Generalized Modified Ratio Estimator for Estimation of Finite Population Mean

Transcript

1 Jourl of Moder Appled Sttstcl Methods Volume Issue Artcle Geerlzed Modfed Rto Estmtor for Estmto of Fte Populto Me Jmbulgm Subrm Podcherr Uverst, Puducherr, Id, drjsubrm@hoo.co. Follow ths d ddtol works t: Prt of the Appled Sttstcs ommos, Socl d Behvorl Sceces ommos, d the Sttstcl Theor ommos Recommeded tto Subrm, Jmbulgm (03) "Geerlzed Modfed Rto Estmtor for Estmto of Fte Populto Me," Jourl of Moder Appled Sttstcl Methods: Vol. : Iss., Artcle 7. DOI: 0.37/jmsm/ Avlble t: Ths Regulr Artcle s brought to ou for free d ope ccess b the Ope Access Jourls t Dgtlommos@WeStte. It hs bee ccepted for cluso Jourl of Moder Appled Sttstcl Methods b uthorzed edtor of Dgtlommos@WeStte.

2 Geerlzed Modfed Rto Estmtor for Estmto of Fte Populto Me over Pge Footote The uthor wshes to record hs grttude d thks to Uverst Grts ommsso (UG) for the fcl ssstce through UG-Mjor Reserch Project. Ths regulr rtcle s vlble Jourl of Moder Appled Sttstcl Methods: ss/7

3 Jourl of Moder Appled Sttstcl Methods November 03, Vol., No., -55. oprght 03 JMASM, Ic. ISSN Geerlzed Modfed Rto Estmtor for Estmto of Fte Populto Me Jmbulgm Subrm Podcherr Uverst Puducherr, Id A geerlzed modfed rto estmtor s proposed for estmtg the populto me usg the kow populto prmeters. It s show tht the smple rdom smplg wthout replcemet smple me, the usul rto estmtor, the ler regresso estmtor d ll the estg modfed rto estmtors re the prtculr cses of the proposed estmtor. The bs d the me squred error of the proposed estmtor re derved d re compred wth tht of estg estmtors. The codtos for whch the proposed estmtor performs better th the estg estmtors re lso derved. The performce of the proposed estmtor s ssessed wth tht of the estg estmtors for cert turl popultos Kewords: prmeters Aulr vrble, bses, turl populto, me squred error, Itroducto osder fte populto U { U, U,, U N } of N dstct d detfble uts. Let be stud vrble wth vlue mesured o U,,, 3,, N gvg vector {,,, N }. The problem s to estmte the populto N me o the bss of rdom smple selected from the N populto U. The smple rdom smple me s the smplest estmtor for estmtg the populto me. If ulr vrble, closel relted to the stud vrble, s vlble the oe c mprove the performce of the estmtor of the stud vrble b usg the kow vlues of the populto prmeters of the ulr vrble. Tht s, whe the populto prmeters of the ulr vrble such s populto me, coeffcet of vrto, coeffcet of kurtoss, coeffcet of skewess etc., re kow, the umber of estmtors vlble the lterture (such s rto, product d ler regresso Dr. Subrm s Assocte Professor d Hed of the Deprtmet of Sttstcs. Eml hm t: drjsubrm@hoo.co..

4 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN estmtors d ther modfctos) perform better th the usul smple rdom smple me uder cert codtos. Amog these estmtors, m reserchers hve used the rto estmtor d ts modfctos for the estmto of the me of the stud vrble (see for emple Ssod d Dwved (98), Kdlr d g (006, 006b), d T (00) d Subrm d Kumrpd (0, 0c)). Before dscussg further the estg estmtors d the proposed estmtors, the ottos to be used ths rtcle re descrbed below: N f /N,, S, S, ρ β β M d B(.) MSE(.) p j Populto sze Smple sze Smplg frcto Stud vrble Aulr vrble Populto mes Smple mes Populto stdrd devtos o-effcet of vrtos o-effcet of correlto betwee d o-effcet of skewess of the ulr vrble o-effcet of kurtoss of the ulr vrble Med of the ulr vrble Bs of the estmtor Me squred error of the estmtor th estg (jth proposed) modfed rto estmtor of I cse of smple rdom smplg wthout replcemet (SRSWOR), the smple me srs s used to estmte populto me, whch s ubsed estmtor, d ts vrce s gve below: V srs S () The rto estmtor for estmtg the populto me of the stud vrble s defed s:

5 JAMBULINGAM SUBRAMANI where R R R () The bs d me squred error of the rto estmtor to the frst degree of ppromto re gve below: ( ρ ) B R (3) ( ρ ) MSE R + (4) The usul ler regresso estmtor together wth ts vrce s gve below: lr + β (5) ( lr ) V ( ρ ) S (6) Ssod d Dwved (98) hve suggested modfed rto estmtor usg the co-effcet of vrto of ulr vrble for estmtg. Whe the coeffcet of kurtoss of ulr vrble s kow, Sgh et l. (004) hs developed modfed rto estmtor. Sgh d Tlor (003) proposed other estmtor for estmtg whe the populto correlto co-effcet betwee d s kow. B usg the populto vrce of ulr vrble, Sgh (003) proposed other modfed rto estmtor for estmtg populto me. More recetl, d T (00) hs suggested other modfed rto estmtor usg the co-effcet of skewess of the ulr vrble, d Subrm d Kumrpd (03) suggested ew modfed rto estmtor usg kow populto med of ulr vrble. Updh d Sgh (999) suggested other modfed rto estmtor usg the ler combto of co-effcet of vrto d co-effcet of kurtoss. Sgh (003) used the ler combto of co-effcet of kurtoss d stdrd devto d co-effcet of skewess d stdrd devto for estmtg the popultos me. Motvted b Sgh (003), d T (00) used the ler combto of co-effcet of kurtoss d co-effcet of 3

6 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN skewess d co-effcet of vrto d co-effcet of skewess. Subrm d Kumrpd (0, 0b, 0c d 03b) suggested modfed rto estmtors usg kow med d co-effcet of kurtoss, med d coeffcet of skewess, med d co-effcet of vrto d med d coeffcet of correlto. More detled dscusso bout the rto estmtor d ts modfcto c be foud Abd d Shhbz (006), Ahmd et l. (009), Al-Jrrh d Al- Hj Ebrhem (0), Bhush (0), ochr (977), Dlbeher d Shoo (994), Dvd d Sukhtme (974), Goodm d Hrtle (958), Gupt d Shbbr (008), Jhjj et l. (006), Kdlr d g (003, 004), Khoshevs et l. (007), Koucu d Kdlr (009), Kulkr (978), Murth (967), Nk d Gupt (99), Olk (958), Pthk (964), Perr (007), R d Sh (980), Redd (973), Robso (987), Se (993), Shbbr d b (003), Shrm d Tlor (00), Sgh d hudhr (986), Sgh (003), Sgh d Espejo (003), Sgh d Aghotr (008), Sgh d Solk (0), Sgh d Tlor (003, 005), Sgh et l. (004, 008), Ssod d Dwved (98), Solk et l. (0), Srvektrm (980), Tlor d Shrm (009), T (965), Updh d Sgh (999) d d T (00). The followg tble cots ll modfed rto estmtors usg kow populto prmeters of the ulr vrble whch some of the estmtors re lred suggested the lterture. The remg estmtors re troduced ths rtcle: Tble. Modfed Rto estmtors wth the costt, the bs, d the me squred errors. Estmtor ostt Bs B(.) Me squred error MSE(.) + + Ssod d Dwved (98) + ( ρ ) + ρ ) + β + β Sgh et l. (004) + β ( ρ ) + ρ ) + β 3 + β d T (00) 3 + β ( 3 ρ 3 ) + 3 ρ 3 ) + ρ 4 + ρ Sgh d Tlor (003) 4 + ρ ( 4 ρ 4 ) + 4 ρ 4 ) 4

7 JAMBULINGAM SUBRAMANI Tble otued Estmtor ostt Bs B(.) Me squred error MSE(.) + S 5 + S Sgh (003) 5 + S ( 5 ρ 5 ) + 5 ρ 5 ) + M d 6 + Md Subrm d Kumrpd (03) 6 + M d ( 6 ρ 6 ) + 6 ρ 6 ) β + 7 β + Updh d Sgh (999) β 7 β + ( 7 ρ 7 ) + 7 ρ 7 ) + β 8 + β Updh d Sgh (999) 8 + β ( 8 ρ 8 ) + 8 ρ 8 ) β + 9 β + β 9 β + ( 9 ρ 9 ) + 9 ρ 9 ) + β 0 + β d T (00) 0 + β ( 0 0ρ ) + 0 0ρ ) ρ + ρ + ρ ρ + ( ρ ) + ρ ) + ρ + ρ + ρ ( ρ ) + ρ ) S + 3 S + S 3 S+ ( 3 3ρ ) + ρ ) + S 4 + S 4 + S ( 4 4ρ ) + 4 4ρ ) Md + 5 M d + Md 5 M + d ( 5 5ρ ) + 5 5ρ ) + M d 6 + M d Subrm d Kumrpd (0c) 6 + M d ( 6 6ρ ) + 6 6ρ ) β + β 7 β + β d T (00) β 7 β + β ( 7 7ρ ) + 7 7ρ ) 5

8 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble otued Estmtor ostt Bs B(.) Me squred error MSE(.) β + β 8 β + β d T (00) β 8 β + β ( 8 8ρ ) + 8 8ρ ) ρ + β 9 ρ + β ρ 9 ρ + β ( 9 9ρ ) + 9 9ρ ) β + ρ 0 β + ρ β 0 β + ρ ( 0 0ρ ) + 0 0ρ ) + β S + β S S S+ β ( ρ ) + ρ ) β + S β + S Sgh (003) β β + S ( ρ ) + ρ ) d + β Md + β 3 M Md 3 M + β d ( 3 3ρ ) + 3 3ρ ) β + M d 4 β + Md Subrm d Kumrpd (0) β 4 β + M d ( 4 4ρ ) + 4 4ρ ) ρ + β 5 ρ + β ρ 5 ρ + β ( 5 5ρ ) + 5 5ρ ) β + ρ 6 β + ρ β 6 β + ρ ( 6 6ρ ) + 6 6ρ ) S + β + β 7 S S 7 S+ β ( 7 7ρ ) + 7 7ρ ) β + S 8 β + S Sgh (003) β 8 β + S ( 8 8ρ ) + 8 8ρ ) d + β Md + β 9 M Md 9 M + β d ( 9 9ρ ) + 9 9ρ ) β + M d 30 β + Md Subrm d Kumrpd (0b) β 30 β + M d ( 30 30ρ ) ρ ) 6

9 JAMBULINGAM SUBRAMANI Tble otued Estmtor ostt Bs B(.) Me squred error MSE(.) S + ρ 3 S + ρ S 3 S+ ρ ( 3 3ρ ) + 3 3ρ ) ρ + S 3 ρ + S ρ 3 ρ + S ( 3 3ρ ) + 3 3ρ ) Md + ρ 33 M d + ρ Md 33 M + ρ d ( 33 33ρ ) ρ ) ρ + M d 34 ρ + Md Subrm d Kumrpd (03b) ρ 34 ρ + M d ( 34 34ρ ) ρ ) Md + S 35 M d + S S + M d 36 S + M d Md 35 M + S S 36 S+ M d d ( 35 35ρ ) ( 36 36ρ ) f ρ ) ρ ) Proposed geerlzed rto estmtor As stted erler, the performce of the estmtor of the stud vrble c be mproved b usg the kow populto prmeters of the ulr vrble, whch re postvel correlted wth tht of stud vrble. The proposed geerlzed modfed rto estmtor for estmtg the populto me s defed s: ( ) ( ) + + α λ p ;,,3,,36 (7) + + α λ The bs d me squred error of the proposed estmtor p hve bee derved (see Apped A) d re gve below: ( ρ ) B p ;,,3,,36 p p (8) 7

10 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN ( ρ ) MSE p ; + p p where p ;,,3,,36 + α λ ( ) (9) where λ, λ β, λ β, 3 λ4 ρ, λ 5 S, λ 6 M d, λ7 / β, λ /, 8 β λ9 / β, λ /, 0 β λ / ρ, λ ρ /, λ 3 / S, λ 4 S /, λ 5 / Md, λ 6 Md /, λ7 β / β, λ8 β / β, λ β / ρ, 9 λ0 ρ / β, λ /, β S λ S / β, λ /, 3 β M d λ4 M d / β, λ β / ρ, 5 λ6 ρ / β, λ /, 7 β S λ8 S / β, λ /, 9 β M d λ30 M d / β, λ3 ρ / S, λ3 S / ρ, λ33 ρ / M d, λ34 M d / ρ, λ 35 S / Md, d λ 36 M / S d Effcec of the proposed estmtor The vrce of SRSWOR smple me srs s gve below: V srs S (0) The bs d me squred error of the usul rto estmtor R to the frst degree of ppromto re gve below: ( ρ ) B R ( ρ ) MSE + R () The bs d the me squred error of the modfed rto estmtors to 36 lsted the Tble re represeted sgle clss s gve below: 8

11 JAMBULINGAM SUBRAMANI + λ ;,,3,,36 + λ ( ρ ) B ;,,3,,36 MSE ( + ρ) where ;,,3,,36 + λ () As dscussed erler, the bs, the me squred error d the costt of the proposed modfed rto estmtor p re gve below: ( ρ ) B p ;,,3,,36 p p ( ) ( ρ ) MSE + where p ;,,3,, α λ p p p (3) where λ, λ β, λ β, 3 λ4 ρ, λ 5 S, λ 6 M d, λ7 / β, λ /, 8 β λ9 / β, λ /, 0 β λ / ρ, λ ρ /, λ 3 / S, λ 4 S /, λ 5 / Md, λ 6 Md /, λ7 β / β, λ8 β / β, λ β / ρ, 9 λ0 ρ / β, λ /, β S λ S / β, λ /, 3 β M d λ4 M d / β, λ β / ρ, 5 λ6 ρ / β, λ /, 7 β S λ8 S / β, λ /, 9 β M d λ30 M d / β, λ3 ρ / S, λ3 S / ρ, λ33 ρ / M d, λ34 M d / ρ, λ 35 S / Md, d λ 36 Md / S From the epressos gve (0) d (3), the codtos (see Apped B ) for whch the proposed estmtor p re more effcet th the smple rdom smplg wthout replcemet (SRSWOR) smple me srs were derved d re: MSE p V r f p ρ (4) 9

12 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN From the epressos gve () d (3), the codtos (see Apped ) for whch the proposed estmtors p re more effcet th the usul rto estmtor R were derved d re: MSE MSE ( ) ρ ρ ether (or) (5) p R p p From the epressos gve () d (3), the codtos (see Apped D) for whch the proposed estmtors p j ; j,,, 5 re more effcet th the estg modfed rto estmtors gve lss, ;,, 3,, were derved d re: MSE MSE ( ) ρ ρ ether (or) (6) pj pj pj The codtos terms of α whch proposed estmtor p performs better th the smple rdom smplg wthout replcemet (SRSWOR) smple me srs were obted d re: λ MSE p V r f α ρ λ (7) From the epresso gve (5), the rge of α whch proposed estmtor p performs better th the usul rto estmtor R s determed d s: MSE MSE ( ) λ ether α ρ λ (or) p R λ ρ α ;,,3,,36 λ (8) 30

13 JAMBULINGAM SUBRAMANI From the epresso gve (6), the rge of α whch proposed estmtor p performs better th the estg modfed rto estmtors lsted Tble s: MSE MSE ( ) λ ether 0 α ρ λ (or) p λ ρ α 0;,,3,,36 λ (9) Prtculr cse: λ ) At α ρ ;,,3,,36, the me squred λ error of the proposed estmtor p ;,,3,,36 equl to the vrce of the SRSWOR smple me srs. λ ) At lmt pot α ρ or the me λ squred error of the proposed estmtor p ;,,3,,36 equl to the me squred error of the usul rto estmtor R λ 3) At lmt pot ρ or 0 the me squred λ error of the proposed estmtor p ;,,3,,36 the me squred error of the estg modfed rto estmtors ;,,3,,36 4) At α ρ ;,,3,,36, the mes squred λ error of the proposed estmtor p ;,,3,,36 equl to the vrce of the usul ler regresso estmtor r 3

14 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Numercl Stud The performce of the proposed geerlzed modfed rto estmtor s ssessed wth tht of the SRSWOR smple me, the usul rto estmtor d the estg modfed rto estmtors lsted Tble for cert turl popultos. I ths coecto, four turl popultos for the ssessmet of the performce of the proposed estmtors wth tht of estg estmtors were cosdered. Populto s tke from Sgh d hudhr (986) gve pge 08; populto d populto 3 re tke from Sgh d hudhr (986) gve pge 77; populto 4 s tke from ochr (977) gve pge 5. The populto prmeters d the costts computed from the bove popultos re gve below Tble, wheres the rge of α whch proposed estmtor performs better th the estg estmtors, the costts, the bses d the me squred errors of the estg d proposed estmtors for the bove popultos re respectvel gve from the Tbles 3 to 8. Tble. Prmeters d costts of the populto Prmeters Populto Populto Populto 3 Populto 4 N ρ S S β () β () M d

15 JAMBULINGAM SUBRAMANI Tble 3. Rge of α whch proposed estmtor performs better th the usul rto estmtor α rge (α L, α u) Estmtor Populto Populto Populto 3 Populto 4 (-, ) (-, ) (-, ) (-, ) p (-, 57.60) (-, ) (-, ) (-,.0738) p (-, ) (-, ) (-, ) (-, ) p3 (-, ) (-, ) (-, ) (-, ) p4 (-, 6.979) (-, 8.65) (-, 06.77) (-, ) p5 (-, 8.40) (-, 8.330) (-,.0853) (-, ) p6 (-, ) (-, ) (-, ) (-, ) p7 (-, 6.95) (-, ) (-, ) (-,.099) p8 (-, ) (-, ) (-, 58.50) (-, ) p9 (-, ) (-, ) (-, ) (-, 5.994) p0 (-, ) (-, ) (-, ) (-, 4.076) p (-, ) (-, ) (-, ) (-, ) p (-, ) (-, ) (-, ) (-, ) p3 (-, ) (-,.88) (-, 79.80) (-, ) p4 (-, ) (-, ) (-, ) (-, ) p5 (-, 6.478) (-,.979) (-, ) (-, ) p6 (-, ) (-, ) (-, ) (-, ) p7 (-, ) (-, ) (-, ) (-, ) p8 (-, 4.405) (-, ) (-, 9.604) (-,.0359) p9 (-, ) (-, ) (-, ) (-, 8.797) p0 (-, ) (-, ) (-, ) (-, 5.509) p 33

16 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 3 cotued. α rge (α L, α u) Estmtor Populto Populto Populto 3 Populto 4 (-, ) (-, ) (-, ) (-, 0.07) p (-, ) (-, ) (-, ) (-, 3.705) p3 (-, ) (-, ) (-, 40.38) (-, 0.88) p4 (-, 48.84) (-, ) (-, ) (-, ) p5 (-, ) (-, ) (-, ) (-, ) p ) (-, ) (-, ) (-, ) p7 (-, ) (-, ) (-, ) (-, ) p8 (-, ) (-, ) (-, ) (-, ) p9 (-, 7.083) (-, 7.908) (-, ) (-, -0.45) p30 (-, ) (-, ) (-, ) (-, ) p3 (-, 4.839) (-, 7.654) (-, ) (-, ) p3 (-, ) (-, ) (-, ) (-, ) p33 (-, 5.740) (-, 7.686) (-, ) (-, -0.76) p34 (-, ) (-, ) (-, ) (-, ) p35 (-, ) (-, ) (-, ) (-, 4.409) p36 34

17 JAMBULINGAM SUBRAMANI Tble 4. Rge of α whch proposed estmtor performs better th the estg modfed rto estmtors α rge (α L, α u) Estmtor Populto Populto Populto 3 Populto 4 w.r.t. (0, ) (0, 383.0) (0, ) (0, 3.09) p w.r.t. (0, 6.33) (0, ) (0, 5.048) (-0.077, 0) p w.r.t. (0, ) (0, ) (0, 9.584) (0, 4.645) p3 3 w.r.t. (0, ) (0, ) (0, ) (0, 3.568) p4 4 w.r.t. (-0.0, 0) (0, ) (0, ) (-.678, 0) p5 5 w.r.t. (0, 0.07) (0, ) (0, 0.863) (-.34, 0) p6 6 w.r.t. (0, ) (0, ) (0, ) (0, 3.68) p7 7 w.r.t. (0, ) (0, 7.47) (0, 4.080) (-0.048, 0) p8 8 w.r.t. (0, ) (0, ) (0, ) (0, 3.398) p9 9 w.r.t. (0, 4.500) (0, ) (0, ) (0, 4.753) p0 0 w.r.t. (0, ) (0, ) (0, ) (0,.806) p w.r.t. (0, ) (0, ) (0, ) (0, 3.76) p w.r.t. (0, ) (0, ) (0, ) (0, ) p3 3 w.r.t. (-0.45, 0) (-0.370, 0) ( , 0) (-.694, 0) p4 4 w.r.t. (0, ) (0, ) (0, ) (0, ) p5 5 w.r.t. (-0.0, 0) (-0.37, 0) (0, 0.058) (-.350, 0) p6 6 w.r.t. (0, 6.005) (0, ) (0, 35.48) (0,.438) p7 7 w.r.t. (0, ) (0, 6.788) (0, 89.86) (0, ) p8 8 w.r.t. (0, ) (0, ) (0, 4.368) (-0.066, 0) p9 9 w.r.t. (0, ) (0, ) (0, ) (0, ) p0 0 w.r.t. (0, ) (0, ) (0, ) (0, 4.880) p 35

18 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 4 cotued. α rge (α L, α u) Estmtor Populto Populto Populto 3 Populto 4 w.r.t. (0, ) (-0.996, 0) (0,.4756) ( , 0) p w.r.t. (0, 973.9) (0, ) (0, 954.0) (0, ) p3 3 w.r.t. (0, ) (-0.995, 0) (0, 3.903) (-0.968, 0) p4 4 w.r.t. (0, ) (0, ) (0, ) (0, 4.565) p5 5 w.r.t. (0, ) (0, ) (0, ) (0, ) p6 6 w.r.t. (0, ) (0, ) (0, ) (0, ) p7 7 w.r.t. (0,.599) (0, ) (0,.34) (-.3376, 0) p8 8 w.r.t. (0, ) (0, ) (0, ) (0, ) p9 9 w.r.t. (0, 3.704) (0, 0.446) (0,.6000) (-.834, 0) p30 30 w.r.t. (0, ) (0, ) (0, ) (0, ) p3 3 w.r.t. (-0.530, 0) (-0.70, 0) ( , 0) (-.65, 0) p3 3 w.r.t. (0, ) (0, ) (0, ) (0, 03.35) p33 33 w.r.t. (-0.366, 0) ( , 0) (-0.633, 0) (-.35, 0) p34 34 w.r.t. (0, 95.66) (0, ) (0, ) (0, 7.673) p35 35 w.r.t. (0, 48.86) (0, ) (0, ) (0, 3.08) p

19 JAMBULINGAM SUBRAMANI Tble 5. ostt, Bs d Me squred error of the Estg d Proposed estmtors for Populto Estmtor B ( (.) ) (.) MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α srs R p p p p p p p p p p p p p p p p p p p9 p 37

20 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 5 cotued. Estmtor p0 ( ) B (.) (.) ( MSE MSE ) ( ) p t α & α L u Bs p t α ( p ) t α MSE t α p p p p p p p p p p p p p p p p36 p 38

21 JAMBULINGAM SUBRAMANI Tble 6. ostt, Bs d Me squred error of the Estg d Proposed estmtors for Populto Estmtor ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α srs R p p p p p p p p p p p p p p p p p p p9 p 39

22 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 6 cotued Estmtor p0 ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α p p p p p p p p p p p p p p p p36 p 40

23 JAMBULINGAM SUBRAMANI Tble 7. ostt, Bs d Me squred error of the Estg d Proposed estmtors for Populto 3 Estmtor ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α srs R p p p p p p p p p p p p p p p p p p p0 p 4

24 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 7 cotued Estmtor p ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α p p p p p p p p p p p p p p p36 p 4

25 JAMBULINGAM SUBRAMANI Tble 8. ostt, Bs d Me squred error of the Estg d Proposed estmtors for Populto 4 Estmtor ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α srs R p p p p p p p p p p p p p p p p p p p9 p 43

26 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 8 cotued Estmtor p0 ( ) B MSE MSE ( ) ( ) p t α & α L u Bs α ( ) p t t α p MSE t α p p p p p p p p p p p p p p p p36 p From the vlues of Tble 5 Tble 8, t s observed tht the bs of the proposed modfed rto estmtor p j ; j,,, 36 s less th the bs of the usul rto estmtor d the estg modfed rto estmtors ;,,3,,36. Smlrl, the me squred error of the proposed modfed rto estmtor p j ; j,,, 36 44

27 JAMBULINGAM SUBRAMANI s less th the vrce of SRSWOR smple me, the me squred error of the usul rto estmtor d the estg modfed rto estmtors p j ; j,,, 36 for ll four popultos. ocluso I ths rtcle, geerlzed modfed rto estmtor hs bee suggested usg the kow populto prmeters of the ulr vrble. Moreover, m modfed rto estmtors hve bee troduced ths rtcle, d hve ot bee dscussed erler the lterture. The bs d me squred error of the proposed geerlzed modfed rto estmtor re obted. Furthermore, the codtos hve bee derved for whch the proposed estmtor s more effcet th the estg estmtors, d t s show tht the SRSWOR smple me, the usul rto estmtor, the ler regresso d the estg modfed rto estmtors re prtculr cses of the proposed estmtor. The performces of the proposed estmtor re lso ssessed for some kow popultos. It s observed tht the bs d the me squred errors of the proposed estmtors re less th the bs d the me squred error of the estg estmtors. Moreover, the proposed estmtor wll be geerlzed modfed rto estmtor for estmtg the populto me of the stud vrble usg the kow populto prmeters of the ulr vrble. Ackowledgemets The uthor wshes to record hs grttude d thks to Uverst Grts ommsso (UG) for the fcl ssstce through UG-Mjor Reserch Project. Refereces Abd,. Z., & Shhbz, M. Q. (006). A comprtve stud of geerlzed rto d regresso estmtors wth ther clsscl couterprts. Pkst Jourl of Sttstcs d Operto Reserch, (), Ahmd, Z., Hf, M., & Ahmd, M. (009). Geerlzed regresso-cumrto estmtors for two-phse smplg usg mult-ulr vrbles. Pkst Jourl of Sttstcs, 5(),

28 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Al-jrrh, J., & Al-Hj Ebrhem, M. (0). A rto estmtor uder geerl smplg desg. Austr Jourl of Sttstcs, 4(), Bhush, S. (0). Some effcet smplg strteges bsed o rto-tpe estmtor. Electroc Jourl of Appled Sttstcl Alss, 5(), ochr, W. G. (977). Smplg Techques. Thrd Edto, Wle Ester Lmted. Dlbeher, M. & Shoo, L. N. (994). omprso of s lmost ubsed rto estmtors. QUESTIIO, 8(3), Dvd, I. P., & Sukhtme, B.V. (974). O the bs d me squre error of the rto estmtor. Jourl of the Amerc Sttstcl Assocto, Theor d Methods Secto, 69(346): Goodm, L. A., & Hrtle, H. O. (958). The precso of ubsed rtotpe estmtors. Jourl of the Amerc Sttstcl Assocto, 53(8): Gupt, S., & Shbbr, J. (008). O mprovemet estmtg the populto me smple rdom smplg. Jourl of Appled Sttstcs, 35(5), Jhjj, H.S., Shrm, M. K., & Grover, L.K. (006). Dul of rto estmtors of fte populto me obted o usg ler trsformto to ulr vrble. Jourl of Jp Sttstcl Socet, 36(), Kdlr,., & g, H. (003). A stud o the ch rto tpe estmtor. Hcettepe Jourl of Mthemtcs d Sttstcs, Vol. 3, Kdlr,., & g, H. (004). Rto estmtors smple rdom smplg. Appled Mthemtcs d omputto, 5, Kdlr,., & g, H. (006). A mprovemet estmtg the populto me b usg the correlto co-effcet. Hcettepe Jourl of Mthemtcs d Sttstcs, 35(), Kdlr,., & g, H. (006b). Improvemet estmtg the populto me smple rdom smplg. Appled Mthemtcs Letters, 9, Khoshevs, M., Sgh, R., huh, P., Sw, N., & Smrdche, F. (007). A geerl fml of estmtors for estmtg populto me usg kow vlue of some populto prmeter(s). Fr Est Jourl of Theoretcl Sttstcs,, 8-9. Koucu, N., & Kdlr,. (009). Effcet Estmtors for the Populto Me. Hcettepe Jourl of Mthemtcs d Sttstcs, 38(),

29 JAMBULINGAM SUBRAMANI Kulkr, S. P. (978). A ote o modfed rto estmtor usg trsformto. Jourl of the Id Socet of Agrculturl Sttstcs, 30(), 5 8. Murth, M. N. (967). Smplg Theor d Methods. lcutt, Id: Sttstcl Publshg Socet. Nk, V. D., & Gupt, P.. (99): A geerl clss of estmtors for estmtg populto me usg ulr formto. Metrk, 38, 7. Olk, I. (958). Multvrte rto estmto for fte popultos. Bometrk, 45, Pthk, P.K. (964). O smplg schemes provdg ubsed rto estmtors. The Als of Mthemtcl Sttstcs, 35(), -3. Perr, P. F. (007). Improved rto-cum-product tpe estmtors. Sttstcs Trsto, 8(), R, S. K., & Sh, A. (980). Effcet fmles of rto d product-tpe estmtors. Bometrk, 67, 5. Redd, V. N. (973). O rto d product methods of estmto. Skh B, 35(3), Robso, J. (987). odtog rto estmtes uder smple rdom smplg. Jourl of the Amerc Sttstcl Assocto, 8 (399), Se, A. R. (993). Some erl developmets rto estmto. Bometrcl Jourl, 35(), 3-3. Shbbr, J., & b, M. Z. (003). Improvemet over trsformed ulr vrble estmtg the fte popultos me. Bometrcl Jourl, 45(6), Shrm, B., & Tlor, R. (00). A ew rto-cum-dul to rto estmtor of fte populto me smple rdom smplg. Globl Jourl of Scece Froter Reserch, 0(), 7-3. Sgh, D., & hudhr, F. S. (986). Theor d lss of smple surve desgs. New Delh: New Age Itertol Publsher. Sgh, G. N. (003). O the mprovemet of product method of estmto smple surves. Jourl of the Id Socet of Agrculturl Sttstcs, 56(3), Sgh, H. P. & Espejo, M. R. (003). O ler regresso d rto-product estmto of fte populto me. The Sttstc, 5,

30 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Sgh, H. P., & Aghotr, N. (008). A geerl procedure of estmtg populto me usg ulr formto smple surves. Sttstcs Trsto, 9(), Sgh, H. P., & Solk, R. S. (0). A ltertve procedure for estmtg the populto me smple rdom smplg. Pkst Jourl of Sttstcs d Operto Reserch, 8(), 3-3. Sgh, H. P., & Tlor, R. (003). Use of kow correlto co-effcet estmtg the fte populto mes. Sttstcs Trsto, 6 (4), Sgh, H. P., & Tlor, R. (005). Estmto of fte populto me wth kow co-effcet of vrto of ulr. Sttstc, o LV,.3, pp Sgh, H. P., Tlor, R., Sgh. S., & Km, J. M. (008). A modfed estmtor of populto me usg power trsformto. Sttstcl Ppers, 49, Sgh, H. P., Tlor, R., Tlor, R. d Kkr, M. S. (004): A Improved Estmtor of Populto Me Usg Power Trsformto. Jourl of the Id Socet of Agrculturl Sttstcs, 58(), Ssod, B. V. S., & Dwved, V. K. (98). A modfed rto estmtor usg co-effcet of vrto of ulr vrble. Jourl of the Id Socet of Agrculturl Sttstcs, 33(), 3-8. Solk, R. S., Sgh, H. P., & Rthour, A. (0). A ltertve estmtor for estmtg the fte populto me usg ulr formto smple surves. ISRN Probblt d Sttstcs, Artcle ID 65768, 4 pp. Srvektrm, T. (980). A dul to rto estmtor smple surves. Bometrk, 37, Subrm, J., & Kumrpd, G. (0). Modfed rto estmtors usg kow med d co-effcet of kurtoss. Amerc Jourl of Mthemtcs d Sttstcs, (4), Subrm, J., & Kumrpd, G. (0b). Estmto of populto me usg kow med d co-effcet of skewess. Amerc Jourl of Mthemtcs d Sttstcs. (5), Subrm, J., & Kumrpd, G. (0c). Estmto of populto me usg co-effcet of vrto d med of ulr vrble, Itertol Jourl of Probblt d Sttstcs, (4), -8. Subrm, J., & Kumrpd, G. (03). A ew modfed rto estmtor for estmto of populto me whe med of the ulr 48

31 JAMBULINGAM SUBRAMANI vrble s kow. Pkst Jourl of Sttstcs d Operto Reserch, 9(), Subrm, J., & Kumrpd, G. (03b). Estmto of Populto Me Usg Kow orrelto o-effcet d Med. Jourl of Sttstcl Theor d Applctos (to pper). Tlor, R., & Shrm, B. (009). A modfed rto-cum-product estmtor of fte populto me usg kow coeffcet of vrto d coeffcet of kurtoss. Sttstcs Trsto-New Seres, 0(), 5-4. T, M. (965). omprso of some rto estmtors. Jourl of the Amerc Sttstcl Assocto, 60, Updh, L. N., & Sgh, H.P. (999). Use of trsformed ulr vrble estmtg the fte populto me. Bometrcl Jourl, V4(5), , Z., & T, B. (00). Rto method to the me estmto usg coeffcet of skewess of ulr vrble. IIA 00, Prt II, IS 06, pp

32 JAMBULINGAM SUBRAMANI Apped A A epresso for the bs d me squred error of the proposed estmtors p j ;,,3,,36 ws derved to frst order of ppromto wth the followg ottos: Let us defe e 0 d e. Further, ( + e0 ) d + e d from the defto of e 0 d e : Ee 0 ( ) [ 0] Ee [ ] E e 0 E e S S S E[ ee 0 ] ρ where, d ρ SS The bs of clss of proposed estmtors p ;,,3,,36 s derved d s: p ( + ( + α) λ );,,3,, α λ ( ) + + p ( + e + ( + α) λ ) ( ( α) λ) p + + e ( + ( + α) λ ) + ( + ( + α) λ ) p where p + e + + α λ ( p ) ( ) p 3 3 ( ) + e p e + e e + p p p p Neglectg the terms more th d order, results ( ( α) λ) 50

33 JAMBULINGAM SUBRAMANI ( ) ( ( 0) )( ) ( 0)( ) e + e p p p + e e + e p p p + e e + e p p p p + e 0 pe pe 0e+ pe + pe 0e Neglectg the terms more th 3 rd order, results p + e0 pe pe 0e+ pe p e 0 pe pe 0e+ pe Tkg epectto o both sdes, results E E e E e E e e + E e ( p ) 0 p p 0 p Bs ( p ) p E e p E e e 0 Bs( p ) p p ρ ( Bs p p p ρ ) Bs ( p ) ( ) where p p ρ p + + α λ ( ) The me squred error of the proposed estmtor p ;,,3,,36 to frst order of ppromto s derved d s: p ( + ( + α) λ );,,3,, α λ ( ) + + p ( + e + ( + α) λ ) ( ( α) λ) p + + e ( + ( + α) λ ) + ( + ( + α) λ ) p where p + e + + α λ ( p ) ( ( α) λ) 5

34 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN + e p ( ) p 3 3 ( ) e + e e + p p p p Neglectg the terms more th st order, results e p ( p ) ( ( 0) )( p ) ( 0)( p ) + e e p + e e p + e e e e e e e e p 0 p p 0 p 0 p p 0 Squrg both sdes ( p ) ( e0 0 ) pe pe e Neglectg the terms more th d order, results e e e e p 0 p p 0 Tkg epectto o both sdes results : ( p ) + 0 p p 0 f ( ) ρ E E e E e E e e MSE p + p p ;,,3,,36 where p + + α λ Apped B ( ) The codtos for whch proposed estmtor p perform better th the SRSWOR smple me re derved d re gve below: MSE V For pj r 5

35 JAMBULINGAM SUBRAMANI ( ( ( + p ) ρ p S ( + ρ ) ( + p ) ρ p ρ p p p p ρ p ρ p ρ Tht s, MSE ( ) V ) Apped p p f ρ p r p The codtos for whch proposed estmtor p perform better th the usul rto estmtor re derved d re gve below: MSE MSE For ( pj ) ( R) ( ( + p ) ( ) ρ p + ρ ( + p ρ p ) ( + ρ ) ρ ρ p p ρ + ρ 0 p p ( p ) ρ p ( ) ( ) ρ odto : 0 ( ) + 0 p p 0 d + ρ 0 p p 53

36 MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN d + ρ p p ρ p d p ρ p odto : d ( + ) ρ ( ) 0 d + ρ 0 p p p p ρ p d p ρ p Tht s, MSE ( p ) MSE ( R) Apped D ether ρ (or) ρ p p The codtos for whch proposed estmtor p perform better th the estg modfed rto estmtors (lss ) re derved d re gve below: MSE MSE ;,,3,,36 For pj ( ( + p ) ( ) ρ p + ρ ( + p ρ p ) ( + ρ ) ρ ρ p p ρ + ρ 0 p p ( p ) 0 ρp p ( p )( ) 0 p + ρ odto : ( p ) ( ) p ρ 0 d

37 JAMBULINGAM SUBRAMANI d + ρ p p ρ p d p ρ p odto : d ( + ) ρ ( ) 0 d + ρ 0 p p p p ρ p d p ρ p MSE MSE ether Tht s, ( p ) ρ ρ p (or) p 55