A Class of Improved Estimators for Estimating Population Mean Regarding Partial Information in Double Sampling

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1 Gloal Journal of Scnc Fronr Rsarch Mahmacs and Dcson Scncs Volum Issu 4 Vrson.0 ar 0 p : Doul Blnd Pr Rvwd Inrnaonal Rsarch Journal Pulshr: Gloal Journals Inc. USA Onln ISSN: & Prn ISSN: A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul B Hna Khan, Salha Shouk & Aamr Sanaullah G Unvrs, Lahor, Paksan Asrac - In hs papr a class of mprovd smaors has n proposd for smang populaon man n wo phas doul samplng whn onl paral nformaon s avalal on hr of wo aular varals. Undr smpl random samplng SRWOR, prssons of man squar rror and as hav n drvd o mak comparson of suggsd class wh wd rang of ohr smaors. mprcal sud has also n gvn usng fv dffrn naural populaons. mprcal sud confrmd ha h suggsd class of mprovd smaors s mor ffcn undr prcn rlav ffcnc PR crron. Kwords : doul samplng, aular varal, paral nformaon, as, man squar rror. GJSFR-F lassfcaon : MS 00 : 6F, 06A06 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Srcl as pr h complanc and rgulaons of : 0. Hna Khan, Salha Shouk & Aamr Sanaullah. hs s a rsarch/rvw papr, dsrud undr h rms of h rav ommons Aruon-Noncommrcal.0 Unpord Lcns hp://cravcommons.org/lcnss/-nc/.0/, prmng all non commrcal us, dsruon, and rproducon n an mdum, provdd h orgnal work s proprl cd.

2 Nos A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Hna Khan α, Salha Shouk σ & Aamr Sanaullah ρ 0 ar Asrac - In hs papr a class of mprovd smaors has n proposd for smang populaon man n wo phas doul samplng whn onl paral nformaon s avalal on hr of wo aular varals. Undr smpl random samplng SRWOR, prssons of man squar rror and as hav n drvd o mak comparson of suggsd class wh wd rang of ohr smaors. mprcal sud has also n gvn usng fv dffrn naural populaons. mprcal sud confrmd ha h suggsd class of mprovd smaors s mor ffcn undr prcn rlav ffcnc PR crron. Kwords : doul samplng, aular varal, paral nformaon, as, man squar rror. h hsor of us of aular nformaon n surv samplng s as old as h hsor of surv samplng. Bowl 96 and Nman 94, 8 provd foundaon sons of modrn samplng hor, dalng wh srafd random samplng. Hansn and Hurw 94 frsl us aular nformaon n slcng sampl wh varng proals. Sndcor and Kng 94, Spurr 95, Frs 96, Unnkrshan and Kun 995, Armsrong and S-Jan 994 provd applcaons of wo phas or doul samplng procdur. h smaon of h populaon man s an unrlnng ssu n samplng hor and svral ffors hav n mad o mprov h prcson of h smaors n h prsnc of mul-aular varals. Mohan 967 suggsd rgrsson cum rao smaor n doul samplng usng wo aular varals. rpah 970 and Das 988 dscr h aular nformaon n four was. Das and rpah 978 na o us populaon varanc of aular varal for smang h populaon varanc. Srvasava and Jhajj 980 also consdr h us of populaon man and varanc of aular varal for smang populaon varanc of h sud varal. Svral ohr auhors hav also usd nformaon on h paramrs of aular varal o fnd mor prcs smas. Rgardng h us of nformaon on,,,, and h rsarchr ma rfrrd o Sar 964, Sngh al. 97, Sn978, Sngh 00, Uphadhaa and Sngh 00, Sngh al. 006, Sngh al 007, and Sngh al. 0. Followng hand 975 and Krgra 980, 984, Sahoo and Sahoo 99 and Sahoo al. 994 hav dscussd a gnral fram work of smaon usng an Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I Auhor α σ : G Unvrs, Lahor, Paksan. Auhor ρ : Naonal ollg of Busnss Admnsraon and conomcs, Lahor, Paksan. 0 Gloal Journals Inc. US

3 addonal aular varal for doul samplng whn h populaon man of h man aular varal s unknown. Krgra 984 dvlopd wo smaors, on s rao-nrgrsson and ohr s rgrsson-n-rgrsson smaor. Mukrj al. 987 dvlopd hr smaors followng Krgra s 984 chnqu. Sahoo s 99 class of smaors covrd a larg numr of smaors. Ro 00 consrucd a rgrsson-p smaor of populaon man of h man varal n h prsnc of avalal knowldg on scond aular varal, whn h populaon man of h frs aular varal was no known. Samuddn and Hanf 007 hav rpord hr dffrn mhods of smaon n doul samplng. hs mhods ar proposd dpndng whhr nformaon of aular varals s avalal or no a frs phas of samplng. Sngh al. 0 proposd chan rao p smaor for populaon man usng som known valus of populaon paramrs of scondar aular varal. L U = U, U,, U n a fn populaon conssng of N uns. L and, h vara of nrs and aular characrscs rspcvl rlad o assum ral non-ngav h valu,, =,,, N wh populaon mans, X and rspcvl. L a smpl random sampl whou rplacmn SRSWOR s drawn n ach phas, h wo phas or doul samplng schm s as follows:. h frs phas sampl S S U of s n s drawn o masur and sa,.. h scond phas sampl S S S of s n n < n s drawn from h frs phas sampl S o masur sa. L,,,, s s s s n n n n and s n For a SRSWOR, w hav som assumpons as followng, N n N n and X X,, 0,,,,. In man praccal suaons vn f X s unknown, nformaon on a scondar aular varal, closl rlad o u compard o rmol rlad o, s radl avalal on all uns of populaon such ha dnos s valu on h un and 4 0 Gloal Journal of Scnc Fronr Rsarch Volum XII Issu rson I V XIV F 0 Gloal Journals Inc. US ar A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul, Nos

4 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul as s known man s Sngh al. 004 and Sngh al.006. For nsanc, f h lmns of populaon ar hospals, and, and ar rspcvl h numr of dahs, numr of pans admd and numr of avalal ds rlang o h h hospal, hn nformaon on s can collcd asl from h offcal rcords of h Halh Dparmn. hs suaon has also n dscussd chand 975, Mukhrg al. 987, Sahoo and Sahoo 99, Ro 00 and among man ohrs. Nos In hs scon w rproduc som wll known rao p smaors for h populaon man avalal for doul samplng undr SRWOR rgardng onl paral nformaon ar avalal. h varanc of h usual unasd smaor undr SRSWOR schm s as; Var. 0 ar 5 Mohan 967 rgrsson o rao smaor [ ].. hand 975 chan rao smaor Krgra 984 rgrsson n rgrsson smaor 5 Bd 985 rao smaor Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I Gloal Journals Inc. US

5 6 Mukhrg a 987 rgrsson smaor Srvasava al 990 rao smaor Sahoo al 994a rgrsson n rgrsson smaor Sngh 00 chan rao p smaor: g Sngh al. 004 gnrald smaor: 0 a a a a Samuddn and Hanf 006 rao cum rgrsson smaor: X.0. Samuddn and Hanf 007 chan rao smaor 6 0 Gloal Journal of Scnc Fronr Rsarch Volum XII Issu rson I V XIV F 0 Gloal Journals Inc. US ar A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos.

6 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul... Sngh al. 007 gnral faml of rao smaors Nos H.P.Sngh and N.Agnhor 008 rao produc smaor ax a.6 a ax mn..7 In scon- mnond smaors hav n wdl usd n h smaon of populaon man n dvrs suaons rgardng paral nformaon. Now followng smaors sad aov, w hav proposd wo smaors n hs scon rgardng h avalal of paral nformaon. On smaor scon-. has n proposd rgardng paral nformaon on man aular varal and ohr smaor scon-. hav n proposd rgardng paral nformaon on scondar varal for doul samplng undr SRWOR. a Proposd smaor n wo phas doul samplng h nw smaor has n proposd for wo phas samplng usng wo aular varals rgardng paral nformaon on man aular varal. h smaor has n convncd Srvasava 97 and Sngh 00 rao smaors. a a. Whr, s an unknown consan whos valus s o sma. a 0, and ar assumd o known as hr ral numrs or Lnar or Non-lnar funcons of som known paramrs of aular varal such as sandard dvaon, coffcn of varaon, skwnss, kuross. Usng h noaons gvn n., s prssd n h form of s and up o h frs dgr of appromaon as: 0 ar 7 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I 0 Gloal Journals Inc. US

7 whr a a,. akng squar and applng pcaon, [gvn n.], h man squar rror of. s oand as:. Dffrnang. wh rspc o and and sng qual ro. W hav: and akng h valus of and n quaon., and smplfng. W g mn. :. mn whr.4 In ordr o drv as of., w agan us. upo h nd ordr of appromaon as:!! Afr applng pcaon and smplfng, h opmum as of. s: 4 4 Bas whr.5. Dducd Faml of A larg numr of smaors hav n dducd as a faml of proposd smaor undr cran chocs of h consans, a, and. hs dducd smaors hav n prsnd n h followng al. 8 0 Gloal Journal of Scnc Fronr Rsarch Volum XII Issu rson I V XIV F 0 Gloal Journals Inc. US ar A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos

8 Dducd smaor a 0 Shkham s 96 rao smaor wo phas produc smaor hand 975 smaor 0 0 a 0 Sng and Upadhaaa s smaor 00 4 Upadhaa and Sngh 00 5 Sngh 00 smaor 6 Sngh 00 smaor 7 Sngh al.007 smaor Anohr proposd smaor for wo phas doul samplng In hs scon anohr smaor dnod has n suggsd for doul samplng usng wo aular varals. h suggsd smaor has n convncd H.P. Sngh and N. Agnhor 008 and Bd 985 rao smaors for wo phas samplng rgardng paral nformaon on scondar aular varal. ax a a ax.6 Whr, and ar h unknown consans whos valus ar o smad. 0, a and ar assumd o known as hr ral numrs or Lnar or Non-lnar funcons of som known paramrs of aular varal as Scon-.. whr ax ax 0 Gloal Journals Inc. US 9 Gloal Journal of Scnc Fronr Rsarch Volum XII Issu rson I V XIV F 0 ar A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos

9 .7 akng squar and applng pcaon, [gvn n.], h man squar rror of.6 s oand as:.8 Dffrnang.8 wh rspc o and and sng qual ro. W hav: and akng h valus of and n quaon.8, and smplfng. W g mn. :. mn.9 In ordr o drv as of.6, w agan us.7 upo h nd ordr of appromaon as: Afr applng pcaon and smplfng, h opmum as of.6 s: Bas.0. Dducd Faml of A larg numr of smaors hav n dducd as a faml of proposd smaor undr cran chocs of h consans, a, and. hs dducd smaors hav n prsnd n h followng al. Dducd smaor a 0 Usual man pr un X Usual rao p 0 0 X Usual produc p Gloal Journal of Scnc Fronr Rsarch Volum XII Issu rson I V XIV F 0 Gloal Journals Inc. US ar A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos

10 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos X Ssoda and Dwvd 98 p smaor 4 X Pand and Du 988 p smaor 5 X Upadhaa and Sngh 999 p smaor 6 X Upadhaa and Sngh 999 p smaor 7 X G.N. Sngh 00 p smaor 8 X G.N. Sngh 00 p smaor 9 X G.N. Sngh 00 p smaor 0 X Sngh and alor 00 p smaor X Sngh and alor 00 p smaor X Sngh al. 004 p smaor X Sngh al. 004 p smaor 4 X han rao p smaor 5 X Produc o raop smaor ar 4 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I In addon o hs smaors a larg numr of smaors can also dducd from h proposd faml of smaors pung valus of, a, and. I s osrvd ha h prsson of h frs ordr appromaon of of h gvn numr of h faml can oand mr susung h valus of, a, and n.8. 0 Gloal Journals Inc. US

11 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul IV. 0 ar 4 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I o anal h prformanc of proposd smaors n comparson o ohr smaors, fv populaon daa ss ar ng consdrd. In wo phas samplng undr SRSWOR, h comparson of proposd smaors and wh rspc o usual unasd smaor, Mohan 967, hand 975, Mukhrg a 987, Srvasava al 990, Sahoo al 994a, Sngh 00, Sngh al. 004, Samuddn and Hanf 006, Samuddn and Hanf 007, and Sngh al. 007 hav n mad rgardng h avalal of paral nformaon onl. h dscrpons of populaons ar gvn low. Populaon-I. Daa usd Andrson 958 : Had lngh of scond son X: Had lngh of frs son : Had radh of frs son N = 5, = 8.84, X =85.7, =5., =0.0546, =0.056, =0.0488, =0.708, =0.69, =0.746, n =0, n =7 Populaon-II. Sourc: Nachshmm, Nr and Kunr. Advancd appld lnar modls, 004 : No of prsons low povr lvl X: No of unmplod prsons : oal populaon N = 440, = 9.50, X = , = 59.7, =.9955, =.750, =.57, = 0.956, = 0.9, = 0.969, n = 88, n =8 Populaon-III. Sourc: Populaon cnsus rpor of Okara dsrc 998, Paksan : Populaon Marc and aov X: Prmar u low Marc : Populaon oh ss N = 00, =4.5, X = 4.58, = , =.85, =.088, = , = 0.894, = 0.84, = 0.94, n = 60, n = Populaon-IV. Sourc: Populaon cnsus rpor of Gujra dsrc 998, Paksan : Populaon Marc and aov X: Prmar u low Marc : Populaon oh ss N = 00, =.5, X = 56.84, = , =.5, = 0.99, = , = 0.97, = 0.89, = 0.97, n = 60, n = Populaon-V. Sourc: Nachshmm, Nr and Kunr. Advancd appld lnar modls, 004 : Grad-pon avrag followng frshman ar : A nranc amnaon scor X: Hgh school class rank as prcnl: lowr prcnl mpl hghr class rank N = 705, =.977, X = 76.95, = 4.54, = 0., = 0.457, = 0.657, = 0.98, = 0.66, = 0.44, n = 4, n =8 Nos al 4. : PR s of dffrn proposd smaors of n doul samplng w.r. smaor Populaon #: 4 5 Usual unasd smaor Mohan Gloal Journals Inc. US

12 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos hand Mukhrg al Srvasava al Sahoo al. 994a Sngh Sngh al Sammudn & Hanf Sammudn & Hanf Sngh al Proposd smaor Proposd smaor V. W hav suggsd wo mprovd smaors and. From al 4., w conclud ha h proposd smaors ar r han usual unasd smaor, Mohan 967, hand 975, Mukhrg a 987, Srvasava al 990, Sahoo al 994a, Sngh 00, Sngh al. 004, Samuddn and Hanf 006, Samuddn and Hanf 007, and Sngh al I s also osrvd ha among h class of suggsd smaors, prforms mor ffcnl cp n populaon-i and populaon- V, n comparson wh proposd smaor and Mukhrg al. 987 rspcvl. I s furhr osrvd ha, Srvasava al. 990 and Sngh al. 007 ar prformd quall. I s also osrvd ha h prformanc of s also farl good hough sms slghl lss ffcn n comparson wh. Hnc proposd smaors ar rcommndd for hr praccal us f onl paral nformaon ar avalal.. Armsrong, J. and S-Jan, H. 99. Gnrald rgrsson smaor for a wophas sampl of a rcords. Surv Mhodolog, 0, Bd, P.K On wo-phas mulvara samplng smaor. Journal of h Indan Soc of Agrculural Sascs, 7, Bowl, A.L. 96. Masurmns of prcson aand n samplng. Bull. Ins. In. Sas.,, hand, L Som Rao-p smaors asd on wo or mor Aular Varals. Unpulshd Ph.D. Dssraon. Iowa Sa Unvrs, Iowa. 5. Das, A.K onruon o h hor of samplng srags asd on aular nformaon. Ph. D. hss Sumd o B.. K.V., Mohanpur, Nada, Ws Bngal, Inda. 6. Das, A.K. and rpah,.p Us of aular nformaon n smang h fn populaon varanc. Sankha, Sr., 40, ar 4 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I 0 Gloal Journals Inc. US

13 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul 0 ar 44 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I 7. Frs, F. 96. lmnar fors samplng. USDA For. Sr. Agrc. Hand.. 8. Hansn, M.H. and Hurw, W.N. 94. On h hor of samplng from fn populaons. Ann. Mah. Sas. 4, Krgra, B A chan rao-p smaor n fn populaon doul samplng usng wo aular varals. Mrka. 7, Krgra, B Rgrsson-p smaor usng wo aular varals and modl of doul samplng from fn populaons. Mrka., Mohan, S omnaon of Rgrsson and Rao sma. J. Ind. Sas. Assoc. 5, Mukrj, R., Rao,.J. and Vjaan, K Rgrsson p smaors usng mulpl aular nformaon. Ausral. J. Sas. 9, Nman, J. 94. On h wo dffrn aspcs of rprsnav mhod: h mhod of srafd samplng and h mhod of purposv slcon. J. Ro. Sas. Soc., 97, Nman, J. 98. onruon o h hor of samplng human populaons. J. Amr. Sas. Assoc.,, Pand, B.N. and Du, Vas 988: Modfd produc smaor usng coffcn of varaon of aular vara, Assam Sascal Rv.,, Ro, D A rgrsson p smas n wo-phas samplng usng wo aular varals. Pak. J. Sas. 9, Sahoo, J. and Sahoo, L.N. 99. A class of smaors n wo-phas samplng usng wo aular varals. J. Ind. Sas. Assoc., Sahoo, J., Sahoo, L. N. and Mohan, S. 994a. A rgrsson approach o smaon n wo-phas samplng usng wo aular varals. urrn Scnc, 65, Sahoo. J., Sahoo, L.N. and Mohan, S An alrnav approach o smaon n wo-phas samplng usng wo aular varals. Bomrcal J.,, Samuddn, M. and Hanf, M smaon of populaon man n sngl and wo phas samplng wh or whou addonal nformaon. Pak. J. Sas.,, Sarls, D h ulaon of a known coffcn of varaon n h smaon procdur. J. Amr. Sas. Assoc., 59, Sn, A.R smaon of h populaon man whn h coffcn of varaon s known. ommun. Sas. hor. Mhod, A 77, Sngh H.P. and alor, R. 00: Us of known corrlaon coffcn n smang h fn populaon man. Sascs n ranson, 64, Sngh H.P. Sngh S. and Km Jong-Mn 006. Gnral famls of chan rao p smaors of h populaon man wh Known coffcn of varaon of h Scond aular varal n wo phas. Journal of h Koran Sascal Soc, 54, Sngh, H.P., alor, R. and Kakaran, M.S. 004: An smaor of Populaon man usng powr ransformaon. Jour. Ind. Soc. Agr. Sascs, 58, Sngh, R. hauhan P. Swan N. and Smarandach F. 0. Ialan journal of pur and appld mahmacs-n, Sngh, G.N. 00. On h us of ransformd aular varal n h smaon of populaon man n wo phas samplng. Sascs n ranson. 5, Sngh, G.N. 00: On h mprovmn of produc mhod of smaon n sampl survs. Jour. Ind. Soc. Agr. Sascs, 56, Nos 0 Gloal Journals Inc. US

14 A lass of Improvd smaors for smang Populaon Man Rgardng Paral Informaon n Doul Nos 9. Sngh, H.P. and Agnhor.N A gnral procdur of smang populaon man usng aular nformaon n sampl surv. Sascs n ranson, 9, Sngh, H.P., Upadhaa, L.N. and handra, P A gnral faml of smaors for smang populaon man usng wo aular varals n wo-phas samplng. Sascs n ranson, 67, Sngh, J. Pand, B.N. and Hrano, K. 97. On h ulaon of a known coffcn of kuross n h smaon procdur of varanc. Ann. Ins. Sa. Mah., 5, Sngh, R., huhan, P. and Sawan, N A Faml of smaors for smang Populaon Man Usng Known orrlaon offcn n wo Phas. Sascs n ranson, 8, Ssoda, B.V.S. And Dwvd, V.K. 98: A modfd rao smaor usng coffcn of varaon of aular varal. Journ. Ind. Soc. Agrl. Sas.,, Sndcor, G.W. and Kng, A.J. 94. Rcn dvlopmns n samplng for agrculural sascs. J. Amr. Sas. Assoc., 7, Spurr, S.H. 95. Fors Invnor. h Ronald Prss ompan. 6. Srvasava, S. Ran, Khar, B.B. and Srvasava, S.R A gnrald chan rao smaor for man of fn populaon. J. Ind. Soc. Agr. Sas, 4, Srvasava, S.K. and Jhajj, H.S A class of smaors usng aular nformaon for smang fn populaon varanc. Sankha, Sr.. 4, Sukham, B.V. 96. Som rao-p smaors n wo-phas samplng, J. Amr. Sas. Assoc., 57, rpah,.p onruons o h samplng hor n mulvara nformaon. Ph.D. hss sumd o Punja Unvrs, Paala, Inda. 40. Unnkrshan, N.K. and Kun, S. 995 Opmal of an analogu of Basu s smaor undr a doul samplng dsgn. Sankha, 57, Upadhaa, L.N. and Sngh, G.N. 00, han p smaors usng ransformd aular varal n wo-phas samplng. Advancs n Modlng and Analss, 8-, Upadhaa, L.N. and Sngh, H.P. 999: Us of ransformd aular varal n smang h fn populaon man. Bomrcal Journal, 45, ar 45 Gloal Journal of Scnc Fronr Rsarch F Volum XII Issu XIV V rson I 0 Gloal Journals Inc. US