A Generalized Class of Ratio-Cum-Dual to Ratio Estimators of Finite Population Mean Using Auxiliary Information in Sample Surveys

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1 Math Sc Lett 5 o 3- (6) 3 Mathematcal Sceces Letters A Iteratoal Joural A Geeralzed lass of ato-um-dual to ato Estmats of Fte Populato Mea Usg Aular Ifmato Sample Surves Housla P Sgh Sura Kat Pal ad Vshal Mehta School of Studes Statstcs Vkram Uverst Ujja-456 Madha Pradesh Ida eceved: 3 Jul 5 evsed: Oct 5 Accepted: 9 Oct 5 Publshed ole: Ma 6 Abstract: I ths paper we have suggested a geeral class of rato-cum-dual to rato tpe estmats of fte populato mea usg a aular varable (sa ) that s crelated wth the varable of terest (sa ) The proposed class of estmats cludes several kow estmats based o trasfmato aular varable The bas ad mea squared err () epressos of the proposed class of estmats have bee obtaed to the frst degree of appromato We have compared the geeralzed rato-cum-dual to rato tpe estmats of fte populato mea to the usual ubased estmat ad varous estg rato product ad rato-cum-product tpe estmats It s foud that the suggested estmats are better tha other estg estmats uder some realstc codtos umercal llustratos are gve suppt of the preset stud Kewds: Aular varable ato-cum-dual to rato tpe estmat Fte populato mea Smple radom samplg as Mea squared err Itroducto I sample surves aular fmato s used at the selecto stage as well as estmato stage to mprove the effcec of the estmats The use of aular fmato at the estmato stage appears to have started wth the wk of ochra (94) Whe the crelato betwee stud varate ad aular varate s postve (hgh) the rato method of estmato s used f estmatg the populato mea The rato method s most effectve f where are coeffcets of varato of respectvel ad s the crelato coeffcet betwee ad O the other had f the crelato s egatve the product method of estmato evsaged b obso (957) ad revsted b Murth (964) s used ad ths s most effectve f Srvekataramaa (98) frst proposed dual to rato estmat ad adopadhaa (98) proposed dual to product estmat Sgh ad Tal (5) Sgh ad Espejo (3) Tal ad Sharma (9) wked o ratocum-product estmats Sharma ad Tal () haudhar ad Sgh () wked o rato dual to rato ad dual to product estmats to estmate the populato mea of the stud varable respodg auth E-mal: surakatpal6676@gmalcom osder a fte populato U u u u uts A sample of sze of sze s draw usg smple radom samplg wthout replacemet (SSWO) method to estmate the populato mea varate of the stud Let the sample meas be the ubased estmats of the populato meas respectvel based o observatos The classcal rato ad product estmats of populato mea of the stud varable are respectvel gve as () ad P () The bases ad mea squared errs (s) of ad P to the frst degree of appromato are respectvel gve as 6 SP atural Sceces Publshg

2 4 P Ad Where S S f ( K) (3) f K (4) f K (5) f K (6) P f K S S S ad S osder a trasfmato Where g The g populato mea S S g g g s a ubased estmat of the ad the crelato betwee Usg the trasfmato of the aular varable ad s egatve o the aular varable Srvekataramaa (98) ad adopadhaa (98) obtaed dual to rato ad dual to product tpe estmats as Ad (7) P (8) The bases ad mea squared errs (s) of ad P to the frst degree of appromato are respectvel gve as f gk (9) H Sgh et al: A geeralzed class of rato-cum-dual to f P Ad f g g K () gg K () f P g g K () Sgh ad Aghotr (8) defed a faml of ratoproduct estmats of populato mea smple radom samplg (SS) as P a b a b (3) a b a b Where a ad b are kow characterzg postve scalars ad s a real costat to be determed such that the of s mmum P P The bas ad of to the frst degree of appromato are respectvel gve as Ad f P f K K (4) P a Where ( a b) K (5) The am of ths paper s to suggest a geeralzed class of rato-cum-dual to rato tpe estmats f populato mea SSWO ad ther propertes are studed uder large sample appromato It s terestg to ote that the proposed geeralzed class of rato-cum-dual to rato tpe estmats cludes several kow estmats based o trasfmato o aular varable A Geeralzed lass of ato-um-dual to ato Tpe Estmats of Fte Populato Mea We suggest a faml of rato-cum-dual to rato tpe estmats SSWO f populato mea as a b a b () a b a b P Where a b are same as defed earler beg a sutabl chose scalar ad 6 SP atural Sceces Publshg

3 Math Sc Lett 5 o 3- (6)/ 5 g g Wth To obta the bas ad of appromato we wrte e Ad Such that E e E e Ad f E e f E e E e e e f K d Epressg () terms of e s we have P g the frst degree of () e [ e ge ] We assume that e From (3) we have P [ so that (3) e s epadable e [ ( e e ) ge ] e [ ge e g e ] e e g g e e g g e ] We assume that the cotrbuto of terms volvg powers e ad hgher tha the secod s eglgble Thus from the above epresso we wrte to a frst appromato P e [ e e [ e e g P g g e e g g e ] g e e g g e ] (4) Takg epectato of both the sdes of (4) we obta the bas of to the frst degree of appromato as P f P K g g f K g g (5) The bas of P f e P gk { K g} s almost ubased whe K g g Squarg both sdes of (4) ad eglectg terms of e s havg power greater tha two we have e e g P g e e g g (6) Takg epectato of both sdes of (6) we get the of to the frst degree of appromato as P f g g g g P K (7) Assumg (whch s tpcal stuato sample surves) mmzg (7) wth respect to we get g K g opt sa (8) g Substtutg the value of opt asmptotcall optmum estmat (AOE) as Popt a b opt a b () elds the a b opt (9) a b Thus the resultg bas ad of gve as f P opt { K( } Kg ad opt respectvel are Popt () f m () P Popt Table shows members of the proposed class of estmats f dfferet choces of ( a b ) P I the Table coeffcet of varato ad coeffcet of kurtoss of a aular varable are kow 6 SP atural Sceces Publshg

4 6 H Sgh et al: A geeralzed class of rato-cum-dual to S o Table : Members of the estmat Estmats ochra (94) estmat Srvekataramaa (98) 3 SD Ssoda ad Dwved (98) estmat SD Shah ad Patel (986) ad Sgh ad Upadhaa (999) estmat SK Sgh et al (4) estmat P SK Dual to Sgh et al (4) estmat UP Upadhaa ad Sgh (999) estmat UP Dual to Upadhaa ad Sgh (999) rato tpe estmat UP Upadhaa ad Sgh (999) estmat UP Dual to Upadhaa ad Sgh (999) rato tpe estmat ST Sharma ad Tal () estmat a b SA a b Sgh ad Aghotr (8) estmat a b 3 SA a b Dual to Sgh ad Aghotr (8) rato tpe estmat f dfferet choces of ( a b ) a Values of costats b ( a b ) a b a b a a b a a b 6 SP atural Sceces Publshg

5 Math Sc Lett 5 o 3- (6)/ Effcec omparsos Uder SSWO varace of sample mea Var f (3) From (7) ad (3) t s foud that the proposed dual to product-cum-dual to rato tpe estmats effcet tha f Var s P s me f f g g g g K P f g g g g K Ths codto holds f g K g ether g g K g g g g Or equvaletl g (K g (K m ma (3) g ( g ( From (5) ad (7) we have that Var P (K ( ether ( ( ( (K ( ( Or equvaletl f (K ( (K ( m ma (33) ( ( ( ( It s observed from () ad (7) that the proposed class of estmats P s me effcet tha the Srvekataramaa s (98) ad adopadhaa s (98) estmat f g( ) g( ) ether ( ( g( ) g( ) ( ( equvaletl g( ) g( ) g( ) g( ) m ma ( ( ( ( (34) It follows from (6) ad (7) that the proposed class of estmats estmat P P f s me effcet tha classcal product ( ( K ether ( ( ( K ( ( ( Or equvaletl ( ( K ( ( K m ma (35) ( ( ( ( From () ad (7) we have that Var f P P g( ) { g( K} ether ( ( { g( K} g( ) ( ( equvaletl g( ) { g( K} g( ) { g( K} m ma (36) ( ( ( ( If we set as () the P reduces to the estmat a b SA (37) a b Whch s due to Sgh ad Aghotr (8) Puttg estmat ( (7) we get the mea squared err of the SA to the frst degree of appromato as SA ( f ) ) From (7) ad (38) we have that f P SA {K g( } ether ( {K g( } ( Or equvaletl ( K) (38) 6 SP atural Sceces Publshg

6 8 {K g( } {K g( } m ma (39) ( ( Iterestg a b a b SA () we get the estmat (3) Whch s due to Sgh ad Aghotr (8) rato tpe estmat Puttg SA (7) we get the mea squared err of the to the frst degree of appromato as SA ( SA ( f ) ) g ( g K) (3) We ote from (7) ad (3) that the suggested class of estmats P wll domate over the estmat ( K ether ( ( K ( equvaletl SA ( K ( K m ma (3) ( ( From (5) ad (7) we ote that f the two costats ad are dfferet the the proposed class of estmats P wll domate over Sgh ad Aghotr s (8) estmat f {( K) ( } ( g ) ether ( ( ( g ) {( K) ( } ( ( equvaletl ( g ) {( K) ( } ( g ) {( K) ( } m ma ( ( ( ( Let (33) (3) ad the the mea squared err of the estmat b ( P P to the frst degree of appromato s gve ( f ) ) ( ) ( ) K) (34) If two costats ( ) are same (e ) the from (7) ad (34) t s observed that the suggested class of estmats P s better tha Sgh ad Aghotr s (8) f H Sgh et al: A geeralzed class of rato-cum-dual to estmat ether P f K {K ( } K (3 4 Partcular ase (35) To llustrate our geeral results we cosder a estmat whch utlzes fmato o ad crelato coeffcet Whe fmato o both ( ) are avalable; we defe the followg class of estmats (just b puttg ad ()) f populato mea : a b ( ) (4) P () F estmat P () the classes of estmats Whch s due to Sgh ad Tal (3) If we set dow to the estmat P() (4) the class of estmats reduce to the P() (4) bols (43) ( ) P Whch s dual s to Sgh ad Tal s (3) estmat P() To the frst degree of appromato the mea squared errs of P() P() ad P() are respectvel gve b ( f ) ( ) P () Ad ( f ) ) ( g ( g K (44) ( P() ( f ) ) ( P() where 4 Effcec omparso ( K) (45) g( g K) (46) I ths secto we have preseted the comparso of the 6 SP atural Sceces Publshg

7 Math Sc Lett 5 o 3- (6)/ 9 suggested class of estmats estmat rato estmat Sgh ad Tal s (3) estmat Sgh ad Tal s (3) P() From (3) ad (44) we ote that P ) Var( ) If ( () g (K ether ( g ) ( (K g ( ) ( ) g g equvaletl P() wth usual ubased dual to rato estmat P() ad dual to g (K g (K m ma (47) ( g ) ( ( g ) ( It s observed from (5) ad (44) that the class of estmats s me effcet tha usual rato P() estmat f (K g ) ( ether ( ( ( (K g ) ( ) ( ) g g equvaletl (K g ) ( (K g ) ( m ma (48) ( ( ( ( We ote from () ad (44) P( ) That ( ) ( ) f (K g g( ) ether ( ( g( ) (K g ( ) ( ) g g Or equvaletl g( ) (K g g( ) (K g m ma (49) ( ( ( ( From (44) ad (45) we ote that the proposed class of estmats s me effcet tha Sgh ad Tal s estmat P() P() f ( ( g K) ether ( g ) ( ( g K) ( ( ( g ) Or equvaletl ( ( g K) ( ( g K) m ma (4) ( g ) ( ( g ) ( It s observed from (44) ad (46) that the proposed class of estmats s better tha the estmat f P() (K ether ( (K ( ) g equvaletl P() (K g (K g m ma (4) ( ( 5 Emprcal Stud To see the perfmace of the proposed class of estmats over the estmat P() ad P() P() we cosder a atural populato data set The descrpto of the populato s gve below: Populato: Murth (967) = Output f 8 factes a rego ad aptal = Fed We have computed the rage of ad fdgs are show Table 5 We have further computed the percet relatve effcec of the proposed class of estmats P() relatve to ad f dfferet P() P() values of b usg the followg fmulae : E PE( P() ) ( ) g g ( ( K [ ( )] K E PE( P() ) ( ) g g ( ( K (5) (5) 6 SP atural Sceces Publshg

8 H Sgh et al: A geeralzed class of rato-cum-dual to Table 5: age of whch the proposed class of estmats Estmat P s better tha P ad P age of P P ( ) (-68 94) ( ) (- 334) (-566 ) Table 5: PEs of the proposed class of estmats Populato E E P values of E 3 wth respect to E 4 P ad P E 5 f dfferet Stads f PEs less tha % E E 3 PE( P() [ g ( g K)] ) ( ) g g ( ( K 4 PE( P() P() [ ) ( ( (53) ( K)] (54) ( ) g g K [ ( )] g g K E5 PE( () () ) (55) P P ( ) g g ( ( K Table 5 ehbts that there s eough scope of choosg the value of scalar f obtag better estmats tha usual ubased estmat rato estmat rato estmat dual to Sgh ad Tal s (3) estmat ad dual to Sgh ad Tal s (3) P() P() It s observed from Table 5 that larger ga effcec ca be observed b usg the proposed class of estmats over P() ad P() eve whe the P() scalar departs from ts true optmum value opt Largest ga effcac s observed at the optmum value of opt Fall our recommedato s to use the proposed class of estmats practce P() 6 ocluso Ths paper addresses the problem of estmatg the populato mea of the stud varable usg fmato o a aular varable A class of ratocum-dual to rato estmats has bee proposed ad ts propertes are studed uder large sample appromato Dfferet estmats of the populato mea have bee detfed as a member of the proposed class of rato-cum- 6 SP atural Sceces Publshg

9 Math Sc Lett 5 o 3- (6)/ dual to rato estmats egos of perfmace have bee obtaed whch suggested class of estmats perfm better tha other estg estmats It s a ufed approach Propertes of several estmats belogg to the proposed class of estmats ca be studed easl I partcular we have studed the propertes of a class of estmats P() based o kow crelato coeffcet betwee the stud varable ad the aular varable A emprcal stud s carred out to judge the merts of the proposed class of estmats P() over other estmats It has bee show emprcall that the suggested class of estmats estg estmats P() s me effcet tha some other Fall we coclude that the proposed class of estmats P ad ts member P() are mmese useful to the researchers ad practtoers egaged ths area Ackowledgemets The auths are thakful to the Edt--hef ad to the aomous leared referees f hs valuable suggestos regardg mprovemet of the paper efereces [] adopadhaa S (98): Improved rato ad product estmats Sakha 4 () [] haudhar S ad Sgh K (): A Effcet lass of Dual to Product-um-Dual to ato Estmat of Fte Populato Mea Sample Surves Global Joural of Scece Froter esearch (3) 5-33 [3] ochra W G (94): Some propertes of estmats based o samplg scheme wth varg probabltes Australa Joural of Statstcs 7-8 [4] Murth M (964): Product method of estmato Sakha [5] obso D S (957): Applcatos of multvarate polkas to the the of ubased rato-tpe estmato Joural of the Amerca Statstcal Assocato [6] Shah D ad Gupta M (986): A geeral modfed dual rato estmat Gujarat Statstcal evew 3 () 4-5 [7] Shah S M ad Patel H (984): ew modfed rato estmat usg coeffcet of varato of aular varable Gujarat Statstcal evew [8] Sgh HP ad uz-espejo M (3): O lear regresso ad rato-product estmato of a fte populato mea The Statstca 5() [9] Sgh HP ad Tal (3): Use of kow crelato coeffcet estmatg the fte populato mea Statstcs Trasto 6(4) [] Sgh H P Tal Tal ad Kakra M S (4): A mproved estmat of populato mea usg power trasfmato Joural of Ida Socet Agrculture Statstcs 58() 3-3 [] Sgh HP ad Tal (5): Estmato of fte populato mea wth kow coeffcet of varato of a aular character Statstca 65(3) 3-33 [] Sgh H P ad Aghotr (8): A geeral procedure of estmatg populato mea usg aular fmato sample surves Statstcs Trasto- ew seres [3] Sgh H P ad Upadhaa L (986): A dual to modfed rato estmat usg coeffcet of varato of aular varable Proceedgs of the atoal Academ of Sceces Ida 56 (A) [4] Ssoda VS ad Dwved VK (98): A modfed rato estmat usg coeffcet of varato of aular varable Joural of Ida Socet Agrculture Statstcs 33() 3-8 [5] Srvekataramaa T (98): A dual to rato estmat sample surves ometrka [6] Tal ad Sharma (9): A modfed rato-cumproduct estmat of fte populato mea usg kow coeffcet of varato ad coeffcet of kurtoss Statstcs Trasto- ew seres 5-4 [7] Upadhaa L ad Sgh H P (999) Use of trasfmed aular varable estmatg the fte populato mea ometrcal Joural Housla P Sgh (Dea Facult of Scece) Profess School of Studes Statstcs Vkram Uverst Ujja MP Ida Hs research terests are the areas of Statstcs Samplg The; Statstcal Iferece-Use of Pr fmato estmato procedure Sura K Pal esearch Scholar Pursug PhD(Statstcs) School of Studes Statstcs Vkram Uverst Ujja M P Ida Vshal Mehta esearch Scholar Pursug PhD (Statstcs) School of Studes Statstcs Vkram Uverst Ujja M P Ida 6 SP atural Sceces Publshg