Appl Math If Sc Lett 4 o 5-33 (6) 5 Appled Mathematcs & Ifmato Sceces Letters A Iteratoal Joural http://ddog/8576/amsl/45 A Geeralzed lass of Dual to Product-um-Dual to Rato Tpe stmats of Fte Populato Mea I Sample Surves Housla P Sgh Sura K Pal ad Vshal Mehta School of Studed Statstcs Vkram Uverst Ujja M P Ida Receved: 3 Jul 5 Revsed: 7 Aug 5 Accepted: 8 Aug 5 Publshed ole: Ja 6 Abstract: I ths paper we have suggested a geeral class of dual to product-cum-dual to rato tpe estmats of fte populato mea usg a aular varable that s crelated wth the varable of terest 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 proposed class of dual to product-cum- dual to rato tpe estmats of fte populato mea to varous estg rato product ad rato-cum-product tpe estmats ad show that the suggested class of estmats s better tha other estg estmats uder some realstc codtos Kewds: Aular varable Dual to product-cum-dual to rato tpe estmat Fte populato mea Smple radom samplg as Mea squared err Itroducto It s well establshed fact that sample surves aular fmato s ofte used to mprove the precso of estmats of populato parameters The use of aular fmato at the estmato stage appears to have started wth the wk of ochra (94) He evsaged the rato estmat to estmate the populato mea total of the stud varate b usg supplemetar fmato o a aular varate postvel crelated wth The rato estmat s most effectve whe the relatoshp betwee stud varate ad aular varate s lear passg through the g ad the mea square err of s proptoal to Whe the aular varable s egatvel crelated wth the stud varate Robso (957) ad Murth (964) proposed the product estmat of the populato mea total I fact f the better utlzato of fmato o a aular varate Murth (964) has suggested the use of Rato estmat R f / / Product estmat P f / / Ubased estmat f / / / Where ) are coeffcets of varato of ( ( ) respectvel ad s the crelato coeffcet betwee ad respectvel osder a fte populato U U U U uts A sample of sze of s draw usg smple radom samplg wthout replacemet (SRSWOR) method to estmate the populato mea of the stud varate 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 are respectvel gve b R () P () The bases ad mea squared errs (s) of R ad P to the frst degree of appromato are respectvel gve b respodg auth e-mal: surakatpal6676@gmalcom 6 SP atural Sceces Publshg
6 R P f K (3) f K (4) where R f K f K (5) f K (6) P S S S S ( ) S ad osder a trasfmato S S S ( )( ) ( ) ( ) g g where g /( ) The ( g) g ubased estmat f betwee ad s a ad the crelato s egatve Usg the trasfmato Srvekataramaa (98) ad adopadhaa (98) obtaed dual to rato ad dual to product estmats respectvel as (7) R P (8) The bases ad mea squared errs (s) of R ad P to the frst degree of appromato are respectvel gve as f R gk (9) f P f g g K () R g g K () f P g g K () Sgh ad Aghotr (8) defed a faml of productcum-rato estmats of populato mea smple radom samplg (SRS) as a b a b (3) a b a b H P Sgh et al: A geeralzed class of dual to where a ad b are kow characterzg postve scalars ad s a real costat to be determed such that the of s mmum The bas ad of to the frst degree of appromato are respectvel gve as f K K (4) where f a a b K (5) The am of ths paper s to suggest a geeralsed class of dual to product-cum-dual to rato estmats f populato mea SRSWOR ad ther propertes are studed uder large sample appromato It s show that the proposed dual to product-cum dual to rato estmat cludes several kow estmats based o trasfmato aular varable A emprcal stud s carred out to dscuss the supert of the proposed class of estmats A Geeralzed lass of Dual to Product- um- Dual to Rato Tpe stmats of Fte Populato Mea We suggest a class of dual to product-cum-dual to rato tpe estmats SRSWOR f populato mea as a b a b a b a b () where ( a b) are same as defed earler beg a sutabl chose scalar ad g g wth g To obta the bas ad of appromato we wrte ad e e such that e e ad f e f e e e to the frst degree of () f K Table shows the members of the proposed class of estmats f dfferet choces of ( a b ) I Table ad respectvel are kow coeffcet of varato ad coeffcet of kurtoss respectvel of a aular varable 6 SP atural Sceces Publshg
Appl Math If Sc Lett 4 o 5-33 (6) / http://wwwaturalspublshgcom/jouralsasp 7 Table : Members of the estmat S o stmats R Dual to rato estmat P Dual to product estmat 3 SD Sgh ad Upadhaa (986) estmat 4 SK Dual to Sgh et al (4) estmat UP 5 Dual to Upadhaa ad Sgh (999) rato-tpe estmat 6 7 8 9 3 UP Dual to Upadhaa ad Sgh (999) product-tpe estmat UP 3 Dual to Upadhaa ad Sgh (999) rato-tpe estmat UP 4 Upadhaa ad Sgh (999) product-tpe estmat S haudhar ad Sgh () estmat a b SA a b Dual to Shah ad Patel (984) ad Sgh ad Aghotr (8) rato-tpe estmat a b 4 SA a b Dual to Sgh ad Aghotr (8) product-tpe estmat 5 TS pressg () terms of e' s we have f dfferet choces of Dual to Tal ad Sharma (9) estmat a ( a b ) Values of costats b ( a b ) a b a b a a b a a b ( e )[ ( ge ) ( )( ge )] (3) We assume that ge so that ge s epadable From (3) we have 6 SP atural Sceces Publshg
8 e [ ( ge g e ) ge e ge g e e ge ge e g e eglectg terms of e s havg power greater tha two we have [ e ge ge e g e ] [ e ge ge e g e ] (4) Takg epectato of both the sdes of (4) we obta the bas of to the frst degree of appromato as f f The bas of gk g f g K g s almost ubased whe g K g ] (5) e K K g Squarg both sdes of (4) ad eglectg terms of e s havg power greater tha two we have e g e g e e (6) Takg epectato of both sdes of (6) we get the of to the frst degree of appromato as f g g K (7) whch s mmzed whe K opt sa g (8) Substtutg the value of () elds the opt asmptotcall optmum estmat (AO) as a b a b opt opt opt (9) a b a b Thus the resultg bas ad of respectvel opt are respectvel gve b f g K m opt opt g () opt f () opt 3 ffcec omparso () Uder SRSWOR the varace of sample mea s H P Sgh et al: A geeralzed class of dual to f Var (3) From (7) ad (3) t s foud that the proposed dual to product-cum-dual to rato tpe estmat effcet tha f Var f f g f Ths codto holds f ether ad ad s me g K g g K K g K g (3) Therefe the rage of uder whch the proposed dual to product-cum-dual to rato tpe estmat effcet tha s s me K K m ma (33) g g () The s of dual to Shah ad Patel (984) ad Sgh ad Aghotr (8) rato(product) tpe estmat as f SA SA s obtaed b puttg g g K (34) SA f SA g g K (35) From (7) ad (34) t s foud that the proposed dual to product-cum-dual to rato tpe estmat s me effcet tha dual to Shah ad Patel (984) ad Sgh ad Aghotr (8) class of rato estmats f f SA Ths codto holds f g SA f g g K 4g g K g K ether K g K (36) g Therefe the rage of uder whch the proposed dual to product-cum-dual to rato tpe estmat s me effcet tha dual to Shah ad Patel (984) ad Sgh ad Aghotr (8) class of rato estmats f SA K K m ma (37) g g () From (7) ad (35) t s foud that the proposed dual to product-cum-dual to rato tpe estmat s 6 SP atural Sceces Publshg
Appl Math If Sc Lett 4 o 5-33 (6) / http://wwwaturalspublshgcom/jouralsasp 9 me effcet tha dual to Sgh ad Aghotr (8) class of product estmats SA f SA Ths codto holds f f g f g g 4g g K g K K ther K K g (38) g Therefe the rage of uder whch the proposed dual to product-cum-dual to rato tpe estmat s me effcet tha dual to Sgh ad Aghotr (8) class of product estmats f SA K K m ma (39) g g Remark 3 The codtos whch suggested dual to product-cum-dual to rato tpe estmat s better tha other estg estmats (dscussed Table ) ca be easl obtaed b puttg dfferet values of a b 4 Illustrato () To llustrate the geeral results we cosder a b a b () we get a class F of estmats f populato mea (4) where ad as are same as defed earler The bas ad of to the large sample appromato are respectvel gve as ( f ) g K g (4) f g g K F we get a estmat f as (43) (44) whch s due to Shah ad Patel (984) ad Sgh ad Upadhaa (986) We ote that the estmat s dual to Ssoda ad Dwved s (98) rato-tpe estmat F reduces to the estmat f as whch s dual to product tpe estmat (45) The bases ad s of ad to the frst degree of appromato are respectvel gve as f g (46) K f g g K (47) f g g K (48) g g K (49) f It s observed from (43) that ( ) Var( ) f ( ) K ether g (4) ( ) K g From () ad (43) we see that the proposed class of estmat ) wll domate over ( Srvekataramaa (98) estmat R K ether g (4) K g From (43) ad (48) we ote that ( ) ( ) f K( ) ether g (4) K( ) g Further from (43) ad (49) we have that ( ) ( ) f K( ) ether g (43) K( ) g () To llustrate the geeral results we cosder a b S a b () we get a F S class of estmats f populato mea as f 6 SP atural Sceces Publshg
3 S S S S (44) where ad The bas ad of are same as defed earler to the frst degree of appromato are respectvel gve as f g K g (45) g f g F (46) K we get a estmat f S S as (47) whch s due to Shah ad Gupta (986) We ote that the estmat s dual to Shah ad Patel (984) rato tpe estmat F reduces to the estmat f as S (48) S whch s dual to product-tpe estmat The bases ad s of ad to the frst degree of appromato are respectvel gve as f g K (49) f g g K (4) f g g K (4) f g g K (4) We ote from (46) that the proposed class of estmats s me effcet tha the usual ubased estmat f ( ) K ether g (43) ( ) K g It follows from () ad (46) that the suggested class of estmats s better tha Srvekataramaa (98) estmat R f H P Sgh et al: A geeralzed class of dual to K ether g K g From (46) ad (4) that ( ) ( ) f K( ) ether g K( ) g (44) (45) Further t observed from (46) ad (4) that ( ) ( f ) K( ) ether g K( ) g 5 mprcal Stud (46) To llustrate the perfmace of the proposed class of estmats ad over usual ubased estmat rato estmat R product estmat P ad we cosder two atural populato data sets Populato I: (Itala bureau f the evrometal protecto-apat Waste 4) =Total amout (tos) of recclable-waste collecto Ital 3 ad = amout (tos) of recclable-waste collecto Ital 3 4 66 3 557 99 9936 4588 4683 Populato II: (Itala bureau f the evrometal protecto- APAT Waste 4) =Total amout (tos) of recclable-waste collecto Ital 3 ad = umber of habtats 3 3 4 6 6 556 554 798 4588 963 We have computed the percet relatve effceces (s) of the estmats ad wth respect to R ad b usg the followg fmulae: 6 SP atural Sceces Publshg
Appl Math If Sc Lett 4 o 5-33 (6) / http://wwwaturalspublshgcom/jouralsasp 3 ( ) g( ) ( ) g K (5) ( [ ( )] g g K R ) g( ) ( ) g K g ) g K g( ) ( ) g K 3 ( ( ) ( ) g ( ) g K (5) (53) (54) [ ( )] g g K (55) ( R ) ( ) g ( ) g K (56) g 3 ( ) g K ( ) g ( ) g K Table 5 shows the percet relatve effceces (s) of ad wth respect to the usual ubased estmat f dfferet value of Table 5: Rage of estmats stmat s better tha whch the proposed class of R ad Rage of Populato I Populato II (-77 5) (-689 5) R (-6 -) (-88 -) (-7 ) (-89 ) We ote that the crelato betwee stud varable ad s postve f both the populato data sets I ad II Owg to ths the proposed classes of estmats are comparable wth usual ubased estmat dual to rato estmat R ad Upadhaa (986) estmat Gupta (986) estmat Table 5: Rage of estmats stmat R ad Shah ad Patel (984) ad Sgh s better tha ad Shah ad whch the proposed class of R ad Rage of Populato I Populato II (-498 5) (-48 5) (-34-734) (-33-548) (-448 ) (-358 ) Tables 5 ad 5 show that there s eough scope of selectg the value of scalar obtag estmats R better tha ad That s eve f the devates from ts true optmum value scalar opt cosderable ga effcec b usg proposed classes of estmats ad over other estg estmats ca be obtaed Largest ga effcec s observed at the optmum value Tables 53 ad 54 ehbt opt of that there s apprecable ga effcec b usg the proposed class of estmats = ; over usual ubased estmat Srvekataramaa s (98) dual to rato estmat Upadhaa (986) estmat R Shah ad Patel (984) ad Sgh ad ad Shah ad Gupta (986) estmat Fdgs closed Tables 5 to 54 are the favour of proposed classes of estmats ad Thus we recommed the proposed classes of estmats ad f ther use practce 6 SP atural Sceces Publshg
3 H P Sgh et al: A geeralzed class of dual to Table 53: s of the proposed class of estmats Populato I 3 wth respect to R ad f dfferet values of Populato II 3-5 566-887 -76 766 6756-686 3-5 98 8-5 3-5 35 9548-5 59 9-7596 365 4686-854 547 45-75 3637 755 9953-75 95 35 64-5 43449 3596 6379-5 5 339 6636-5 84776377 5454-5 94 997 575 36 947 6 556 5 6489 9 5 747 5 5-63 783746 4387 4757-595 395 335 6778 Stads f s less tha % Table 54: s of the proposed class of estmats Populato I 3 wth respect to R ad Populato II f dfferet values of -5-45 43-489 7-4 38-45 4639-375 663-448 4897-356 73 94-3 67 3447 7445-35 385 374-3 38 3598 7694-33 639 9 76-493 554 39698-3 633 3 87-75 499 69986 53-93 35 68-5 773 36633 7997-8839 757 4936-46475 446 364-75 733 8 3734-75 3796 5 6-546 6 694-5 436 6364-5 573 457-5 8738 65-5 49 94 4866 63 5 75 83 5 35 5 677 5-4 783758 43874 573-79 395 335 6963 Stads f s less tha % 6 ocluso Refereces 3 Ths paper proposed a geeral class of dual to product-cumdual to rato tpe estmat estmat based o the trasfmato a aular varable smple radom samplg The evsaged class of estmats cludes several kow estmats based o trasfmato aular varable The bas ad epressos of the proposed class of estmats have bee obtaed uder large sample appromato It s terestg to ote that ths stud ufes several estmats wth ther propertes at oe place mprcal stud also shows the supert of the proposed class of estmats over other estg estmats Ackowledgemets The auths are thakful to the dt--hef ad to the aomous leared referees f hs valuable suggestos regardg mprovemet of the paper [] adopadhaa S (98): Improved rato ad product estmats Sakha 4() 45-49 [] haudhar S ad Sgh K (): A effcet class of dual to product-cum-dual to rato estmat of fte populato mea sample surves Global Joural of Scece Froter Research (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 6 69-74 [5] Robso D S (957): Applcatos of multvarate polkas to the the of ubased rato-tpe estmato Joural of the Amerca Statstcal Assocato 5 5 5 6 SP atural Sceces Publshg
Appl Math If Sc Lett 4 o 5-33 (6) / http://wwwaturalspublshgcom/jouralsasp 33 [6] Shah D ad Gupta M R (986): A geeral modfed dual rato estmat Gujarat Statstcal Revew 3 () 4-5 [7] Shah S M ad Patel H R (984): ew modfed rato estmat usg coeffcet of varato of aular varable Gujarat Statstcal Revew 45-54 [8] Sgh H P ad Aghotr (8): A geeral procedure of estmatg populato mea usg aular fmato sample surves Statstcs Trasto- ew seres 9 7-87 [9] 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) 4 336-34 [] Sgh H P Tal R Tal R ad Kakra M S (4): A mproved estmat of populato mea usg power trasfmato Joural of Ida Socet Agrculture Statstcs 58() 3-3 [] 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 [] Srvekataramaa T (98): A dual to rato estmat sample surves ometrka 67 99-4 [3] Tal R 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 [4] Upadhaa L ad Sgh H P (999): Use of trasfmed aular varable estmatg the fte populato mea ometrcal Joural 4 67-636 Housla P Sgh (Dea Facult of Scece) Profess School of Studed Statstcs Vkram Uverst Ujja MP Ida Hs research terests are the areas of Statstcs Samplg The ad Statstcal Iferece Sura K Pal Research Scholar Pursug PhD(Statstcs) School of Studed Statstcs Vkram Uverst Ujja M P Ida Vshal Mehta Research Scholar Pursug PhD(Statstcs) School of Studed Statstcs Vkram Uverst Ujja M P Ida 6 SP atural Sceces Publshg