An Improved Double Sampling Regression Type Estimator of Population Mean Using Auxiliary Information

Transcription

1 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) A Improvd Doubl amplg Rgrsso Tp stmator of Populato Ma Usg Auxlar Iformato Push Msra Dpartmt of tatstcs, D.A.V.(P.G.) ollg, Dhradu- 8, Uttarahad (Ida) Abstract: Ths papr provds a w mprovd doubl samplg rgrsso tp stmator of populato ma usg auxlar formato. Its bas ad ma squar rror ar foud ad ts comparatv stud wth som of th wll ow stmators s do. From umrcal xampls t s also obsrvd that th proposd mprovd rgrsso tp stmator prforms bttr tha th xstg stmators. Kwords: Auxlar Varabl, Bas, Ma quard rror ad ffcc.. Itroducto: Oft, statstcas ma us of th formato avalabl o a auxlar varabl wth th varabl udr stud for mprovg th ffcc of a stmator. For bttr udrstadg o ma s ochra (977), Ds Raj (968), Murth (967), Muhopadha (), gh & haudhar (997) ad uhatm t, al. (98). It s wll ow that th us of auxlar formato sampl survs rsults substatal mprovmt th prcso of th stmators of th populato paramtrs ad w ow that somtms paramtrs of th auxlar varabls ar ot ow advac th doubl or two phas samplg tchqu s usd. I doubl samplg or two-phas samplg tchqu, w frst ta a prlmar larg sampl of sz (calld frst phas sampl) from a populato of sz ad th a sub-sampl of sz (calld scod phas sampl) s draw from th frst phas sampl of sz usg smpl radom samplg wthout rplacmt at both th phass. At frst phas sampl of sz, ol th auxlar varabl b obsrvd but at th scod phas sampl of sz, th stud varabl ad th auxlar varabl both ar obsrvd. Lt b th populato ma of stud varabl ad b th populato ma of auxlar varabl x., ad b th populato corrlato coffct btw ad x. r s Also lt rs,,,,,,. Lt th frst phas sampl of sz b x, x,..., x o x ad th scod phas sampl of sz b, x,, x,...,, x o varabls, x wth th frst phas sampl ma st Jul, 7 Pag o:

2 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) x x stmator of populato ma ad th scod phas sampl ma ad ma x x rspctvl o ad x. For smplct, t s assumd that s larg ough as compard to so that ft populato corrcto trms ma b gord. A w mprovd doubl samplg rgrsso tp stmator rprstd b ŷ for stmatg th populato ma s proposd as s ˆ b (.) x x whr b s a stmat of th chag wh x s crasd b ut.. Bas ad Ma quar rror of th Proposd stmator: I ordr to obta bas ad ma squar rror of th proposd stmator, lt us dot b x x s s x sx (.) so that gorg ft populato corrcto, for smplct w hav (.) x st Jul, 7 Pag o:

3 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) (.) Th proposd w mprovd doubl samplg rgrsso tp stmator rprstd b ŷ for stmatg th populato ma gv (.) s s ˆ b x x (.) I trms of s,,,,, ; th abov proposd doubl samplg rgrsso tp stmator up to trms of ordr O ˆ whr. rducs to (.5) Tag xpctato o both th sds of (.5), th bas of ŷ up to trms of ordr b Bas ŷ ˆ (.6) O s gv st Jul, 7 Pag o:

4 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) st Jul, 7 Pag o: ow squarg both sds of (.5) ad tag xpctato, th ma squar rror of ŷ up to trms of ordr O s gv b M ŷ ˆ usg valus of th xpctato gv (.) ad (.), w hav M ŷ 5 6 (.7) whch attas th mmum for th optmum valu 5 6 (.8) ubsttutg th valu of gv b (.8) (.7), w gt th mmum ma squard rror of ŷ to b M m ŷ (.9). ffcc omparso: () Gral stmator of ma cas of RWOR: Th gral stmator of ma cas of RWOR s wor ˆ wth M ˆ (.) It s clar that th proposd stmator s mor ffct tha th stmator wor ŷ basd o smpl radom samplg wh o auxlar formato s usd. () Usual doubl samplg rgrsso stmator: Th usual doubl samplg rgrsso stmator s x x b ld wth M ld (.)

5 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) It s clar that th proposd stmator s mor ffct tha th usual doubl samplg rgrsso stmator whr th auxlar formato alrad s us.. mprcal tud: To llustrat th prformac of th proposd stmator, lt us cosdr th followg data Populato I: ochra (977, Pag umbr- 8) : Paraltc Polo ass placbo group x : Paraltc Polo ass ot oculatd group = , = , = 9.98, = 6.79, = 5.777, =.867, = 9.868, = 9.5, = , = , =.58859, x = , x = 8.777, =.757, =.7959, =.76769, x= 8.995, =.755, =.567, =.7689, =, =5 (sa). M =.9 ad M ŷm=.8678 wor ld M ˆ =.98698, PR of th proposd stmator ŷ ovr ŷ wor = PR of th proposd stmator ŷ ovr ld =.786. Populato II: Muhopadha (, Pag umbr - ) : Qualt of raw matrals ( lahs of bals) x : umbr of labourrs ( thousads) = 97.75, = 9.95, = 6.75, = , = , = 556.5, = 87.65, = 896.9, = 58.55, = 58., =.5, x =.95, =.8667, x x = , =.657, =.68576, =, =5 (sa). wor ld M ˆ =.575, = , = , =.9, =.9879, M = ad M ŷm = PR of th proposd stmator ŷ ovr ŷ wor = PR of th proposd stmator ŷ ovr ld =.77. Populato III: Murth (967, Pag umbr - 98) : umbr of absts x : umbr of worrs = , =.655, = 5.6, = , = 5.9, = , = 8.575, = 88.69, = , = , = , x = , x = 6.665, = , = , = , x= , =.56867, = , =.89886, =, =5 (sa). st Jul, 7 Pag o: 5

6 ITRATIOAL RARH JOURAL OF MULTIDIIPLIAR TUDI Vol., Issu 7, Jul, 7 I (Ol): Impact Factor:.599(GIF),.679(IIF) wor ld M ˆ = , M = ad M ŷm = PR of th proposd stmator ŷ ovr ŷ wor = PR of th proposd stmator ŷ ovr ld = oclusos: Th ma squar rror of th proposd stmator ŷ s mmzd for th optmum valu 6 5 ad th mmum ma squard rror of ŷ s gv b M m ŷ (5.). Th optmum valu volvg som uow paramtrs ma ot b ow advac for practcal purposs; hc th altratv s to rplac th uow paramtrs of th optmum valu b thr ubasd stmators gvg stmator dpdg upo stmatd optmum valu. (5.) Rfrcs:. ochra, W.G. (977): amplg Tchqus, rd dto, Joh Wl ad os, w or.. Ds Raj (968): amplg Thor, McGraw- Hll, w or.. Murth, M (967): amplg Thor ad Mthods, st dto, alcutta tatstcal Publshg oct, Kolata, Ida.. Muhopadha, Parmal (): Thor ad Mthods of urv ad amplg, d dto, PHI Larg Prvat Lmtd, w Dlh, Ida. 5. gh, Daroga ad haudhar, F.. (997): Thor ad Aalss of amplg urv Dsgs, w Ag Itratoal Publshrs, w Dlh, Ida. 6. uhatm, P. V., uhatm, B. V., uhatm,. Ad Aso,. (98): amplg Thor of urvs wth Applcatos, rd dto, Ams, Iowa (UA) ad Ida oct of Agrcultural tatstcs, w Dlh, Ida. st Jul, 7 Pag o: 6