SAMPLING STRATEGIES FOR FINITE POPULATION

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1 ajsh Sgh Flor Saradach (dors ajsh Sgh Flor Saradach (dors SMING STTEGIES FO FINITE OUTION USING UXIIY INFOMTION Th Educaoal ublshr olubus, 5

2 Salg Srags or F oulao Usg uxlar Iorao ajsh Sgh Flor Saradach (dors SMING STTEGIES FO FINITE OUTION USING UXIIY INFOMTION ars b Sach Mal, ajsh Sgh, Flor Saradach, B. B. har,. S. Jha, Usha Srvasava, Habb Ur. ha.

3 ajsh Sgh Flor Saradach (dors Th Educaoal ublshr Z ublshg hsaa v. olubus, Oho, US Eal: o@duublshr.co ISBN Th uhors, Th Edors, Th ublshr, 5.

4 Salg Srags or F oulao Usg uxlar Iorao ajsh Sgh Dar o Sascs, BHU, Varaas (U.., Ida Edor Flor Saradach har o Dar o Mahacs, Uvrs o Nw Mxco, Gallu, US Edor SMING STTEGIES FO FINITE OUTION USING UXIIY INFOMTION Th Educaoal ublshr olubus, 5

5 ajsh Sgh Flor Saradach (dors

6 Salg Srags or F oulao Usg uxlar Iorao os Forword... 7 Gralzd Fal O Esaors For Esag oulao Ma Usg Two uxlar rbus... 9 bsrac... 9 words Iroduco So Esaors raur.... Th Suggsd lass o Esaors.... Ercal Sud Doubl Salg Esaor d Two-has Salg ocluso... rcs... Gral rocdur o Esag oulao Ma Usg Iorao o uxlar rbu... bsrac... words.... Iroduco.... roosd Esaor.... Mbrs o h al o saor o ad hr Bass ad MSE Ercal sud... 8 ocluso... 9 rcs... 9 Esao o ao ad roduc o Two oulao Mas Usg uxlar haracrs h rsc o No sos... bsrac... words... Iroduco... Esao o ao ad roduc o wo oulao as... as. Th as o ol sos:... as. Icol sos h Sal du o No-rsos:... rcs... 6 O Th Us o oc o Varao ad, Esag Ma o a F oulao... 9 bsrac... 9 words... 9 Iroduco... 9 Esaors ad hr Ma Squar Error... 9 rcs... 5

7 ajsh Sgh Flor Saradach (dors Sud o Irovd ha ao-cu-grsso Esaor or oulao Ma h rsc o No- sos or Fxd os ad Scd rcso... 5 bsrac... 5 words... 5 Iroduco... 5 Th Esaors... 6 Ma Squar Errors o h Sud Esaor... 8 Ercal Sud... 5 ocluso... 5 rcs

8 Salg Srags or F oulao Usg uxlar Iorao Forword Th rs boo as o rs so rovd saors usg auxlar ad arbu orao cas o sl rado salg ad srad rado salg ad so cass wh o-rsos s rs. Ths volu s a collco o v ars, wr b sv co-auhors (lsd h ordr o h ars: Sach Mal, ajsh Sgh, Flor Saradach, B. B. har,. S. Jha, Usha Srvasava ad Habb Ur. ha. Th rs ad h scod ars dal wh h robl o sag h oulao a wh so orao o wo auxlar arbus ar avalabl. I h hrd ar, robls rlad o sao o rao ad roduc o wo oulao a usg auxlar characrs wh scal rrc o o-rsos ar dscussd. I h ourh ar, h us o coc o varao ad sha arars ach srau, h robl o sao o oulao a has b cosdrd. I h h ar, a sud o rovd cha rao-cu-rgrsso saor or oulao a h rsc o o-rsos or xd cos ad scd rcso has b ad. Th auhors ho ha h boo wll b hlul or h rsarchrs ad suds ha ar worg h ld o salg chqus. 7

9 ajsh Sgh Flor Saradach (dors 8

10 Salg Srags or F oulao Usg uxlar Iorao Gralzd Fal O Esaors For Esag oulao Ma Usg Two uxlar rbus Sach Mal, ajsh Sgh ad Flor Saradach Dar o Sascs, Baaras Hdu Uvrs Varaas-5, Ida har o Dar o Mahacs, Uvrs o Nw Mxco, Gallu, US orrsodg auhor, rsghsa@gal.co bsrac Ths ar dals wh h robl o sag h oulao a wh so orao o wo auxlar arbus ar avalabl. class o saors s dd whch cluds h saors rcl roosd b Mal ad Sgh (, Na ad Gua (996 ad Sgh al. (7 as arcular cass. I s show ha h roosd saor s or c ha h usual a saor ad ohr xsg saors. Th sud s also xdd o wo-has salg. Th rsuls hav b llusrad urcall b ag rcal oulao cosdrd h lraur. words Sl rado salg, wo-has salg, auxlar arbu, o bsral corrlao, h corrlao, cc.. Iroduco Thr ar so suaos wh lac o o auxlar arbu, w hav orao o wo qualav varabls. For llusrao, o sa h hourl wags w ca us h orao o aral saus ad rgo o rsdc (s Gujra ad Sagha (7, ag-. Hr w assu ha boh auxlar arbus hav sgca o bsral corrlao wh h sud varabl ad hr s sgca h-corrlao (s Yul (9 bw h auxlar arbus. Th us o auxlar orao ca cras h rcso o a saor wh sud varabl Y s hghl corrlad wh auxlar varabls X. I surv salg, auxlar varabls ar rs or o rao scal varabls (.g. co, ouu, rcs, coss, hgh ad raur bu sos a rs h or o qualav or oal scal such as sx, rac, color, rlgo, aoal ad gograhcal rgo. For xal, al worrs ar oud o ar lss ha hr al courars do or o-wh worrs ar oud o ar lss ha whs (s Gujra ad Sagha (7, ag. Na ad Gua (996 roducd a rao saor wh h sud varabl ad h auxlar arbu ar osvl corrlad. Jhajj al. (6 suggsd a al o saors or h oulao a sgl ad wo-has salg wh h sud varabl 9

11 ajsh Sgh Flor Saradach (dors ad auxlar arbu ar osvl corrlad. Shabbr ad Gua (7, Sgh al. (8, Sgh al. ( ad bd-elaah al. ( hav cosdrd h robl o sag oulao a Y ag o cosdrao h o bsral corrlao bw auxlar arbu ad sud varabl.. So Esaors raur I ordr o hav a sa o h sud varabl, assug h owldg o h oulao rooro, Na ad Gua (996 ad Sgh al. (7 rscvl, roosd ollowg saors: (. (. x x (. (. Th Bas ad MSE xrsso s o h saor s (=,,, u o h rs ordr o aroxao ar, rscvl, gv b B Y b (.5 B Y b Y B b Y B b MSE Y b MSE Y b (.6 (.7 (.8 (.9 (.

12 Salg Srags or F oulao Usg uxlar Iorao MSE b Y (. MSE b Y (. whr,, Y N S, N,S N - N j j N j j j j,, (j ; S, Y S, S S S j j b j j j j., b b b b s s s ad s b h sal h-covarac ad hcorrlao bw ad rscvl, corrsodg o h oulao h-covarac ad h-corrlao N N S. S S S ad Mal ad Sgh ( roosd saors 5 ad 6 as 5 (. 6 x x (. whr,, ad ar ral cosas. Th Bas ad MSE xrsso s o h saor s 5 ad 6 u o h rs ordr o aroxao ar, rscvl, gv b Y ( B b b 5 (.5

13 ajsh Sgh Flor Saradach (dors Y B( b b 6 (.6 Y ( MSE b b 5 (.7 β β β β β β Y MSE( b φ b 6 (.8. Th Suggsd lass o Esaors Usg lar cobao o,,, w d a saor o h or H w (. Such ha, w ad w (. Whr,, α α ad β β ( ( ( ( x ( ( ( ( x whr,, w dos h cosas usd or rducg h bas h class o saors, H dos h s o hos saors ha ca b cosrucd ro,, ad dos h s o ral ubrs (or dal s Sgh. al (8. lso,,,...,8 ar hr ral ubrs or h ucos o h ow arars o h auxlar arbus. Exrssg rs o s, w hav α α β β w w φ φ Y w x θ θ x θ θ (. whr,

14 Salg Srags or F oulao Usg uxlar Iorao θ θ φ φ r xadg, Subracg Y ro boh sds o h quao (. ad glcg h r havg owr grar ha wo, w hav θ β θ β w φ α φ α w Y Y (. Squarg boh sds o (. ad h ag xcaos, w g MSE o h saor u o h rs ordr o aroxao, as 5 T w w T T w w T w T w Y MSE (.5 whr, 5 5 w w (.6 ad b b 5 b b φ φ φ φ θ β θ β φ α φ α β φ θ α θ φ β α θ β α αβ θ φ θ β β c β θ c β θ φ φ α α α φ α φ (.7

15 ajsh Sgh Flor Saradach (dors. Ercal Sud ar dd as: Daa: (Sourc: Govr o asa ( Th oulao cosss rc culvao aras 7 dsrcs o asa. Th varabls Y= rc roduco ( os, wh o o =.98 o durg, = roduco o ars whr rc roduco s or ha os durg h ar, ad = rooro o ars wh rc culvao ara or ha ha durg h ar. For hs daa, w hav N=7, Y =6., =.7, =.5, b =.6, b =.67, =.889. S =7., S =.59, S =.8, Tabl.: E o dr saors o Y wh rsc o. HOIE OF SES, wh w w w α α E S N b b b 78. N.95 b - N N.78 - N N.68 b b - N N N b N b N N.9 Wh, w w w

16 Salg Srags or F oulao Usg uxlar Iorao β β E S N b b b.8 N b - N N N N 8.56 b b - N N N b N b N N 8.57 Wh, α α w w w β also,,...,8 β E = Doubl Salg I s assud ha h oulao rooro or h rs auxlar arbu s uow bu h sa s ow or h scod auxlar arbu. Wh s uow, s so s sad ro a rlar larg sal o sz o whch ol h arbu s asurd. Th a scod has sal o sz (< s draw ad Y s obsrvd. j j,(j,. Th saor s,, ad wo-has salg a h ollowg or d (5. 5

17 ajsh Sgh Flor Saradach (dors 6 d (5. d x (5. d x (5. Th bas ad MSE xrssos o h saors d, d, d ad d u o rs ordr o aroxao, ar rscvl gv as b d Y B (5.5 b d Y B (5.6 b d Y B (5.7 b d Y B (5.8 MSE b d Y (5.9 MSE d Y (5. MSE b d Y (5. MSE b d Y (5. whr, j j S J,, S j j! j, N.

18 Salg Srags or F oulao Usg uxlar Iorao 7 Th saor s 5 ad 6, wo-has salg, as h ollowg or d5 (5. d6 x x (5. Whr,, ad ar ral cosas. Th Bas ad MSE xrsso s o h saor s d5 ad d6 u o h rs ordr o aroxao ar, rscvl, gv b b b d5 Y B (5.5 b b 6 d Y B (5.6 b b 5 d Y MSE (5.7 (5.8 Y MSE b b 6 d 6. Esaor d Two-has Salg Usg lar cobao o,,, d w d a saor o h or H h d d (6. Such ha, h ad h (6. whr,, d ad d ( ( ( ( x ( ( ( ( x whr,, h dos h cosas usd or rducg h bas h class o saors, H dos h s o hos saors ha ca b cosrucd ro,, d ad

19 ajsh Sgh Flor Saradach (dors 8 dos h s o ral ubrs (or dal s Sgh. al. (8. lso,,,...,8 ar hr ral ubrs or h ucos o h ow arars o h auxlar arbus. Exrssg d rs o s, w hav - φ φ φ h h Y θ θ x θ x θ h (6. r xadg, subracg Y ro boh sds o h quao (6. ad glcg h rs havg owr grar ha wo, w hav d θ θ θ h φ φ φ h Y Y (6. Squarg boh sds o (6. ad h ag xcaos, w g MSE o h saor u o h rs ordr o aroxao, as 5 d h h h h h h Y MSE (6.5 whr, 5 5 h h (6.6 ad b b 5 b b φ θ θ φ φ φ θ - θ φ θ θ φ φ (6.7 Daa: (Sourc: Sgh ad haudhar (986,. 77. Th oulao cosss o wha ars vllags cra rgo o Ida. Th varabls ar dd as: = ara udr wha cro ( acrs durg 97. = rooro o ars udr wha cro whch hav or ha 5 acrs lad durg 97. ad

20 Salg Srags or F oulao Usg uxlar Iorao = rooro o ars udr wha cro whch hav or ha acrs lad durg 97. For hs daa, w hav N=, Y =99., =.6765, =.75, S =56.6, S =.59, S =.55, b =599, b =.559, =.75. Tabl 6.: E o dr saors o Y wh rsc o HOIE OF SES, wh h h h E S b b N b N 6.9 b N N. N N 5.7 b b N N N b N 9 b N N 5.86 Wh, h h h E S b b - N b N 5.9 b - N N N N 5.8 b b - N N

21 ajsh Sgh Flor Saradach (dors - N b N b N N 5.8 Wh, h h h 7. ocluso also,,...,8 E =5.8 I hs ar, w hav suggsd a class o saors sgl ad wo-has salg b usg o b sral corrlao ad h corrlao coc. Fro Tabl. ad Tabl 6., w obsrv ha h roosd saor ad d rors br ha ohr saors cosdrd hs ar. rcs. bd-elaah,.m. El-Shr, E.. Mohad, S.M. bdou, O. F.,, Irov sag h oulao a sl rado salg usg orao o auxlar arbu. l. Mah. ad o. do:.6/j.ac.9... Govr o asa,, ros ra roduco b Dsrcs (Msr o Food, grculur ad vsoc Dvso, Ecooc Wg, asa.. Gujara, D. N. ad Sagha, 7, Basc coorcs. Taa McGraw Hll.. Jhajj, H.S., Shara, M.. ad Grovr,.., 6, al o saors o oulao a usg orao o auxlar arbu. a. Jour. o Sa., (, Mal, S. d Sgh,.,, Fal O Esaors O oulao Ma Usg Iorao O o B-Sral d h-orrlao oc. Ir. Jour. Sa. d Eco. (accd. 6. Na,V.D ad Gua,.., 996, o o sao o a wh ow oulao rooro o a auxlar characr. Jour. Id. Soc. gr. Sa., 8(, Shabbr, J. ad Gua, S., 7, O sag h oulao a wh ow oulao rooro o a auxlar varabl. a. Jour. o Sa., (, Sgh, D. ad haudhar, F. S., 986, Thor ad alss o Sal Surv Dsgs (Joh Wl ad Sos, NwYor. 9. Sgh,., auha,., Sawa, N. ad Saradach, F., 7, uxlar orao ad a ror valus cosruco o rovd saors. assac Hgh rss.. Sgh,. hauha,. Sawa, N. Saradach, F., 8, ao saors sl rado salg usg orao o auxlar arbu. a. J. Sa. Or. s. ( Sgh,., uar, M. ad Saradach, F.,, ao saors sl rado salg wh sud varabl s a arbu. WSJ (5: Yul, G. U., 9, O h hods o asurg assocao bw wo arbus. Jour. o Th oal Soc. 75, d

22 Salg Srags or F oulao Usg uxlar Iorao Gral rocdur o Esag oulao Ma Usg Iorao o uxlar rbu Sach Mal, ajsh Sgh ad Flor Saradach Dar o Sascs, Baaras Hdu Uvrs Varaas-5, Ida har o Dar o Mahacs, Uvrs o Nw Mxco, Gallu, US orrsodg auhor, rsghsa@gal.co bsrac Ths ar dals wh h robl o sag h oulao a wh so orao o auxlar arbu s avalabl. I s show ha h roosd saor s or c ha h usual a saor ad ohr xsg saors. Th rsuls hav b llusrad urcall b ag rcal oulao cosdrd h lraur. words Sl rado salg, auxlar arbu, o b-sral corrlao, rao saor, cc.. Iroduco Th us o auxlar orao ca cras h rcso o a saor wh sud varabl s hghl corrlad wh auxlar varabl x. Thr ar a suaos wh auxlar orao s avalabl h or o arbus,.g. sx ad hgh o h rsos, aou o l roducd ad a arcular brd o cow, aou o ld o wha cro ad a arcular var o wha (s Jhajj. al. (6. osdr a sal o sz draw b sl rado salg whou rlac (SSWO ro a oulao o sz N. ad do h obsrvaos o varabl ad rscvl or =; h N h u ( =,,..., N. h u o h oulao osssss arbu = ; ohrws. = ad a=, do h oal ubr o us h oulao ad sal rscvl ossssg arbu. =/N ad =a/ do h rooro o us h oulao ad sal rscvl ossssg arbu. Na ad Gua (996 roducd a rao saor osvl corrlad. Th saor NG wh h sud varabl ad h auxlar arbu ar NG s gv b

23 ajsh Sgh Flor Saradach (dors NG wh MSE MSE( NG S S S (. (. N Y whr,, N N N S Y, S N N, S Y. N N (or dals s Sgh al. (8 Jhajj. al. (6 suggsd a al o saors or h oulao a sgl ad wo has salg wh h sud varabl ad auxlar arbu ar osvl corrlad. Shabbr ad Gua (7, Sgh. al. (8 ad bd-elaah. al. ( hav cosdrd h robl o sag oulao a Y ag o cosdrao h o bsral corrlao coc bw auxlar arbu ad sud varabl. Th objcv o hs arcl s o suggs a gralsd class o saors or oulao a Y ad aals s rors. urcal llusrao s gv suor o h rs sud.. roosd Esaor, bg a suabl chos scalar, ha as valus ad. Th q N, ad Q (N, b B N whr q,q,b ad b. N Movad b Bd (996, w d a al o saors or oulao a Y as w w b q Q (. whr w, w ad ar suabl chos scalars. To oba h Bas ad MSE o h saor, w wr

24 Salg Srags or F oulao Usg uxlar Iorao Y, s, s S, S, b such ha E(, =,,, ad E(, N E(, N E(, E(, N N b E(, N b Exrssg (. rs o s, w hav Yw w N (. W assu ha ad Na, so ha ( ad N ar xadabl. Exadg h rgh had sd o (. ad rag rs u o scod owrs o s,w hav Y Y[w N N N w ] (. N Tag xcao o boh sds o (., w g h bas o o h rs dgr o aroxao as : B( Yw w N N w b N (. Squarg boh sds o (. ad glcg rs o s havg owr grar ha wo, w hav

25 ajsh Sgh Flor Saradach (dors N N N w Y Y N w w w N N N w N w (.5 Tag xcao o boh sds o (.5, w g h MSE o o h rs dgr o aroxao as: 5 w w w w w w Y MSE( (.6 whr, N N b N N N b 5 N whr,. b Th MSE( s sd or (.7 w w 5 (.8 w w 5

26 Salg Srags or F oulao Usg uxlar Iorao. Mbrs o h al o saor o ad hr Bass ad MSE Tabl.: Dr brs o h al o saors o hoc o scalars w w Esaor w w w w Sarls (96 saor q w Q w - 5, Na ad Gua (996 saor b - 6 Sgh. al. (8 saor w w 7 w w b w 8 w b w w 9 w b b grsso saor Th saor s a ubasd saor o h oulao a Y ad has h varac Var S (. 5

27 ajsh Sgh Flor Saradach (dors To, h rs dgr o aroxao h bass ad MSE s o rscvl gv b s, =,,..., ar Yw B (. B Yw w b N N B Yw wb B Y B 5 b Y ( b b 6 b Yw B 7 b Yw B 8 w b Yw B B 9 Y b Th corrsodg MSE s wll b MSE MSE MSE MSE w Y w w Y w Y w w Y 5 (. (. (.5 (.6 (.7 (.8 (.9 (. (. (. (. (. 6

28 MSE MSE MSE MSE MSE Y Salg Srags or F oulao Usg uxlar Iorao 5 (.5 w w w w w w (.6 6 Y 7 Y w 5 8 w Y w w 9 Y 5 5 Th MSE s o h saors o, =,,,7,8,9 wll b sd rscvl, or 5 (.7 (.8 (.9 w (. w (. w 5 w ( ( ( ( w ( ( ( w ( w 5 ( ( ( 5( ( (. (. (. (.5 Thus h rsulg u MSE o, =,,,7,8,9 ar, rscvl gv b 7

29 ajsh Sgh Flor Saradach (dors..... MSE MSE MSE MSE MSE Y Y Y 7 Y 8 Y 5. MSE 9 Y (.6 (.7 (.8 (.9 (. (.. Ercal sud Th daa or h rcal sud s a ro aural oulao daa s cosdrd b Suha ad Suha (97: = Nubr o vllags h crcls ad = crcl cossg or ha v vllags N 89,Y.6,.6, b.766,.6, ,.8, 6.75,.7 I h Tabl. rc rlav ccs (E s o varous saors ar coud wh rsc o. 8

30 Salg Srags or F oulao Usg uxlar Iorao Tabl.: E o dr saors o Y wh rsc o. Esaor E s ocluso Th MSE valus o h brs o h al o h saor hav b obad usg (.6. Ths valus ar gv Tabl.. Wh w xa Tabl., w obsrv h suror o h roosd saors, 7, 8, 9 ad ovr usual ubasd saor,,, Na ad Gua (996 saor 5 ad Sgh. al. (8 saor 6. Fro hs rsul w ca r ha h roosd saors 8 ad 9 ar or c ha h rs o h saors cosdrd hs ar or hs daa s. W would also l o rar ha h valu o h. MSE(, whch s qual o h valu o h MSE o h rgrsso saor s.98. Fro Tabl. w oc ha h valu o MSE o h saors 8 ad 9 ar lss ha hs valu, as show Tabl.. Fall, w ca sa ha h roosd saors 8 ad 9 ar or c ha h rgrsso saor or hs daa s. rcs. bd-elaah,.m. El-Shr, E.. Mohad, S.M. bdou, O. F. (: Irov sag h oulao a sl rado salg usg orao o auxlar arbu. l. Mah. ad o. do:.6/j.ac.9... Bd,.. (996. Ec ulzao o auxlar orao a sao sag. Bo. Jour, 8: Jhajj, H.S., Shara, M.. ad grovr,.. (6 : al o saors o oulao a usg orao o auxlar arbu. a. Jour. o Sa., (, -5.. Na,V.D. ad Gua,..(996: o o sao o a wh ow oulao rooro o a auxlar characr. Jour. o h Id. Soc. o gr. Sa., 8(, Sarls, D.T. (96: Th ulzao o ow coc o varao h sao rocdur. Jour. o h r. Sa. ssoc., 59,

31 ajsh Sgh Flor Saradach (dors 6. Sgh,. hauha,. Sawa, N. Saradach, F. (8: ao saors sl rado salg usg orao o auxlar arbu. a. J. Sa. Or. s. ( Shabbr, J. ad Gua, S.(7: O sag h oulao a wh ow oulao rooro o a auxlar varabl. a. Jour. o Sa., (, Suha,.V. ad Suha, B.V. (97: Salg hor o survs wh alcaos. Iowa Sa Uvrs rss, s, U.S..

32 Salg Srags or F oulao Usg uxlar Iorao Esao o ao ad roduc o Two oulao Mas Usg uxlar haracrs h rsc o No sos B. B. har Dar o Sascs, Baaras Hdu Uvrs, Varaas (U., Ida Eal: bbhar56@ahoo.co bsrac Th auxlar orao s usd crasg h cc o h saors or h arars o h oulaos such as a, rao, ad roduc o wo oulao as. I hs cox, h sao rocdur or h rao ad roduc o wo oulao as usg auxlar characrs scal rrc o h o rsos robl has b dscussd. words uxlar varabl, MSE, o rsos, SS, cc. Iroduco Th us o auxlar orao sal survs h sao o oulao a, rao, ad roduc o wo oulao as has b sudd b dr auhors b usg dr sao rocdurs. Th rvw wor hs oc has b gv b Trah al. (99 ad har (. I h rs cox h robls o sao o rao ad roduc o wo oulao as hav b cosdrd dr suaos scall h rsc o o rsos. Esao o ao ad roduc o wo oulao as as. Th as o ol sos: Sgh (965,69, ao ad arra (968, Shahoo ad Shahoo (978, Trah (98, a ad Sgh (985 ad har (987 hav roosd saors o rao ad roduc o wo oulao as usg auxlar characrs wh ow a. Sgh (98 has roosd h cas o doubl salg or h sao o rao ad roduc o wo oulao a. har (99(a has roosd a class o saors or ad usg doubl salg sch, whch ar gv as ollows: v u ad gw, u, (

33 ajsh Sgh Flor Saradach (dors such ha,, g,,, ad, ad g, whr v, w x u. Hr, ad x do h sal a o sud characrs, ad x auxlar characr x basd o a sub sal o sz ( ad x s sal a o x basd o a largr sal o sz draw b usg SSWO hod o salg ro h v u g w, u wh rsc o oulao o sz N. Th rs aral drvavs o, ad v ad w ar dod b v, u ad g w, u rscvl. Th uco v, u ad g w, u also sasd so rgular codos or cou ad xsc o h ucos. Th sal sz or rs has ad scod has sal whch a b ro h rs has sal or dd o rs has sal draw ro h rag ar o h oulao ( N. Sgh al. (99 hav xdd h class o saors roosd b har (99(a ad roosd a w class o saor or, whch s gv as ollows: g h ˆ u, v ( whr ˆ x sx, u ad v, s x x, sx ar sal a ad sal x sx a squar o auxlar characr basd o ad us rscvl., whr x ad Srvasava al. (988,89 hav suggsd cha rao saors or ad. Whch ar gv as ollows: ˆ ad Y ˆ Y ( ˆ ad Y ˆ Y ( Y Furhr Sgh al. (99 hav gv a gral class o saors ˆ hˆ, u vad h hˆ, u, v h, such ha h,, ad h, ˆ, u v ad h, u, v h,,, whr ˆ sas h rgular codos. ˆ, (5 u ad v. Th ucos Y har (99(b hav roosd h class o saors or usg ul-auxlar characrs wh ow as. whch ar gv as ollows: h ˆ u u u h ˆ,... u ad gˆ, u, (6

34 such ha codos. h ad g, Salg Srags or F oulao Usg uxlar Iorao ˆ, whr u h ad g, u ˆ sasg so rsodg Furhr, har (99(a has roosd a class o saors or usg ulauxlar characrs wh uow as, h class o saors s gv as ollows: such ha g,, whr u g ˆ, u x, u u, u... u x ad us or auxlar characrs x,,,...., (7, x ad x ar sal a basd o Slarl, har (99 hav roosd class o saors or usg auxlar characrs wh ow ad uow oulao a ad sudd hr rors. Furhr, har (99 has roosd a gralzd class o saor or a cobao o roduc ad rao o so oulao as usg ul-auxlar characrs. Th ararc cobao s gv b: Y, Y, Y,..., Y, (8 Y, Y, Y,..., Y whch s h roduc o rs oulao as Y, Y, Y,..., Y dvdd b roduc o oulao as Y,..., gv b ˆ,,,...,, (9,,,...,, Y, YM Y rscvl. Th covoal saor or s I s ora o o ha or, ;, ;, ; Y ; Y Y, Y, ; Y Y Y, Y,, ; Y Y Y, Y, Y, Usg auxlar characrs class o saors s gv b: x, x,..., x X, X,..., X h wh ow oulao as u h ˆ, (

35 ajsh Sgh Flor Saradach (dors such ha h, whr u u u,... Th uco hu, u,... u hu x, u ad u,,,...,. X sasd h ollowg rgular codos: u, assu valus aboudd closd covx sub s G o dsoal ral sac coag h o u. b I G, h uco h u s couous ad boudd. c Th rs ad scod aral drvavs o h u xss ad ar couous ad boudd G. For wo auxlar varabls s oud ha h lowr bod o h varac o h class o a Whavr b h sal chos saors s sa as gv b h saors roosd b Sgh (969 ad Shah ad Shah (978. Hc s rard ha h class o saors wll aa lowr boud or a squar rror h scd ad rgular codos ar sasd. Furhr, har (99b hav roosd h class o wo has salg saors or h cobao o roduc ad rao o so oulao as usg ul-auxlar characrs wh uow oulao as, whch s gv as ollows: whr v v v,... v,, Such ha h ad v v x v h ˆ, (,,,... x. h sass so rgularl codos. as. Icol sos h Sal du o No-rsos: I cas o o-rsos o so us slcd h sal, Has ad Hurwz (96 hav suggsd h hod o sub salg ro o-rsods ad roosd h saor or oulao a. Furhr, har al. ( hav roosd so w saors hs suao o sub salg ro o-rsods. har & ad ( ad har & Sha ( hav roosd h class o saors or rao ad roduc o wo oulao as usg auxlar characr wh ow oulao a h rsc o o-rsos o h sud characrs, whch s gv as ollows: such ha h ad h u h, whr,,,,, ( u x u, X x u ad X, ad x ar sal as or, ad x characrs roosd b Has ad Hurwz (96 basd o r us ad x s h sal a basd o us. har & Sha ( hav roosd a cobd class o saors or rao ad roduc o wo oulao a h rsc o o-rsos wh ow oulao a X. Ths s a or gral class o saors or ad udr so scd ad rgular codos. har al. ( (a hav roosd a rovd class o saors or. I hs cas, h rovd class o saors or usg auxlar characr wh ow oulao a X h rsc o o rsos s gv as ollows:

36 Salg Srags or F oulao Usg uxlar Iorao u such ha g,, g ad g g, u x o,. Th uco v u. Th uco v, g v,,, (,, whr v, u x ad g,, assus osv valus a ral l coag h g, s assud o b couous ad boudd a ral l ad u s rs ad scod ordr aral drvavs xss. Th rs aral drvav o g v, u, a h o, wh rsc o v ad u s dod b g, ad g,. Th scod ordr aral drvav o g v, u, o, s dod b g, g ad h class o saors whr w, w, w, w, w,,, ar gv as ollows: wh rsc o v, v ad u, ad u a h g, rscvl. So brs o wvu, wv wu, w v w vu,,, ( ad,, ar cosas. Furhr h class o saor roosd b har ad Sha ( s or c ha h saor roosd b har ad ad (. Furhr, har ad Sha ((a, b hav roosd wo has salg saors or rao ad roduc o wo oulao as h rsc o o-rsos. har ad Sha ((a,b hav roosd a or gral class o wo has salg saors or ad. whch ar gv as ollows: such ha g, ad, basd o u T g v,,,, (5 g, whr us. Th uco v u such ha g, ad g, so rgularl codos. u v, u, x x x u ad x s sal a x g, sas so rgularl codos. T g w,,,, (6, whr w, u, x x u x x ad w u g, sas har al. ( hav roosd wo gralzd cha saors T g ad T g or usg auxlar characrs h rsc o o-rsos, whch ar gv as ollows: whr ˆ T g ˆ x x z Z ad ad, ad ˆ x z T g, (7 x Z, ar suabl cosas. I has b obsrvd ha du o us o addoal auxlar characr wh ow oulao a alog wh h a auxlar characr, h roosd class o saors T g ad T g ar or c ha h 5

37 ajsh Sgh Flor Saradach (dors corrsodg gralzd saors or usg h a auxlar characr ol h cas o wo has salg h rsc o o rsos or xd sal szs (, ad also or xd cos (. I s also s ha lss cos s currd or T g ad T g ha h cos currd h gralzd saor or h cas o wo has salg h rsc o o rsos or scd rcso ( V V. Furhr, gralzd cha saors or rao ad roduc o wo oulao as hav b rovd b ug ˆ ad ˆ lac o ˆ ad ˆ h roosd saors o ad. Furhr, har al. ( (b hav roosd h rovd class o cha saors or rao o wo oulao as usg wo auxlar characrs h rsc o o-rsos. Th class o saors s gv as ollows: ˆ, u v,,, (8 c, such ha,, ad,, uco, u, v, whr ˆ, u, x x ˆ,, sass so rgular codos. x u ad x z v. Th Z har ad Sha (7 hav roosd saor or usg ul-auxlar characrs wh ow oulao a h rsc o o-rsos. Th class o saors s gv as ollows: such ha g (, whr u ad x j (,,...,, u j ad u X j ˆ g ( u,,, (9 x do h colu vcors u, u,..., u ad j j j,,..., X j. 6 ( rovd udr class o saors or usg ul-auxlar varabls usg doubl salg sch h rsc o o-rsos has b roosd b har ad Sha ( ad suds hr rors. har ad Sha ( hav xdd h class o saor roosd b har ad Sha ( ad roosd a wdr class o wo has salg saors or usg ulauxlar characrs h rsc o o-rsos. rcs. Has, M. H. ad Hurwz, W. N. (96: Th robl o o-rsos sal survs. Jour. r. Sa. ssoc.,, har, B. B. (987: O odd class o saors o rao ad roduc o wo oulao as usg auxlar characr. roc. Mah. Soc, B.H.U., -7.. har, B. B. (99: gralzd class o saors or cobao o roducs ad rao o so oulao as usg ul-auxlar characrs. J. Sa. s.,, -8.. har, B. B. (99 (a: Drao o sal szs or a class o wo has salg saors or rao ad roduc o wo oulao as usg auxlar characr. Mro (Ial, XIX, (-,

38 Salg Srags or F oulao Usg uxlar Iorao 5. har, B. B. (99 (b: O gralzd class o saors or rao o wo oulao as usg ul-auxlar characrs. lgarh J. Sa., har, B. B. (99: O class o saors or roduc o wo oulao as usg ul-auxlar characrs wh ow ad uow as. Id. J. l. Sa.,, har, B. B. (99(a: class o wo has salg saors or h cobao o roduc ad rao o svral oulao as usg ul-auxlar characrs. roc. Na. cad. Sc., Ida, 6 (,. II, har, B. B. (99(b: O a class o wo has salg saors or rao o wo oulao as usg ul-auxlar characrs. roc. Na. cad. Sc., Ida, 6(a, III, har, B. B. (: Esao o oulao arars usg h chqu o sub salg ro o rsods sal survs- vw. roc. Na. cad. Sc. Sc, 8 (, 7-.. har, B. B. (-5: lcaos o sascs bo-dcal sccs. raja, Scal Issu o Scc & Tcholog, Vol. - 6 (,.. har, B. B. ad ad, S.. (: class o saors or rao o wo oulao as usg auxlar characr rsc o o-rsos. J. Sc. s., 5, 5-.. har, B. B. ad Sha,.. (a: Esao o h rao o wo oulaos as usg auxlar characr wh uow oulao a rsc o o rsos. rog. Mahs. Vol.- 6. No. (,, har, B. B. ad Sha,.. (b: O class o wo has salg saors or h roduc o wo oulao as usg auxlar characr rsc o o rsos. roc. o Vh raoal sosu o ozao ad Sascs hld a MU, lgarah, -.. har, B. B. ad Sha,.. ( (a: Esao o oulao rao usg wo has salg rsc o o rsos. lgarh J. Sa., har, B. B. ad Sha,.. ( (b: O h gral class o wo has salg saors or h roduc o wo oulao as usg h auxlar characrs h rsc o o-rsos. Id.J. l. Sascs., 8, har, B. B. ad Sha,.. (7: Esao o h rao o h wo oulao as usg ul- auxlar characrs rsc o o-rsos. I Sascal chqus l sg, rlabl, salg hor ad qual corol dd b B. N. ad Narosa ublshg hous, Nw Dlh, har, B. B. ad Sha,.. (: O class o saors or h roduc o wo oulao as usg auxlar characr rsc o o-rsos. Ir. Tras. l. Sc., (, har, B. B. ad Sha,.. ( (a: obd class o saors or rao ad roduc o wo oulao as rsc o o-rsos. I. Jour. Sas. ad Eco., 8(S, har, B. B. ad Sha,.. ( (b: Irovd classs o rao o wo oulao as wh doubl salg h o-rsods. Sasa-Sascs & Ecoo Jour. 9(, har, B. B. ad Sha,.. (: class o wo has salg saor or rao o wo oulaos as usg ul-auxlar characrs h rsc o o rsos. Sa. Tras. Nw srs, 5, (, har, B. B. ad Srvasava, S.. (999: class o saors or rao o wo oulao as ad as o wo oulaos usg auxlar characr. J. Na. cad. Mah., -.. har, B. B. ad Srvasava, S. a. (998: obd gralzd cha saors or rao ad roduc o wo oulao as usg auxlar characrs. Mro (Ial VI (-,

39 ajsh Sgh Flor Saradach (dors. har, B. B., Jha,. S. ad uar,. (: Irovd gralzd cha saors or rao ad roduc o wo oulao as usg wo auxlar characrs h rsc o orsos. Iraoal J. Sas & Ecoocs, (, 8-.. har, B. B., ad, S.. ad uar,. ( (a: Irovd class o saors or rao o wo oulao as usg auxlar characr rsc o o-rsos. roc. Na. cad. Sc. Ida, 8(, har, B. B., uar,. ad Srvasava, U. ( (b: Irovd classs o cha saors or rao o wo oulao as usg wo auxlar characrs h rsc o orsos. I. Jour. dv. Sas. & rob. (, har, B. B., Srvasava, U. ad uar. (: ha saors or rao o wo oulao as usg auxlar characrs h rsc o o rsos. J. Sc. s., BHU, 56, har, B. B., Srvasava, U. ad uar. (: Gralzd cha saors or rao o wo oulao as usg wo auxlar characrs h rsc o o-rsos. Iraoal J. Sas & Ecoocs, (, ao, J. N.. ad arra, N.. (968: O doubl rao saors. Saha Sr..,, a, S.. ad Sgh,.. (985: So saors or h rao ad roduc o oulao arars. Jour. Id. Soc. gr. Sa., 7, -.. Shah, S. M. ad Shah, D. N. (978: ao cu roduc saors or sag rao (roduc o wo oulao arars. Saha Sr..,, Sgh, M.. (969: oarso o so rao cu roduc saors. Saha, Sr. B,, Sgh,.. (98b: O sag rao ad roduc o oulao arars. alcua Sa. ssoc.,, Sgh, V.., Sgh, Har. ad Sgh, Housla. (99: Esao o rao ad roduc o wo oulao as wo has salg. J. Sa. la. I., Sgh, V.., Sgh, Har., Sgh, Housla. ad Shula, D. (99: gral class o cha saors or rao ad roduc o wo oulao as o a oulao. ou. Sa.- TM. (5, Srvasava, a S., har, B. B. ad Srvasava, S.. (988: O gralzd cha saor or rao ad roduc o wo oulao as usg auxlar characrs. ssa Sa. vw, (, Srvasava, a S., Srvasava, S.. ad har, B. B. (989: ha rao saor or rao o wo oulao as usg auxlar characrs. ou. Sa. Thor Mah.(US, 8(, Trah, T.. (98: gral class o saors or oulao rao. Saha Sr..,, Trah, T.., Das,.. ad har, B. B. (99: Us o auxlar orao sal survs - rvw. lgarh J. Sa.,,

40 Salg Srags or F oulao Usg uxlar Iorao O Th Us o oc o Varao ad, Esag Ma o a F oulao B. B. har,. S. Jha ad U. Srvasava Dar o Sascs, B.H.U, Varaas-5 Sascs Sco, MMV, B.H.U, Varaas-5 bbhar56@ahoo.co bsrac I hs ar h us o coc o varao ad sha arars ach srau, h robl o sao o oulao o a has b cosdrd. Th xrsso o a squard rror o h roosd saor s drvd ad s rors ar dscussd. words uxlar orao, MSE, coc o varao, srau, sha arar. Iroduco Th us o ror orao abou h oulao arars such as coc o varao, a ad swss ad uross ar vr usul h sao o h oulao arar o h sud characr. I agrculural ad bologcal suds orao abou h coc o varao ad h sha arars ar o avalabl. I hs arars ra ssall uchagd ovr h ha h owldg abou h such cas a roabl b usd o roduc ou sas o h arars (S ad Grg (975. Sarls (96, 67 ad Hrao (97 hav roosd h us o coc o varao h sao h oulao a. Sarl ad Iaraach (99 hav suggsd h us o uross h sao o varac. S (978 has roosd h saor or oulao a usg h ow valu o coc o varao. I Srad rado salg, h hor has b dvlod o rovd h ou saor T o h oulao a basd o sal a ro ach srau. W xd b cosrucg a saor T usg h coc o varao ad sha arar (,,... ro ach srau ad dscuss s usulss. W also d saors, T ad T wh h cocs o varao ar uow bu sha arars ar ow ad wh hr h cocs o varao ar ow or h sha arars ar ow. Esaors ad hr Ma Squar Error N dos h sz o h h srau ad dos h sz o h sal o b slcd ro h h srau ad h b h ubr o sraa wh 9

41 ajsh Sgh Flor Saradach (dors h N h N ad, ( whr N ad do h ubr o us h oulao ad sal rscvl. xrssd as j b h j h u o h h srau. Th h oulao a h Y N ca b Y N Y, ( whr N ad Y s h oulao a or h h srau. N us b slcd ro h h srau ad h corrsodg salg a ad sal varac b dod b ad ad h s rscvl. Th h sa o Y N s gv b T ( h V(T h (..c s gord, ( whr s h oulao varac o h h srau. as : oc o varao ad h sha arars ar ow. W dd ad xcao o T s gv b T E(T h { ( s } h h h Y N { Y { Y { Y O( (5 V ( s ( Y ( 8 V ( s ( ( 8 } } ( ( } 8 (

42 Salg Srags or F oulao Usg uxlar Iorao whr s h asur o uross h h srau. (6 Th bas T s o ordr MSE(T ad wll b glgbl or larg s. Th a squar rror o h saor s h ( / { ( ( } O( / Msg (7 wh rsc o, w g h ou valu o s gv b whr s h asur o uross h h srau.. (7 o, (8 O ug h ou valu o o ro (8 (7 ad o slcao w g MSE(T Th valu o h { } O( / ( ( o wll b lss ha o or. (9, whch ls ha h dsrbuo s ar oral, oso, gav boal ad Na I. Th valu o wll b qual o o or Th valu o wll b grar ha o or o o, whch s ru or gaa ad xoal dsrbuo., whch s ll o h dsrbuo o logoral or vrs Gaussa. I s as o s ha T wll alwas b or c ha T or, jusg h us o T h cas o ar oral, oso, gav boal, Na I ad logoral or vrs Gaussa dsrbuo. T s quall c T, ad so or gaa or xoal dsrbuo o a us T or T. Ths shows ha roosd saor T s uorl suror o h saor T, hough a coaravl hgh cc a b s ar oral, oso, gav boal ha logoral or vrs Gaussa dsrbuo.

43 ajsh Sgh Flor Saradach (dors as : s ar uow, s ad s ar ow. Wh s ar uow, w us hr sas c basd o a largr sal o sz ro a rvous occaso. Now w d a saor T or N Y gv b h s c } ( { T ( Th a squar rror o h saor T as gv b h c V } ( {( ( ( / MSE(T, ( whr ( } {( ( c V. Th ou valu o s gv b ( ( ( ( ( o c V c V. ( I s as o s ha h o c V c V ( ( ( ( ( / MSE(T. ( I a b rard ha ( drs ro (9 b a sgl r ( c V boh uraor ad doaor. Th aur o h saor T s slar o T ad s MSE wll covrg o ( T MSE or ( c V. as : s, s ad s ar uow: Wh s, s ad s ar o ow h h ca b sad o h bass o a largr sal o sz... ro h as daa ad w a hav h saor or h oulao a N Y gv b h s c } ˆ ( { ˆ T, (

44 Salg Srags or F oulao Usg uxlar Iorao ˆ ˆ ˆ whr ˆ o. ˆ ˆ ˆ ˆ I s as o s ha h MSE(T wll b sa as MSE(T bcaus ar sag h uow arars h cosa o, h MSE wll ras uchagd u o h rs o O ( (Srvasava ad Jhajj (98. rcs. Sarls, D. T. (96: Th ulzao o coc o varao h sao rocdur. Jour. o r. Sa. ssoc., 59, Sarls, D. T. (967: o o h us o a aroxal ow coc o varao. Th r. Sasca,, -.. S,.. ad Grg, T. M. (975: Esao o a oulao a havg qual coc o varao o succsso occasos. Bull. I. Sa. Is., 6, -.. S,.. (978: Esao o h oulao a wh h coc o varao s ow. ou. Sa. Thor Mh., 7,, Srvasava, S.. ad Jhajj, H.S. (98: class o saors o h oulao as usg ul- auxlar orao. alcua Sa. ssoc. Bull,, Sarls, D. T. ad Iaraach (99: o o a saor or varac ha ulzd h uross. r. Sa., (, Hrao (97: Usg so aroxal ow coc o varao sag a. roc. Is. Sa. Mah, (, 6-6.

45 ajsh Sgh Flor Saradach (dors

46 Salg Srags or F oulao Usg uxlar Iorao Sud o Irovd ha ao-cu-grsso Esaor or oulao Ma h rsc o No- sos or Fxd os ad Scd rcso B. B. har, Habb Ur ha ad U. Srvasava Dar O Sascs, B.H.U, Varaas-5 Sascs Sco, MMV, B.H.U, Varaas-5 bbhar56@ahoo.co bsrac I hs ar, a sud o rovd cha rao-cu rgrsso saor or oulao a h rsc o o-rsos or xd cos ad scd rcso has b ad. Thorcal rsuls ar suord b carrg ou o urcal llusrao. words Sl rado salg, o rsos, xd cos, rcso. Iroduco I h ld o soco, coocs, rsarchs ad agrculurs h robl arss du o o-rsos whch rdl occur du o o a ho, lac o rs, call bac c. I hs xrsso a rocdur o sub salg ro o rsods was suggsd b Has ad Hurwz (96. Th us o auxlar orao h saors o h oulao arars hav hld crasd h cc o h roosd saor. Usg auxlar characr wh ow oulao a o h saors hav b roosd b ao (986,9 ad har ad Srvasava (996,997. Furhr, har ad Srvasava (99,995,har al. (8, Sgh ad uar (, har ad uar (9 ad har ad Srvasava( hav roosd dr s o saors or h sao o oulao a h rsc o o-rsos cas o uow oulao a o h auxlar characr. I h rs ar, w hav sudd a rovd cha rao-cu-rgrsso saor or oulao a h rsc o o-rsos hav roosd b har ad ha ( h cas o xd cos ad scd rcso. I h rs sud w hav obad h ou sz o rs has sal ( ad scod has sal ( s draw ro h oulao o sz N b usg SSWO hod o salg cas o xd cos ad also cas o scd rcso V V. Th xrsso or h u MSE o h saor has b obad or h ou valus o ad cas o xd cos. Th xrsso or u cos or h saor has also b obad 5

47 ajsh Sgh Flor Saradach (dors cas o scd rcsov V. rcal sud has b cosdrd o obsrv h rors o h saor cas o xd cos ad also cas o scd rcso. Th Esaors Y, X ad Z do h oulao a o sud characr, auxlar characr x ad addoal auxlar characr z havg jh valuy j, X j ad Z j : j,,,..., N. Suosd h oulao o sz N s dvdd N rsodg us ad N o rsodg u. ccordg o Has ad Hurwz a sal o sz s a ro oulao o sz N b usg sl rado salg whou rlac (SSWO sch o salg ad has b obsrvd ha us rsod ad us do o rsod. ga b ag xra or, a sub sal o sz r( s draw ro o-rsodg u ad collc orao o r us or sud characr. Hc h saor or Y basd o r us o sud characr s gv b: ( whr ad ar h rsodg ad o-rsodg us a sal o sz slcd ro oulao o sz N b SSWO hod o salg. ad ar h as basd o ad r us slcd ro o-rsodg us b SSWO hods o salg. Slarl w ca also d saor or oulao a X o auxlar characr x basd o ad r u rscvl, whch s gv as; Varac o h saors ad V x x x ( ad x ar gv b W ( ( S S ( ( W ( ( x S x S x( V ( N whr, W N N, (, S ( S ad ( S x, S x ( ar oulao a squars o ad x or r oulao ad o-rsodg ar o oulao. I cas wh h oulao as o h auxlar characr s uow, w slc a largr rs has sal o sz us ro a oulao o sz N us b usg sl rado sal whou rlac (SSWO hod o salg ad sa X b x basd o hs us. Furhr scod has sal o sz (.. < s draw ro us b usg SSWO hod o salg ad varabl udr vsgao s asurd rsodg ad o-rsodg us. ga a sub sal o sz r ( /, s draw ro o-rsodg us ad collc orao o r us b rsoal rvw. 6

48 Salg Srags or F oulao Usg uxlar Iorao I hs cas wo has salg rao, roduc ad rgrsso saors or oulao a Y usg o auxlar characr h rsc o o-rsos hav b roosd b har ad Srvasava (99,995 whch ar gv as ollows: x T (5 x T b x (6 x whr x x x, x Ŝ x ad S ˆx ar sas o j x j S x ad Sˆ x, x x j, b ˆ j Sx, sx x x S x basd o r us. Th covoal ad alrav wo has salg rao saors suggsd b har ad Srvasava ( whch ar as ollows: x T, x whr ad ar cosas. x T (7 x Sgh ad uar ( hav roosd drc saor usg auxlar characr h rsc o o-rsos whch s gv as ollows: whr ad x x T 5 (8 x x ar cosas. I cas wh X s o ow ha w a us a addoal auxlar characr z wh ow oulao a Z wh h assuo ha h varabl z s lss corrlad o ha x., ( z x, x ad z ar varabls such ha z s or char ha x. Followg had (975, so saors hav b roosd b rgra (98,8, Srvasaava al. (99 ad har & uar (. I h cas o o-rsos o h sud characr, h cha rgrsso ad gralzd cha saors or h oulao a h rsc o o-rsos hav b roosd b har & uar ( ad har al. (. rovd cha rao-cu-rgrsso saor or oulao a h rsc o o-rsos hav b roosd b har & ha (, whch s gv as ollows: q x x b Z z x Z T 6 bx xz x z (9 whr ad q ar cosas. bx ad b xz ar rgrsso cocs. Z ad z oulao a ad sal a basd o rs has sal o sz us slcd ro oulao o sz N b SSWO hod. 7

49 ajsh Sgh Flor Saradach (dors Ma Squar Errors o h Sud Esaor Usg h larg sal aroxaos, h xrssos or h a squar rrors o h saor roosd b har & ha ( u o h rs o ordr ( ar gv b x x x x x x x x MSE( T6 V Y b X Y XYb XYb Y q b Z Y q YZb YZqb N W z xz z z xz z xz z Y x( bxx x( Y x( XYbx x( XYbx x( ( Th ou valus o ad q ad h valus o rgrsso coc ar gv as ollows: q o W Y Xb Y Xb x x x x( x x( ( W Y x Y x( Yz Zbxzz o, ( Yz b x Y X x x ad b X xz x x ( Z z Ma squar rrors o h saorst, T, T, T adt 5 ar gv as ollows: MSE T W ( V ( Y x x x( ( ( x W ( MSE( T V ( Y x B x( B x( W ( x x( MSE ( T V ( Y (6 W ( x x( MSE ad ( T V ( Y x ( (5 (7 W ( MSE( T5 Y ( x ( x( ( (8 whr W ( V ( Y ( ad Y x B X x 8

50 Salg Srags or F oulao Usg uxlar Iorao Drao o, ad or h Fxd os us assu ha b h oal cos (xd o h surv aar ro ovrhad cos. Th xcd oal cos o h surv aar ro ovrhad cos s gv as ollows: W ( W, (9 whr : h cos r u o obag orao o auxlar characr x a h rs has. : h cos r u o obag orao o addoal auxlar characr z a h rs has. : h cos r u o alg qusoar/vsg h u a h scod has. : h cos r u o collcg, rocssg daa obad ro rsodg us. : h cos r u o obag ad rocssg daa (ar xra ors or h sub salg us. Th xrsso or, MSE( T6 ca b xrssd rs o D, D, D ad D whch ar h cocs,, ad rscvl. Th xrsso o MSE( T N 6 s gv as ollows: D D D D MSE ( T6, ( N For obag h ou valus o,, or h xd cos, w d a uco whch s gv as: 6 MSE( T, ( whr s h agrag s ullr. W drag wh rsc o,, ad quag zro, w g ou valus o, ad.whch ar gv as ollows: o o D ( D W, ( o D W o, ( ad o DW, ( D ( W whr W D ( D od W, (5 o 9

51 ajsh Sgh Flor Saradach (dors 5 Th u valu o ( 6 T MSE or h ou valus o, ad h xrsso ( 6 T MSE, w g: N D W W D D D T MSE o o 6 ( ( (, (6 Now glcg h r o O ( N, w hav 6 ( ( ( o o W W D D D T MSE (7 Drao o ad, or h Scd rcso V V V b h scd varac o h saor 6 T whch s xd advac, so w hav N D D D D V, (8 To d h ou valus o,, ad u xcd oal cos, w d a uco whch s gv as ollows: ( ( ( 6 V MSE T W W, (9 whr s h agrag s ullr. r drag wh rsc o,, ad quag o zro, w d h ou valu o, ad whch ar gv as; ( D o, ( o o o W W D D, ( ad ( W D D W o, ( whr N D V W W D D D o o ( (, ( Th u xcd oal cos currd o h us o 6 T or h scd varac V wll b gv as ollows:

52 Salg Srags or F oulao Usg uxlar Iorao 6 W D ( ( D od W o, ( D V N Now glcg h rs o O ( 6 N, w hav W D ( ( D D o W o, (5 V Ercal Sud To llusra h rsuls w us h daa cosdrd b har ad Sha (7.Th dscro o h oulao s gv blow: Th daa o hscal growh o ur soco-cooc grou o 95 schoolchldr o Varaas udr a IM sud, Dar o darcs, B.H.U., durg 98-8 has b a udr sud. Th rs 5% (.. chldr us hav b cosdrd as orsodg us. Hr w hav a h sud varabl (, auxlar varabl (x ad h addoal auxlar varabl (z ar a as ollows: : wgh ( g. o h chldr. x : sull crcurc ( c o h chldr. z : chs crcurc ( c o h chldr. Th valus o h arars o h, x ad z characrs or h gv daa ar gv as ollows: Y 9.968, Z 5.76, X 55.86,.56,.6,.586, (.75, z(.78, x(.5, z.8, x.86, xz.97, xz(.57, W.5, W.7, N 95, 5 z x Tabl. lav cc ( % o h saors wh rsc o =s., c =s..9, c =s.., c =s., c =s., c =s. 5. (or h xd cos Esaors o o o Ecc (.8 T (.7 T. 7 9 (.9 5

53 ajsh Sgh Flor Saradach (dors T (.7 T (.89 T (.5 T (.5 Fgurs arhss gv h MSE (.. Fro abl, w obad ha or h xd cos h sud saor T6 s or c coarso o h saors, T, T, T, T ad T 5. Tabl. Excd cos o h saors or h scd varac V. 56 : ( c =s..9, c =s.., c =s., c =s. 5, c =s. 5 Esaors o o o Excd os ( s T T. 5 5 T T T T Fro abl, w obad ha or h scd varac h sud saor T6 has lss cos coarso o h cos currd h saors, T, T, T, T ad T 5. 5

54 Salg Srags or F oulao Usg uxlar Iorao ocluso Th orao o addoal auxlar characr ad ou valus o cras h cc o h sud saors coarso o corrsodg saors cas o h xd cos ad scd rcso V V. rcs. had,. (975: So rao- saors basd o wo or or auxlar varabls. h.d. Thss subd o Iowa Sa Uvrs, s, IOW.. Has, M. H. ad Hurwz, W. N. (96: Th robl o o-rsos sal survs. J. r. Sas. ssoc.,, rgra, B. (98: cha rao saor oulao doubl salg usg wo auxlar varabls. Mra, 7, 7-.. rgra, B. (98: grsso - saors usg wo auxlar varabls ad odl o doubl salg ro oulaos. Mra,, har, B. B. ad Srvasava, S. (99: Esao o oulao a usg auxlar characr rsc o o-rsos. Na. cad. Sc. rs, Ida, 6(, har, B. B. ad Srvasava, S. (995: Sud o covoal ad alrav wo has salg rao, roduc ad rgrsso saors rsc o o-rsos. roc. Na. cad. Sc., Ida, Sc. 65(a II, har, B. B. ad Srvasava, S. (996: Trasord roduc saors or oulao a rsc o socor obsrvaos. roc. Mah. Soc. B. H.U.,, har, B. B. ad Srvasava, S. (997: Trasord rao saors or oulao a rsc. ou. Sas. Thor Mh., 6(7, har, B.B. ad Sha,.. (7: Esao o h rao o h wo oulaos as usg ul-auxlar characrs h rsc o o-rsos. I Sascal Tchqu Tsg, labl, Salg Thor ad Qual orol. Edd b B.N. ad, Narosa ublshg hous, Nw Dlh, har, B.B., uar ua, Sha.., ad S..(8: Two has salg saors or oulao a usg auxlar characr rsc o o-rsos sal survs. Jour. Sc. s., BHU, Varaas, 5: har, B.B. ad uar, S. (9: Trasord wo has salg rao ad roduc saors or oulao a h rsc o orsos. lgarh J. Sas., 9, har, B.B. ad Srvasava, S. (: Gralzd wo has salg saors or h oulao a h rsc o orsos. lgarh. J. Sas.,, har, B. B. ad uar, S. (: ha rgrsso saors usg addoal auxlar varabl wo has salg h rsc o o rsos. Na. cad. Sc. rs, Ida,, No. ( &, har, B.B. ad uar, S. (: gralzd cha rao saor or oulao a usg coc o varaos o h sud varabl. Na. cad. Sc. rs, Ida, (9-, har, B.B., Srvasava, U. ad alsh uar (: Gralzd cha saors or h oulao a h rsc o o-rsos. roc. Na. cad. Sc., Ida, 8(, III har, B.B., ad ha, H. U. (: rovd cha rao-cu-rgrsso saor or oulao a h rsc o o-rsos. I. J. gr. Sa. Sc., (,

55 ajsh Sgh Flor Saradach (dors 7. ao,.s..s.(986:ao sao wh sub-salg h o-rsods. Surv Mhodolog. (: ao,.s..s.(99: ao ad rgrsso Esaors wh Sub-salg o h orsods.i:, Gur E.,Uulur V, dors. Daa Qual orol hor ad ragacs, Marcl Dr, Nw Yor Sgh, H.. ad uar, S. (: Esao o a rsc o o-rsos usg wo has salg sch. Sascal ars, 5, Srvasava, S.., har, B.B. ad Srvasava, S..(99: gralsd cha rao saor or a o oulao. Jour. Id. Soc. gr. Sa., (,

56 Salg Srags or F oulao Usg uxlar Iorao Th rs boo as o rs so rovd saors usg auxlar ad arbu orao cas o sl rado salg ad srad rado salg ad so cass wh o-rsos s rs. Ths volu s a collco o v ars, wr b sv co-auhors (lsd h ordr o h ars: Sach Mal, ajsh Sgh, Flor Saradach, B. B. har,. S. Jha, Usha Srvasava ad Habb Ur. ha. Th rs ad h scod ars dal wh h robl o sag h oulao a wh so orao o wo auxlar arbus ar avalabl. I h hrd ar, robls rlad o sao o rao ad roduc o wo oulao a usg auxlar characrs wh scal rrc o o-rsos ar dscussd. I h ourh ar, h us o coc o varao ad sha arars ach srau, h robl o sao o oulao a has b cosdrd. I h h ar, a sud o rovd cha rao-cu-rgrsso saor or oulao a h rsc o o-rsos or xd cos ad scd rcso has b ad. Th auhors ho ha h boo wll b hlul or h rsarchrs ad suds ha ar worg h ld o salg chqus. 55