Jourl of Moder Appled Sttstcl Methods Volume Issue Artcle 7 --03 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: http://dgtlcommos.we.edu/jmsm 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/38378760 Avlble t: http://dgtlcommos.we.edu/jmsm/vol/ss/7 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.
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: http://dgtlcommos.we.edu/jmsm/vol/ ss/7
Jourl of Moder Appled Sttstcl Methods November 03, Vol., No., -55. oprght 03 JMASM, Ic. ISSN 538 947 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..
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:
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
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
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
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 ) + 30 30ρ ) 6
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 ) + 33 33ρ ) ρ + M d 34 ρ + Md Subrm d Kumrpd (03b) ρ 34 ρ + M d ( 34 34ρ ) + 34 34ρ ) 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 + 35 35ρ ) + 36 36ρ ) 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
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
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,,36 + + 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
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
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
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 70 34 34 49 5 0 0 0 96.7000 856.48 85.64 7.7959 75.67 08.884 9.944 03.49 ρ 0.793 0.449 0.4453 0.987 S 60.474 733.407 73.34 3. 0.654 0.856 0.856 0.9634 S 40.857 50.5060 5.05 04.405 0.8037 0.705 0.753.0 β () 7.095 0.0974 3.757 7.54 β ().9507 0.978.83.553 M d.5000 50.0000 4.500 64.0000 3
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 (-, 396.64) (-, 403.475) (-, 38.496) (-, 4.3885) p (-, 57.60) (-, 975.6396) (-, 43.568) (-,.0738) p (-, 574.6399) (-, 963.549) (-, 36.997) (-, 5.9068) p3 (-, 538.6966) (-, 6454.993) (-, 367.830) (-, 4.8665) p4 (-, 6.979) (-, 8.65) (-, 06.77) (-, -0.8508) p5 (-, 8.40) (-, 8.330) (-,.0853) (-, -0.7566) p6 (-, 99.584) (-, 39.699) (-, 7970.040) (-, 4.5893) p7 (-, 6.95) (-, 436.6645) (-, 34.779) (-,.099) p8 (-, 74.4479) (-, 3935.639) (-, 58.50) (-, 33.7053) p9 (-, 46.648) (-, 34.834) (-, 05.6054) (-, 5.994) p0 (-, 07.957) (-, 806.368) (-, 95.756) (-, 4.076) p (-, 36.454) (-, 4650.7357) (-, 74.709) (-, 5.0607) p (-, 96799.666) (-, 605657.49) (-, 337.3735) (-, 605.6390) p3 (-, 5.4070) (-,.88) (-, 79.80) (-, -0.8490) p4 (-, 69754.435) (-, 6036.59) (-, 30486.7557) (-, 983.8649) p5 (-, 6.478) (-,.979) (-, 84.758) (-, -0.7536) p6 (-, 307.77) (-, 90.8696) (-, 50.3764) (-, 3.6769) p7 (-, 4083.80) (-, 87.8790) (-, 5077.098) (-, 50.8800) p8 (-, 4.405) (-, 336.58) (-, 9.604) (-,.0359) p9 (-, 093.4555) (-, 68.634) (-, 348.6757) (-, 8.797) p0 (-, 9.3463) (-, 4477948.800) (-, 6496.03) (-, 5.509) p 33
MODIFIED RATIO FOR ESTIMATION OF FINITE POPULATION MEAN Tble 3 cotued. α rge (α L, α u) Estmtor Populto Populto Populto 3 Populto 4 (-, 55.563) (-, 0.8775) (-, 398.688) (-, 0.07) p (-, 97.8366) (-, 446894.9330) (-, 66.5069) (-, 3.705) p3 (-, 64.5737) (-, 0.8838) (-, 40.38) (-, 0.88) p4 (-, 48.84) (-, 330.3074) (-, 605.940) (-, 5.7807) p5 (-, 300.486) (-, 634.000) (-, 477.5430) (-, 34.7834) p6 808.043) (-, 44637.055) (-, 0473.799) (-, 70.090) p7 (-, 4.5508) (-, 7.8444) (-, 5.8339) (-, -0.6635) p8 (-, 69939.477) (-, 444637.376) (-, 94.638) (-, 44.0376) p9 (-, 7.083) (-, 7.908) (-, 3.7007) (-, -0.45) p30 (-, 6876.3557) (-, 97664.4899) (-, 54359.58) (-, 655.5450) p3 (-, 4.839) (-, 7.654) (-, 46.7706) (-, -0.8535) p3 (-, 8707.40) (-, 968397.9653) (-, 5567.3306) (-, 04.4570) p33 (-, 5.740) (-, 7.686) (-, 49.3569) (-, -0.76) p34 (-, 967.5869) (-, 888.7708) (-, 57.7007) (-, 8.5485) p35 (-, 300.7996) (-, 908.988) (-, 697.704) (-, 4.409) p36 34
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, 343.3398) (0, 383.0) (0, 57.768) (0, 3.09) p w.r.t. (0, 6.33) (0, 953.809) (0, 5.048) (-0.077, 0) p w.r.t. (0, 54.473) (0, 757.363) (0, 9.584) (0, 4.645) p3 3 w.r.t. (0, 485.690) (0, 640.8577) (0, 68.9984) (0, 3.568) p4 4 w.r.t. (-0.0, 0) (0, 0.4679) (0, 0.6773) (-.678, 0) p5 5 w.r.t. (0, 0.07) (0, 0.4777) (0, 0.863) (-.34, 0) p6 6 w.r.t. (0, 9857.594) (0, 49.4093) (0, 433.435) (0, 3.68) p7 7 w.r.t. (0, 87.6545) (0, 7.47) (0, 4.080) (-0.048, 0) p8 8 w.r.t. (0, 670.6504) (0, 375.5608) (0, 69.738) (0, 3.398) p9 9 w.r.t. (0, 4.500) (0, 934.048) (0, 35.4434) (0, 4.753) p0 0 w.r.t. (0, 965.8454) (0, 608.9539) (0, 7.6363) (0,.806) p w.r.t. (0, 83.888) (0, 4439.3079) (0, 787.755) (0, 3.76) p w.r.t. (0, 96744.7709) (0, 605435.848) (0, 6599.484) (0, 604.348) p3 3 w.r.t. (-0.45, 0) (-0.370, 0) (-0.0477, 0) (-.694, 0) p4 4 w.r.t. (0, 69699.5893) (0, 603399.7594) (0, 4999.754) (0, 98.5405) 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, 888.0870) (0, 35.48) (0,.438) p7 7 w.r.t. (0, 409.336) (0, 6.788) (0, 89.86) (0, 49.5634) p8 8 w.r.t. (0, 77.050) (0, 343.6647) (0, 4.368) (-0.066, 0) p9 9 w.r.t. (0, 0868.8640) (0, 464.746) (0, 9009.665) (0, 6.8585) p0 0 w.r.t. (0, 36.679) (0, 447777.3738) (0, 399.5909) (0, 4.880) p 35
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, 7.4608) (-0.996, 0) (0,.4756) (-0.9064, 0) p w.r.t. (0, 973.9) (0, 446673.496) (0, 954.0) (0, 30.3989) p3 3 w.r.t. (0, 34.4533) (-0.995, 0) (0, 3.903) (-0.968, 0) p4 4 w.r.t. (0, 370.96) (0, 40.396) (0, 49.3999) (0, 4.565) p5 5 w.r.t. (0, 948.590) (0, 600.05) (0, 667.5443) (0, 33.4704) p6 6 w.r.t. (0, 807.00) (0, 44595.77) (0, 549.989) (0, 78.7848) p7 7 w.r.t. (0,.599) (0, 0.4053) (0,.34) (-.3376, 0) p8 8 w.r.t. (0, 69884.49) (0, 44445.8999) (0, 4444.804) (0, 439.737) p9 9 w.r.t. (0, 3.704) (0, 0.446) (0,.6000) (-.834, 0) p30 30 w.r.t. (0, 68.5087) (0, 97443.098) (0, 4840.398) (0, 654.04) p3 3 w.r.t. (-0.530, 0) (-0.70, 0) (-0.6684, 0) (-.65, 0) p3 3 w.r.t. (0, 8707.953) (0, 96876.5684) (0, 45644.6094) (0, 03.35) p33 33 w.r.t. (-0.366, 0) (-0.7090, 0) (-0.633, 0) (-.35, 0) p34 34 w.r.t. (0, 95.66) (0, 683.093) (0, 83.533) (0, 7.673) p35 35 w.r.t. (0, 48.86) (0, 70.4493) (0, 34.789) (0, 3.08) p36 36 36
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 - - 94.0466 94.0466 - - - R - 0.6946 73.0773 73.0773 - - - 0.9954 0.684 7.4673 7.4673 0.448 0.69 60.973 p 0.96 0.6076 68.0853 68.0853 0.945 0.9 55.598 p 0.9890 0.6695 7.673 7.673 0.545 0.79 59.45 p3 0.9959 0.685 7.53 7.53 0.44 0.68 60.60 p4 0.5544 0.06 44.058 44.058 0.567 0.0003 44.053 p5 0.5906 0.09 44.080 44.080 0.5666 0.0009 44.054 p6 0.9994 0.693 7.9906 7.9906 0.389 0.6 60.7979 p7 0.950 0.5880 66.999 66.999 0.3069 0.85 54.570 p8 0.9977 0.6893 7.763 7.763 0.45 0.64 60.5354 p9 0.9863 0.6635 7.7 7.7 0.584 0.83 58.8657 p0 0.9938 0.6803 7.439 7.439 0.474 0.7 59.9443 p 0.9948 0.688 7.3894 7.3894 0.457 0.70 60.089 p.0000 0.6946 73.079 73.079 0.380 0.60 60.8934 p3 0.5000 0.054 44.738 44.738 0.5595 0.007 44.0353 p4.0000 0.6946 73.07 73.07 0.380 0.60 60.895 p5 0.5369 0.064 44.707 44.707 0.5659 0.005 44.057 p6 0.9797 0.6485 70.40 70.40 0.68 0.89 57.945 p7 0.9984 0.69 7.867 7.867 0.403 0.63 60.6553 p8 0.9474 0.578 66.446 66.446 0.33 0.79 54.0709 p9 p 37
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 0.9994 0.6933 7.9986 7.9986 0.388 0.6 60.807 0.9997 0.6940 73.0387 73.0387 0.383 0.60 60.8536 p 0.8983 0.477 6.060 6.060 0.3747 0.60 49.7965 p 0.9997 0.6939 73.035 73.035 0.384 0.60 60.8465 p3 0.90 0.506 6.3499 6.3499 0.3596 0.0 50.738 p4 0.9850 0.6604 7.099 7.099 0.604 0.84 58.67 p5 0.9979 0.6898 7.790 7.790 0.4 0.64 60.5687 p6 0.9999 0.6945 73.0667 73.0667 0.380 0.60 60.886 p7 0.708 0.60 47.006 47.006 0.536 0.098 44.43 p8 0.9999 0.6944 73.0650 73.0650 0.380 0.60 60.884 p9 0.7378 0.08 48.596 48.596 0.564 0.044 44.4309 p30.0000 0.6946 73.0733 73.0733 0.379 0.60 60.8938 p3 0.4757-0.070 45.333 45.333 0.557 0.03 44.0594 p3.0000 0.6946 73.077 73.077 0.380 0.60 60.893 p33 0.57 0.045 44.49 44.49 0.56 0.0048 44.096 p34 0.9934 0.6796 7.0 7.0 0.478 0.7 59.896 p35 0.995 0.6834 7.430 7.430 0.453 0.69 60.47 p36 p 38
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 - - 066.0800 066.0800 - - - R - 4.694 0539.700 0539.700 - - - 0.9966 4.33 054.50 054.50 0.39 0.485 0098.8070 p 0.9995 4.63 0535.860 0535.860 0.68 0.47 03.590 p 0.9953 4.070 0505.3560 0505.3560 0.340 0.4903 0085.4900 p3 0.9979 4.406 053.670 053.670 0.97 0.4795 0.9870 p4 0.58 0.533 885.750 885.750 0.594 0.006 8834.090 p5 0.580 0.58 885.340 885.340 0.59 0.03 8834.00 p6 0.9658 3.8 098.4430 098.4430 0.835 0.588 9794.7990 p7 0.9994 4.607 0534.540 0534.540 0.7 0.479 09.5790 p8 0.9965 4.3 053.6700 053.6700 0.3 0.4854 0097.970 p9 0.9935 4.83 049.3780 049.3780 0.37 0.4977 0066.90 p0 0.994 4.676 0483.9890 0483.9890 0.39 0.504 0053.6950 p 0.9970 4.95 057.5830 057.5830 0.3 0.483 003.8680 p.0000 4.69 0539.030 0539.030 0.60 0.470 036.5390 p3 0.5000 0.538 884.8000 884.8000 0.535 0.003 8833.9850 p4.0000 4.69 0539.00 0539.00 0.60 0.470 036.5380 p5 0.5008 0.50 884.360 884.360 0.536 0.0098 8833.980 p6 0.9995 4.630 0535.7860 0535.7860 0.68 0.47 03.4760 p7 0.954 3.673 00.4740 00.4740 0.0 0.633 9695.0 p8 0.9990 4.555 053.6900 053.6900 0.77 0.4746 05.380 p9 p 39
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 0.9784 3.9839 0385.0680 0385.0680 0.68 0.556 99.890.0000 4.693 0539.480 0539.480 0.59 0.4700 036.7600 p 0.9 0.450 080.5970 080.5970 0.7 0.638 9646.3850 p.0000 4.693 0539.480 0539.480 0.59 0.4700 036.7600 p3 0.95 0.4530 078.990 078.990 0. 0.643 9643.640 p4 0.9897 4.38 0464.6450 0464.6450 0.438 0.530 005.640 p5 0.9978 4.400 053.690 053.690 0.98 0.4797 0.4600 p6.0000 4.690 0539.0430 0539.0430 0.60 0.470 036.4470 p7 0.5758 0.6 8847.930 8847.930 0.5303 0.06 8834.0370 p8.0000 4.690 0539.040 0539.040 0.60 0.470 036.4460 p9 0.5767 0.73 8848.480 8848.480 0.530 0.069 8834.0440 p30.0000 4.69 0539.660 0539.660 0.60 0.4700 036.6350 p3 0.3840 0.559 9009.4490 9009.4490 0.496 0.888 8847.7480 p3.0000 4.69 0539.660 0539.660 0.60 0.4700 036.6350 p33 0.3848 0.54 9007.5850 9007.5850 0.49 0.870 8847.4570 p34 0.995 4.054 0504.490 0504.490 0.343 0.4908 0084.940 p35 0.9953 4.058 0504.70 0504.70 0.34 0.4906 0084.5400 p36 p 40
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 - - 379.4085 379.4085 - - - R -.6938 375.879 375.879 - - - 0.9636.59 365.6490 365.6490 0.096 0.33 354.406 p 0.846 0.973 337.49 337.49 0.84 0.67 38.8736 p 0.9440.478 360.507 360.507 0.7 0.653 346.3464 p3 0.978.5835 369.68 369.68 0.0658 0.0994 36.08 p4 0.5704 0.57 305.3883 305.3883 0.4979 0.039 304.944 p5 0.5833 0.543 305.99 305.99 0.4944 0.099 304.54 p6 0.9900.647 37.936 37.936 0.0435 0.069 367.006 p7 0.803 0.8 39.76 39.76 0.3340 0.97 3.877 p8 0.9690.5385 367.90 367.90 0.087 0.0 356.8370 p9 0.970.3383 356.36 356.36 0.560 0.873 340.698 p0 0.98.34 354.938 354.938 0.647 0.98 338.484 p 0.97.549 367.7088 367.7088 0.0787 0.54 357.885 p 0.9975.680 375.09 375.09 0.090 0.0475 37.034 p3 0.5000 0.005 304.857 304.857 0.5060 0.000 304.748 p4 0.9974.6803 375.053 375.053 0.093 0.0479 370.9498 p5 0.53 0.04 304.895 304.895 0.5060 0.000 304.748 p6 0.8636.0586 34.7007 34.7007 0.537 0.96 3.894 p7 0.9843.644 37.3460 37.3460 0.054 0.084 364.476 p8 0.9940.6634 374.000 374.000 0.0357 0.0576 369.66 p0 p 4
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 0.9877.634 37.986 37.986 0.0478 0.075 365.8606 0.838 0.99 335.3364 335.3364 0.965 0.3 37.089 p 0.987.68 37.30 37.30 0.049 0.0769 365.550 p3 0.839 0.958 336.7384 336.7384 0.87 0.57 38.694 p4 0.885.39 345.787 345.787 0.6 0.7 37.90 p5 0.985.6000 370.545 370.545 0.0597 0.093 36.796 p6 0.996.6737 374.680 374.680 0.038 0.057 370.54 p7 0.609 0.94 307.397 307.397 0.4844 0.0360 304.38 p8 0.9959.676 374.60 374.60 0.03 0.053 370.38 p9 0.633 0.505 308.09 308.09 0.4790 0.0446 304.39 p30 0.9985.686 375.3879 375.3879 0.070 0.0444 37.58 p3 0.376 0.75 309.497 309.497 0.4703 0.0577 304.5507 p3 0.9984.6858 375.3647 375.3647 0.07 0.0447 37.5383 p33 0.3839 0.609 308.5583 308.5583 0.4766 0.048 304.430 p34 0.9498.445 36.9937 36.9937 0.7 0.563 348.600 p35 0.9546.468 363.505 363.505 0.087 0.48 350.567 p36 p 4
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 - - 448.5780 448.5780 - - - R - 0.54 8.369 8.369 - - - 0.9903 0.44 7.7773 7.7773 0.93 0.0 6.363 p 0.93 0.008 6.333 6.333 0.9344 0.0000 6.307 p 0.9786 0.676 7.984 7.984 0.933 0.0076 6.39 p3 0.9906 0.56 7.7934 7.7934 0.930 0.0 6.364 p4 0.4970 0.843 0.9857 0.9857 0.796 0.5790 36.9976 p5 0.67 0.7587 66.0867 66.0867 0.866 0.345.9799 p6 0.9987 0.488 8.780 8.780 0.9300 0.059 6.404 p7 0.939 0.0055 6.39 6.39 0.9344 0.0000 6.307 p8 0.9957 0.364 8.0897 8.0897 0.9304 0.045 6.387 p9 0.9789 0.686 7.095 7.095 0.933 0.0076 6.330 p0 0.990 0.37 7.7674 7.7674 0.93 0.00 6.36 p 0.9907 0.6 7.7996 7.7996 0.930 0.0 6.364 p 0.9999 0.538 8.3558 8.3558 0.998 0.065 6.4 p3 0.5000 0.846 09.6730 09.6730 0.734 0.573 36.46 p4 0.9998 0.536 8.350 8.350 0.998 0.065 6.4 p5 0.600 0.7553 65.890 65.890 0.886 0.3397.7747 p6 0.9687 0.88 6.84 6.84 0.933 0.0046 6.35 p7 0.997 0.43 8.775 8.775 0.930 0.05 6.395 p8 0.9309 0.05 6.366 6.366 0.9344 0.0000 6.307 p9 p 43
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 0.9987 0.490 8.805 8.805 0.9300 0.060 6.405 0.9993 0.53 8.369 8.369 0.999 0.06 6.408 p 0.88 0.85 7.698 7.698 0.934 0.009 6.353 p 0.9989 0.495 8.887 8.887 0.999 0.060 6.405 p3 0.937 0.0383 6.874 6.874 0.9343 0.0004 6.307 p4 0.978 0.66 7.84 7.84 0.933 0.0074 6.38 p5 0.9958 0.369 8.0976 8.0976 0.9304 0.045 6.388 p6 0.9998 0.533 8.3483 8.3483 0.998 0.065 6.4 p7 0.690 0.653 45.7570 45.7570 0.8706 0.53 8.47 p8 0.9997 0.58 8.3398 8.3398 0.998 0.064 6.40 p9 0.784 0.4563 7.3969 7.3969 0.903 0.085 6.59 p30 0.9999 0.538 8.3560 8.3560 0.998 0.065 6.4 p3 0.494 0.8433.997.997 0.753 0.5877 37.886 p3 0.9999 0.536 8.353 8.353 0.998 0.065 6.4 p33 0.67 0.7637 67.4673 67.4673 0.837 0.3534.307 p34 0.9844 0.909 7.4704 7.4704 0.937 0.0097 6.343 p35 0.994 0.99 7.9954 7.9954 0.9306 0.038 6.379 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
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, (), 63-68. Ahmd, Z., Hf, M., & Ahmd, M. (009). Geerlzed regresso-cumrto estmtors for two-phse smplg usg mult-ulr vrbles. Pkst Jourl of Sttstcs, 5(), 93-06. 45
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(), 05-5. Bhush, S. (0). Some effcet smplg strteges bsed o rto-tpe estmtor. Electroc Jourl of Appled Sttstcl Alss, 5(), 74-88. 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), 369-375. 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): 464-466. Goodm, L. A., & Hrtle, H. O. (958). The precso of ubsed rtotpe estmtors. Jourl of the Amerc Sttstcl Assocto, 53(8): 49-508. Gupt, S., & Shbbr, J. (008). O mprovemet estmtg the populto me smple rdom smplg. Jourl of Appled Sttstcs, 35(5), 559 566. 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(), 07 9. Kdlr,., & g, H. (003). A stud o the ch rto tpe estmtor. Hcettepe Jourl of Mthemtcs d Sttstcs, Vol. 3, 05-08. Kdlr,., & g, H. (004). Rto estmtors smple rdom smplg. Appled Mthemtcs d omputto, 5, 893-90. Kdlr,., & g, H. (006). A mprovemet estmtg the populto me b usg the correlto co-effcet. Hcettepe Jourl of Mthemtcs d Sttstcs, 35(), 03-09. Kdlr,., & g, H. (006b). Improvemet estmtg the populto me smple rdom smplg. Appled Mthemtcs Letters, 9, 75-79. 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(), 7-5. 46
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, 54-65. 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(), 5-69. 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), 307-36. Robso, J. (987). odtog rto estmtes uder smple rdom smplg. Jourl of the Amerc Sttstcl Assocto, 8 (399), 86-83. 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), 73 79. 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), 67 75. Sgh, H. P. & Espejo, M. R. (003). O ler regresso d rto-product estmto of fte populto me. The Sttstc, 5, 59-67. 47
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(), 7 87. 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), 555-560. Sgh, H. P., & Tlor, R. (005). Estmto of fte populto me wth kow co-effcet of vrto of ulr. Sttstc, o LV,.3, pp 30-33. Sgh, H. P., Tlor, R., Sgh. S., & Km, J. M. (008). A modfed estmtor of populto me usg power trsformto. Sttstcl Ppers, 49, 37 58. 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(), 3-30. 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, 99 04. Subrm, J., & Kumrpd, G. (0). Modfed rto estmtors usg kow med d co-effcet of kurtoss. Amerc Jourl of Mthemtcs d Sttstcs, (4), 95-00. Subrm, J., & Kumrpd, G. (0b). Estmto of populto me usg kow med d co-effcet of skewess. Amerc Jourl of Mthemtcs d Sttstcs. (5), 0-07. 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
JAMBULINGAM SUBRAMANI vrble s kow. Pkst Jourl of Sttstcs d Operto Reserch, 9(), 37-45. 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, 94-307. Updh, L. N., & Sgh, H.P. (999). Use of trsformed ulr vrble estmtg the fte populto me. Bometrcl Jourl, V4(5), 67-636., Z., & T, B. (00). Rto method to the me estmto usg coeffcet of skewess of ulr vrble. IIA 00, Prt II, IS 06, pp. 03. 49
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,,36 + + 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
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,,36 + + α λ ( ) + + p ( + e + ( + α) λ ) ( ( α) λ) p + + e ( + ( + α) λ ) + ( + ( + α) λ ) p where p + e + + α λ ( p ) ( ( α) λ) 5
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
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
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 + 0 54
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