Pa. J. Statst. 5 Vol. 3(4), 353-36 A CLASS OF TRANSFORMED EFFICIENT RATIO ESTIMATORS OF FINITE POPULATION MEAN Sajjad Ahmad Khan, Hameed Al, Sadaf Manzoor and Alamgr Department of Statstcs, Islama College, Peshawar, Pastan Department of Statstcs, Unverst of Peshawar, Pastan Correspondng author Emal: sajjadhan@cp.edu.p ABSTRACT In ths paper we have proposed a class of transformed effcent estmators of fnte populaton mean b modfng Mohant and Sahoo (995) transformaton. The appromate bas and MSE of the proposed class of estmators have been found. Condtons under whch the proposed class of transformed estmators performs good, have been obtaned. Eamples are gven based on real lfe data to llustrate the results. KEYWORDS Aular Informaton, Transformaton, Effcenc, Mean Square Error.. INTRODUCTION In ths paper we have proposed an effcent estmator for estmatng the mean of the fnte populaton under smple random samplng wthout replacement samplng and stratfed random samplng schemes. Motvated b Subraman and Kunarpendan (), Bed (996), Mohant and Sahoo (995), Shabbr and Gupta (), we have made use of aular nformaton b ntroducng some transformaton. The Classcal estmator of the mean of the fnte populaton Y s. Varance of s V C, where n N (.) Cochran (94) ntroduced the tradtonal rato tpe estmator of populaton mean, as cr X wth the followng bas and MSE and cr (.) B Y C C (.) cr MSE Y C C C C. (.3) 5 Pastan Journal of Statstcs 353
354 A Class of Transformed Effcent Rato Estmators of Fnte Populaton Mean S S where C s the coeffcent of varaton of varable Y, C s the coeffcent Y X S of varaton of X, C s the coeffcent of covarance of X and Y and YX C s the coeffcent of correlaton between Y and X. CC The classcal regresson estmator gven b Hansen and Harwtz (943) s lr b X (.4) wth the followng mean square error lr MSE Y C (.5) Ths s a well-establshed fact that regresson estmator s effcent when the lnear relatonshp passes n the neghborhood of orgn. and An eponental rato tpe estmator due to Bhal and Tuteja (99) s gven b BT ep X X The bas and MSE due to 3C C B BT Y 8 BT s gven b MSE Y C C 4 C BT (.6) (.7) (.8) Mohant and Sahoo (995), ntroduces the followng transformaton n rato estmator of populaton mean and where t Z z t z U u M m m and u m s the mnmum and M M M m. s the mamum value of aular varable wth n the data.
Khan, Al, Manzoor and Alamgr 355 The bases and MSEs of t and t are gven b z z B t Y S S. (.9) u u B t Y S S. (.) z z MSE t Y C C C. (.) u u MSE t Y C C C. (.) where C C C C and c m M, c X X C C z,, u z c c c Shabbr and Gupta () proposed the followng transformed eponental rato tpe of estmator A a SG X ep A a (.3) where a NX and A X NX Wth the followng MSE MSE SG C C 8 N Y 4 N C where the optmum value of and s gven b (.4) C 8 N C Y C and X N N C.. SUGGESTED TRANSFORMATION Mohant and Sahoo (995) suggested the followng transformaton of the aular varable
356 A Class of Transformed Effcent Rato Estmators of Fnte Populaton Mean and or and (.) where m and M. Further we can wrte X X V and U. V f C U f Further we wrte V n and, c U n c where c and c. (3.) X X Obvousl, for the above transformaton one has to fnd out two features of aular varable, such as and for estmator of fnte populaton mean. One of the dsadvantage of the above transformaton s that, onl one constant are tang part n the reducton of MSE whle the other has no contrbuton at all. In order to remove the etra feature from estmator and to ncorporate onl those feature whch are tang contrbuton n the reducton of MSE, we precede as followng. We wrte (.) as c or Xc X where s constant or some functon of aular nformaton. The proposed transformaton s, or X c, and C E X, where 3. PROPOSED ESTIMATOR IN SIMPLE RANDOM SAMPLING X cx. Followng Bhal and Tuteja (99) and Shabbr and Gupta (), we suggest the followng class of estmators X pr X ep,,,3,4 (3.) X where X Xc and. X c, or E X Further c C, c, c3, c4.,,3,4. 3
Khan, Al, Manzoor and Alamgr 357 where & are sutable constant to be determned so that MSE s mnmum, where c s to be chosen, so that the transformed data results hgh gan n effcenc. We wll use the followng terms to fnd the Bas and MSE due to our proposed estmator e such that and Y e Y, E e E e E e e X e X or X c X,, C or E e e E e C E e C C C. or or Then we can wrte (3.) as follows e e pr Y e X e ep c c e 3e pr Y e X e 8 c c Terms wth power hgher than two can be gnored, we have e ee 3e pr Y Y e c c 8c We can defne the Bas due to our proposed estmator b pr pr Bas E Y e X e c 3C C C Bas pr Y X 8c c t For MSE, Squarng and tang epectaton of equaton (3.3), we have (3.) (3.3) (3.4) e ee 3e 3e E pr Y Y Y e X e c c 8c c (3.5) Snce pr pr MSE E Y
358 A Class of Transformed Effcent Rato Estmators of Fnte Populaton Mean C C 3C C MSE pr Y C c c 8 c c C C X C YX YX C c c (3.6) The optmum values of & can be obtaned b dfferentatng MSE of pr wth respect to the & b Dfferentatng (3.6) w.r.to and equatng to zero Y C 3C C c c 8c MSE pr C 3C c we get 3C YX C c Dfferentatng (3.6) w.r.to and equatng to zero, we get C C X C YX YX C c c MSE pr (3.7) we get (3.8) Solvng equaton (3.7) & (3.8) smultaneousl to get the optmum values of &. After smplfcaton we have C c 8 Y 3 C opt and opt opt C X c c C Substtutng these values n (3.6), we get MSE pr C Y C 8c C (3.9)
Khan, Al, Manzoor and Alamgr 359 4. THEORETICAL COMPARISON OF THE PROPOSED ESTIMATOR IN SRSWOR In ths secton we wll compare the MSE of our proposed estmator wth the estng estmator dscussed here above. ) MSE MSE If pr C 8c C C C ths s hold true for all choces of c. ) MSE MSE If pr cr C 8c C C C CC C 3) MSE MSE If pr lr C 8c C C C 4) MSE MSE If pr BT C 8 C c C C C 4 C 5) MSE MSE If pr SG
Parameters 36 A Class of Transformed Effcent Rato Estmators of Fnte Populaton Mean C C C 8c 8 C N 4 C C N C 8 C N c c 4 N C Condtons to 5 wll alwas hold true for all tpes of real data especall for some choces of c. 5. NUMERICAL COMPARISON OF THE PROPOSED ESTIMATOR FOR SRS In ths secton we wll mae an assessment of the effcenc of our proposed estmator under SRS usng data sets from real lfe eamples. It should be noted that f we put c N or X X NX then our proposed estmator wll reduces to Shabbr and Gupta () estmator. Populaton Source: US Agrcultural Statstcs (995) Y ; Speces group n 995 X ; Speces group n 995 Populaton 3 Source: Pastan MFA (4) Y ; Dstrct wse tomato producton (n tons), n Pastan (3) X ; Dstrct wse tomato producton () Populaton Source: Agrculture Statstcs () Y ; The State wse producton of major spces n thousand metrc tons of Inda. (,) X ; The State wse producton of major spces n thousand metrc tons of Inda.(,) Populaton 4 Source: Murth (967) page 8 N 69 97 9 8 N 7 5 3 Y 454.9 335.6 84.5 5.864 X 455.6 35.8 38.476.664 C.483.37893.753.757 C.3756.3.856.354.9.987.4453.953
Khan, Al, Manzoor and Alamgr 36 Estmators Table Mean Square Error of the Proposed Estmator for c C, c, c3, c4 3 and the Estng Estmators Populaton Populaton Populaton 3 Populaton 4 7995.485 954.736 54789.37 67.86 cr 38874.563 38.8 458.7 96.84 lr 3779. 5.63 3963.5 887.57 t 3673.3 6.77 5453.9 9339.943 t 36544.483 733.54 589.63 9473.9 BT 3836.836 78.85 7768.63 9574.837 SG 384.536 98.8 39463.855 8766.5 Proposed Class of Estmators pr 3876.86 96.99 3936.7 876.599 pr 7493.55 657.4 567.6 867.6 pr3 364.3 863.47 3379. 87.855 pr4 3484.3 956.834 3887.953 8753.95 CONCLUSION From the above analss t s clear that our proposed estmators for varous value of transformer c C, c, c3, c4 are superor to all other estmators 3 consdered n ths paper. Hence the transformaton results consderable reducton n MSE, as obvous from the above table and hgh gan n effcenc. Smlar strateg can be used for further optmzaton of estng estmators or for the development of new estmators.
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