whe the secod phase sample of size is subsample of the first phase of size. Case II: whe the secod phase sample of size is draw idepedetly of the first phase sample of size. I this case, we have CASE I III. Bias, mse ad Optimum Value of Ȳ i Case wd i E (e 0 =E(e =E(e 2 =E(e 3 =E(e 4 =0; E ( ( f e 2 0 = CY 2 ; E ( ( f e 2 = C 2 X; E ( ( f e 2 3 = C 2 Z; E ( ( f e2 2 = E (e e 2 = C 2 X; E ( ( f e4 2 = E (e3 e 4 = C 2 Z; ( ( f f E (e 0 e = ρ YX C Y C X ; E (e 0 e 2 = ρ YX C Y C X ; ( ( f f E (e 0 e 3 = ρ YZ C Y C Z ; E (e 0 e 4 = ρ YZ C Y C Z ; ( f E (e e 3 = ρ XZ C X C Z ; ( f E (e e 4 =E(e 2 e 3 =E(e 2 e 4 = ρ XZ C X C Z (3 (2 where f = N is the samplig fractio, f = N, C 2 Y = S2 Y Ȳ 2, C 2 Z = S2 Z Z 2. C2 X = S2 X X 2, 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Takig expectatios o both the sides ad usig the results of (3 i (2, we get the bias of Ȳ w(d as I. B w(d =Ȳ I where f =, 2 K 3 = C 2 X + C2 Z ( 2K XZ. [α 2 { C 2 X + C 2 Z ( 2K XZ } + α { C 2 X ( 2K YX (4 CZ 2 ( 2K YZ }] ( f [α =Ȳ 2 K 3 + α (K K 2 ] (5 2 Now from equatio (2, we have K YX = ρ YX C Y C X, K = C 2 X ( 2K YX, K 2 = C 2 Z ( 2K YZ, Ȳ w(d Ȳ = Ȳ [e 0 α (e e 2 e 3 + e 4 ]. Squarig both the sides ad takig expectatios i the above equatio ad usig the resultsof(3,wegetthemeasquareerrorofȳ w(d as M w(d I ( f =Ȳ 2 =Ȳ 2 ( CY 2 + Ȳ 2 f [α { 2 C 2 C 2 Y + Ȳ 2 where S = C YX C YZ, C YZ = ρ YZ C Y C Z, C YX = ρ YX C Y C X Differetiatig M w(d w.r.t α ad equatig to zero, we get α = S K 3. to the first degree of approximatio X + CZ 2 ( 2K XZ } 2α (C YX C YZ ] [α ] 2 K 3 2αS (6 (7 Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 73 Now puttig the optimum value of α from (7 i the proposed estimator (, we get the asymptotically optimum estimator(aoe as ( w(d x =ȳ I(opt x Iα(opt z. z 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Therefore, after puttig the value of α i (4 ad (6, we obtai the optimum bias ad MSE of Ȳ w(d respectively as Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 74 B w(d = Ȳ Iα(opt 2 S K 3 [K K 2] where K = C 2 X ( K YX, K 2 = C 2 Z ( K YZ. M ( ( w(d f = Ȳ 2 CY 2 Iα(opt Ȳ 2 f S 2. (8 K 3 Remark For α =, the estimator reduces to ratio cum product estimator i double samplig. The bias ad MSE of Ȳ (d ad (6 as follows where ad B MSE (d = Ȳ I (d = I are obtaied by puttig α =i relatio (4 K = C 2 X ( K YX where K 4 = C 2 Z ( 2K XZ +2K YZ. [K C XZ + C YZ ] (9 ( SY 2 + Y f 2 (K + K 4 (0 Remark 2 For α =ad whe the auxiliary variate z is ot used, i.e if z is ozero costat, the proposed estimator reduces to the usual ratio estimator i two phase samplig. The bias ad mea square error of Ȳ d R ca be obtaied by puttig α =ad omittig the terms of z i equatio (4 ad (6, respectively as B ( ( ȲR d = Ȳ f K I ad MSE ( ( ( f ȲR d = Ȳ C 2 I Y + Ȳ f K. ( 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Remark 3 For α =ad whe the auxiliary variate x is ot used, i.e if x is o-zero, the proposed estimator reduces to the usual product estimator i two phase samplig. The bias ad mea square error of Ȳ d P ca be obtaied by puttig α =ad omittig I. the terms of x i equatio (4 ad (6, respectively as B ( ( ȲP d = Ȳ f C I YZ ad where K 5 = C 2 X (+2K YX. M ( ( ȲP d = Y f 2 I IV. a with sample mea per uit estimator Efficiecy Comparisos Compairiso of the optimum proposed estimator ȳ ( CY 2 + Y f 2 K 5 (2 w(d Iα(opt The MSE of sample mea ȳ uder SRSWOR samplig scheme is give by V (ȳ = From equatio (3 ad (8, we observed if K 3 > 0 i.e K XZ < /2. V (ȳ M b with ratio estimator i double samplig From equatio ( ad (8, we observed M ( Ȳ d R S 2 Y. (3 w(d = S2 > 0. IαI(opt K 3 (4 ] ( ( w(d M [Ȳ = Ȳ 2 f K + S2 > 0. (5 Iα(opt K 3 Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 75 if K > 0, K 3 > 0 i.e. K YX < /2, K XZ < /2. 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig c with product estimator i double samplig From (2 ad (8, we observed M ( ] ( [ ] ȲP d w(d MSE [Ȳ = Ȳ 2 f K 5 + S2 > 0 (6 Iα(opt K 3 if K 3 > 0, K 5 > 0 i.e. K XZ < /2. d with ratio cum product estimators i double samplig Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 76 From (0 ad (8, we observed M ( Ȳ d ] [ ] w(d M [Ȳ = Ȳ 2 K + K 5 + S2 > 0 (7 Iα(opt K 3 if K > 0, K 3 > 0, K 5 > 0 i.e. K YX < /2, K XZ < /2, K XZ K YZ < /2. Now we state the theorem Theorem 4 To the first degree of approximatio, the proposed class of estimators Ȳ w(d Rd uder the optimality (7 is cosider to be more efficiet tha Ȳ d R, Ȳ d P, Ȳ d ad ȳ uder the give coditios K, K 3, K 4, ad K 5 >0, where K = C 2 X ( 2K YX, K 3 = C 2 X + C2 Z ( 2K XZ, K 4 = C 2 Z ( 2K XZ +2K YZ ad K 5 = C 2 X (+2K YX. V. Bias, mse ad Optimum Value of Ȳ w(d i Case ii I this case, we have E (e 0 =E(e =E(e 2 =E(e 3 =E(e 4 =0; CY 2 ; E ( ( f e 2 = E ( ( f e 2 0 = E ( e2 2 = E (e e 2 = E (e 0 e = E (e e 3 = C 2 X; E ( e 2 3 CX; 2 E ( e4 2 = E (e3 e 4 = ρ YX C Y C X ; E (e 0 e 3 = ρ XZ C X C Z ; E (e 2 e 4 = ( f = ρ YZ C Y C Z ; ρ XZ C X C Z ; C 2 Z; C 2 Z; E (e 0 e 2 =E (e 0 e 4 =E (e e 2 =E (e e 4 =E (e 3 e 2 =E (e 3 e 4 =0. (8 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Takig expectatios i (2 ad usig the results of (8, we get the bias of Ȳ w(d first degree of approximatio as to the I. where B w(d = Ȳ [ α 2 N K 3 + α { f ( }] CX 2 CZ 2 f S. II f = f, f = f 2, N = 2 ( + 2. N Squarig ad takig expectatios i both the sides of (2 ad usig the results of (8, we obtai the MSE of to the first degree of approximatio as M w(d w(d = Ȳ 2 f CY 2 + Ȳ 2 2 [ ] α 2 N K 3 αf S. (9 II Miimizatio of (9 is obtaied with optimum value of α as α = f S 2N K 3 = α II(opt. (20 Substitutig the value of α from (20 i ( gives the AOE of ( as } ( w(d x {Ȳ =ȳ II(opt x. z z αii(opt. (2 Thus, the resultig bias ad MSE of (2 are, respectively give as where B w(d = Ȳ f [ S f C f ] S IIα(opt 2N K 3 2N K M C = C 2 X C2 Z ad w(d = Ȳ 2 f CY 2 Ȳ 2 f 2 S 2. IIα(opt 2N K 3 Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 77 For simplicity, we assume that the populatio size N is large eough as compared to the sample sizes ad so that the fiite populatio correctio (FPC terms /N ad 2/N are igored. 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Igorig the FPC i (9, the MSE of M w(d II reduces to [ w(d = Ȳ C2 Y II + Ȳ 2 α 2 ( 2 + K 3 α S ] which is miimized for Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 78 α = S ( + K 3 = α II(opt (say (22 Substitutig the value of α from (22 i (, we obtaied AOE of ( as Therefore, the resultig MSE of M w(d = Ȳ w(d is [ x. z ] α II(opt. x z w(d = Ȳ 2 C2 Y IIα(opt Ȳ 2 S 2. (23 ( + K 3 Remark 5 For α =, the proposed estimator reduces to ratio cum product estimator i double samplig ad MSE is give as M [ ( w(d = Ȳ 2 C2 Y IIα(opt + Ȳ 2 2 2 + K 3 S ]. (24 Igorig the FPC, the variace of ȳ uder SRSWOR is give by ad the MSE of ( Ȳ Rd ad ( Ȳ II Pd are give by II V (y II = Ȳ 2 C2 Y. (25 M ( Ȳ Rd [ = Ȳ 2 C2 Y II + Ȳ 2 C 2 X ( 2 2 ] K YX (26 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig ad respectively. [ Pd II = Ȳ 2 C2 Y C + Ȳ 2 2 X + ( 2 CZ 2 + 2C ] YZ (27 VI. Efficiecy Copmparisos I. Compairiso of the optimum proposed estimator a with sample mea per uit estimator From (25 ad (23, we observed that V (ȳ II M if K 3 > 0 i.e K XZ < /2. b with ratio estimator i double samplig From (26 ad (23, we observed that if M ( Ȳ Rd M w(d II w(d IIα(opt w(d = Ȳ 2 S 2 > 0 (28 IIα(opt ( + K 3 IIα(opt [ ] = Ȳ 2 S 2 + C 2 ( + K XP > 0 (29 3 K 3 > 0, P>0 i.e K XZ < /2, K YX < 2, where P = 2 ( K YX. c with product estimator i double samplig From (27 ad (23, we observed that M ( Ȳ Pd M w(d II IIα(opt = Ȳ 2 [ Q C2 Z 2 + ] S 2 > 0 (30 ( + K 3 Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 79 if K 3 > 0 i.e K XZ < /2, where Q = C 2 X + C2 Z 2 +3C YX. 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig d with ratio cum product estimator i double samplig From (24 ad (23, we observed that Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 80 M ( Ȳ d M w(d II IIα(opt if K 3 > 0 i.e K XZ < /2. VII. [( = Ȳ 2 + K 3 S ] 2 + S 2 > 0 ( + K 3 Coclusio We have developed a efficiet class of ratio-cum-product estimators i two phase samplig. The comparative study shows that the proposed estimator Ȳ w(d established their superiority over sample mea Ȳ, ratio estimator Y d R, product estimatory P d ad ratio-cum-product estimator Y d Refereces Référeces Referecias (3 i two-phase samplig uder the give coditios. Hece from the resultig equatio (4, (5, (6 ad (7, we coclude that uder the give coditios the proposed estimator is cosider to be the best estimator. Choudhury, S. ad Sigh, B. (20. A improved class of ratio-cum-product estimator of fiite populatio mea i simple radom samplig. Iteratioal Joural of Statistics ad Aalysis, (4:393 403. Cochra, W. (940. The estimatio of the yields of cereal experimets by samplig for the ratio of grai to total produce. Joural of Agricultural Sciece, 30:262 275. Murthy, M. (967. Samplig: theory ad methods. Series i probability ad statistics. Statistical Publishig Society. Neyma, J. (938. Cotributio to the theory of samplig huma populatios. Joural of the America Statistical Associatio, 33(20:0 6. Robso, D. (957. Applicatios of multivariate polykays to the theory of ubiased ratio-type estimatio. Joural of America Statistical Associatio, 52(280:5 522. 204 Global Jourals Ic. (US
Improved Class of Ratio -Cum- Product Estimators of Fiite Populatio Mea i two Phase Samplig Shah, S. ad Shah, D. (978. Ratio cum product estimators for estimatig ratio (product of two populatio parameters. Sakhya, 40:56 66. I. Sharma, B. ad Tailor, R. (200. A ew ratio-cum-dual to ratio estimator of fiite populatio mea i simple radom samplig. Global Joural of Sciece Frotier Research, 0(:27 3. Sigh, H. P. ad Tailor, R. (2005. Estimatio of fiite populatio mea usig kow correlatio. Statistica, Ao LXV, 65:407 48. Sigh, M. (967. Ratio cum product method of estimatio. Metrika, 2(:34 42. Tailor, R. ad Sharma, B. (2009. A modified ratio-cum-product estimator of fiite populatio mea usig kow coefficiet of variatio ad coefficiet of kurtosis. Statistics i Trasitio-ew series, Jul-09, 0(:5 24. Global Joural of Sciece Frotier Research F Volume XIV Issue II V ersio I Year 204 8 204 Global Jourals Ic. (US