Available olie at www.ia.org.i/jia JOURA OF THE IDIA OIETY OF AGRIUTURA TATITI 64() 00 55-60 Variace Etimatio for te Regreio Etimator of te Mea i tratified amplig UMMARY at Gupta * ad Javid abbir Departmet of Matematic ad tatitic, Uiverity of ort arolia at Greeboro, Greeboro, 740, UA Departmet of tatitic, Quaid-i-Azam Uiverity, Ilamabad 4530, Pakita Received 8 Jauary 00; Revied 9 April 00; Accepted 30 April 00 I ti paper, we propoe a cla of etimator for variace of eparate regreio etimator of mea i tratified amplig ad derive it propertie uder large ample approximatio. Te propoed cla of etimator perform better ta te traditioal regreio etimator ad te Wu (985) etimator. Mea quare error of differet etimator are compared umerically alo uig tree differet data et from te literature. Keyword: eparate regreio etimator, tratificatio, Bia, Mea quare error, Efficiecy.. ITRODUTIO et U be a fiite populatio of ize. Te tudy variable ad te auxiliary variable are deoted by y ad x repectively ad te populatio i partitioed ito o-overlappig trata accordig to ome caracteritic. Te ize of te t tratum i (,,..., ) uc tat. A tratified ample of ize i draw from ti populatio ad let be ample ize from t tratum uc tat. Te obervatio o y ad x correpodig to i t uit of t tratum (,,, ) are y i ad x i repectively. et y ad x be ample mea ad Y ad be populatio mea of y ad x repectively i t tratum. uppoe y W y ad t x Wx are t tratified ample mea ad Y WY ad W are populatio mea of y ad x repectively, were W / i kow tratum weigt. et y ( ) x i i ( y y i ) ad i x x be ample variace ad y ( y Y i ) ad i x ( x i ) be populatio variace of y i ad x repectively i t tratum. Fially, let yx * orrepodig autor : at Gupta E-mail addre : gupta@ucg.edu
56 at Gupta / Joural of te Idia ociety of Agricultural tatitic 64() 00 55-60 ( y y )( x x ) ad i i i i yx ( y Y )( x ) be ample ad populatio i i covariace repectively i t tratum. We aume tat all parameter correpodig to auxiliary variable x are kow. I ubequet preetatio, we igore fiite populatio correctio term ( / ) for computatioal eae. A well-kow etimator of Y i eparate regreio etimator y W { y + b ( x )}, were b i ample regreio coefficiet. Variace of y i give by V ( y ) W () were yx ( y x ) ( ρ ) / y ρ i populatio correlatio coefficiet betwee y ad x i t tratum. Te primary objective of ti paper i to preet a etimator of V ( y ) ad compare it wit ome of te kow etimator. A obviou etimator of V ( y ) i give by v W r () were r yx ( y x ) ( )/ y i ample correlatio coefficiet betwee y ad x i t tratum. Propertie of v ca be derived eaily oce we defie te followig error term: e 0 e y y y x x x x, e ad e 3 yx It ca be verified tat E(e i ) 0 (i 0,,, 3). Alo up to firt order of approximatio, we ave te yx yx followig expectatio tat ca be derived eaily o te lie of ukatme et al. (997): 0 E( ε ) E( ε ) E(e 0 e i ) E( ε ε ) 0 3 ( λ ), 40 ( λ ), 04 E( ε ) x E( ε ) 3 ρ l x, E(e 0 e ) E( ε ε 3 ) E( ε ε ) were 3 l pq ad m pq ρ 3, x ρ ρ µ 3 pq p/ q/ 0 0 µ µ i E( ε ε ) (l ) λ p ( y Y ) ( x ) i i ow writig v i term of e, we ave v ( + ε ) yx 3 W ( + ε ) y 0 ( + ε ) x q 03 (3) From (3), te bia ad ME of v are give by were B(V ) W A (4) / y ρ A (l 04 ) + 3 ρ ρ x
at Gupta / Joural of te Idia ociety of Agricultural tatitic 64() 00 55-60 57 ad ME(v ) were W 4 4 y ( λ ) ρ ρ 4 3 + B + 40 B ( λ ) + 4( λ ρ 04 ) ( λ ρ ) 4 3 ad ( λ ) ( λ ρ ) 3 (5) May autor ave preeted etimator tat exploit a variety of iformatio available from te auxiliary variable. Tee iclude Da ad Tripati (98), rivatava ad Jajj (980, 983), Wu (985), Praad ad ig (990, 99). I particular, Wu (985) a preeted a etimator of V ( y ) wic i give by W ( r ) x (6) v W ( ) g y were g i a appropriately coe cotat. Propertie of ti etimator will be dicued i te ext ectio.. PROPOED ETIMATOR Our propoed etimator of V ( y ) i a exteio of te Wu (985) etimator were we exploit iformatio o bot i give by ad. Te ew etimator x cotat, uc a te coefficiet of variatio ( x ), coefficiet of kurtoi (l 04 ) or coefficiet of correlatio (r ). ote tat (i) For k 0 ad k 0, we ave v P v. (ii) For k g, a k 0 ad k 0, we ave v P v W. ow we derive te propertie of v P uder large ample approximatio. Writig v P i term of e, we ave v P were ( + ε ) yx 3 W ( + ε ) y 0 ( + ε ) x k ( + φ ε ) ( + φ ε ) k f ( + a ) k ad f ( + b x x k) From (8), te bia of v P i give by B(v p ) W y ( λ ) ( ) ( ρ ) k k φ φ + k k + φ 03 x x k ( k )( λ ) + + 04 (8) + ρ k x k 3 φ φ + ρ ρ v p W + a + b k x k ( ) x a + k + b x k r y (7) were k i (i, ) are cotat woe value are to be determied ad a k ad b k are ome kow uit free k k k φ λ k ( λ ) 03 x 04 3 ( λ ρ 04 ) + ρ { k k ( ) } φ λ + φ λ x (9)
58 at Gupta / Joural of te Idia ociety of Agricultural tatitic 64() 00 55-60 ubtitutig k g, k 0 ad a k 0 b k i.e. (f f ) i (9), we ca get te bia expreio for te Wu (985) etimator a B(v w ) W + ρ y x ρ g λ ( ρ ) ( + ) g g x g λ + 3 03 x ρ ( ) λ g λ 04 x ρ Alo from (8), te ME of v p i give by ME(v P ) ME(v ) were ad E + W ( ρ ) 3 { k φx kφ λ04 4 4 y + ( ) + k k φ φ λ03 x + ( ρ ) } { k φ D k φ E } λ D λ ρ λ ρ x x 03 (0) () ( ) 3 λ ρ ( λ ) ρ 04 From (), optimum value of k i (i, ) are give by k ad k ( ) ' & ( 04 03 x x 04 03 φ φ ( ρ )( ) E & ' x 03 x x 04 03 ( ρ )( ) ubtitutig optimum value of k i (i, ) i (), we get miimum ME of v P a ME(v P ) mi ME(v ) W 3 4 4 y D E D ( λ ) x 03 + x x { λ λ 04 03 } () ote tat ubtitutig k g, k 0 ad a k 0 b k, i.e. (f f ) i (), we ca get te ME of v W a ME(v W ) ME(v ) + g g x W 3 D ( ρ ) 4 4 ( ρ ) y From (3), te optimum value of g i (3) * g D ( ) x. Puttig optimum value of g i (3), we ca get te miimum ME of v W a ME(v W ) mi ME(v ) W 4 4 D ( 3 y x ) (4) ote tat te optimum value of k i (i, ) ivolve ukow parameter. Wu (985) poited out tat, if computatioal eae i ot a iue ad ample ize i uc tat te aymptotic reult become effective, te etimator baed o etimated optimal value of k i (i, ) ca be ued. 3. OMPARIO OF ETIMATOR We ow compare te propoed etimator wit two oter etimator dicued above. It i eay to verify tat (i) ME(v P )mi < ME(v ) if 4 D ( E D ) x 03 W λ + > 0 3 y x { λ λ x 04 03 }
at Gupta / Joural of te Idia ociety of Agricultural tatitic 64() 00 55-60 59 (ii) ME(v P ) mi < ME(v W ) mi if > 0 { λ λ } 4 ( E D ) W λ x 03 3 y x 04 03 oditio (i) ad (ii) alway old true becaue λ 04 03 ( λ ) ³ 0 (ee Jajj et al. 005). We ue te followig expreio for obtaiig te percet relative efficiecie wit repect to v : PRE ME( v ) ME( v ) i 00, i, W, P Te reult reported i Table are baed o tree data et give i te Appedix. Table. Percet relative efficiecy of differet etimator wit repect to v Etimator Data Data Data 3 v 00.000 00.000 00.000 v W 8.460 4.73.970 v P 64.43 56.05 36.557 Reult i Table ow tat te performace of propoed etimator i better ta te oter competig etimator. Ti wa clearly expected baed o oditio (i) ad (ii) above wic alway old true. Tu utilizig te iformatio o populatio variace i additio to te populatio mea of x ca furter improve efficiecy of te Wu (985) etimator. AKOWEDGEMET Te autor are takful to te referee for teir valuable uggetio tat reulted i igificat improvemet i te preetatio of ti paper. REFEREE Da, A.K. ad Tripati, T.P. (98). A cla of amplig trategie for populatio mea uig iformatio o mea ad variace of a auxiliary caracter. Tecical Report o. 3/8, tat. ad Mat. Diviio, II, alcutta. Jajj, H.., arma, M.K. ad Grover,.K. (005). A efficiet cla of cai etimator of populatio variace uder ub-amplig ceme. J. Jap. tat. oc., 35, 73-86. Praad, B. ad ig, H.P. (990). ome improved ratio-type etimator of fiite populatio variace i ample urvey. omm. tatit.- Teory Metod, 9, 7-39. Praad, B. ad ig, H.P. (99). Ubiaed etimator of fiite populatio variace uig auxiliary iformatio i ample urvey. omm. tatit.- Teory Metod,, 367-376. ig, R. ad Magat,.. (996). Elemet of urvey amplig. Kluwer Academic Publier. rivatava,.k. ad Jajj, H.. (980). A cla of etimator uig auxiliary iformatio for etimatig te fiite populatio variace. akya, 4, 87-96. rivatava,.k. ad Jajj, H.. (983). A cla of etimator of mea ad variace uig auxiliary iformatio we correlatio coefficiet i kow. Biom. J., 5, 40-407. ukatme, P.V., ukatme, B.V., ukatme,. ad Aok,. (997). amplig Teory of urvey wit Applicatio. 3 rd ed. Iowa tate Uiverity Pre, Ame, Iowa, U..A. ad Idia ociety of Agricultural tatitic, ew Deli, Idia. Wu,.F.J. (985). Variace etimatio for te combied ratio ad combied regreio etimator. J. Roy. tatit. oc., 47, 47-54.
60 at Gupta / Joural of te Idia ociety of Agricultural tatitic 64() 00 55-60 APPEDI Data : [ource: ig ad Magat (996, p. )] y : leaf area for te ewly developed trai of weat ad x: weigt of leave. 39,, 3, 3 4, 3, 4, 4, 5, 3 3 tratum Value of Parameter o. Y y x r l l l 03 l l 04 l 40 l 3 l 3 5.75 03.40 6.066460 0.893 0.9037 0.49305 0.509785 0.35990.9346.7483.939455.05797.8797 8.94 0.90 5.953 0.073375 0.9540 0.996098 0.08555.03465.970998 3.436904.9897 3.096674.930390 3 5.84 04.30 6.49630 0.9378 0.96689 0.0576 0.977 0.083846.53437.895550.344898.67595.398860 Data :[ource:ig ad Magat (996, p. 9)] y : juice quatity ad x: weigt of cae (Kg). 5, 6,, 3 7, 3, 0, 3, 4, 3 3 tratum Value of Parameter o. Y y x r l l l 03 l l 04 l 40 l 3 l 3 35.00 366.67 8.6496 0.4888 0.945563 0.57673 0.6493 0.45984.664.86563.343750.58679.8869 99.7 30.83 4.40968 0.3953 0.94896 0.9857 0.973885 0.94658 3.379509 3.68973 3.79407 3.77747 3.548659 3 80.7 37.4 0.59 0.6953 0.7533.03540 0.89565 0.85880.377 3.30635.3948.48754.7034 Data 3 :[ource:ig ad Magat (996, p. 0)] y : total umber of milc cow i 993 ad x: total umber of milc cow i 990. 4, 7,, 3 5, 3, 0, 3, 5, 3 tratum Value of Parameter o. Y y x r l l l 03 l l 04 l 40 l 3 l 3 7.43 5.9 3.8863 0.994 0.765459 0.44840 0.449445 0.03884.34807.849759.655367.09983.36937 9.58 7.5 3.9044 0.3860 0.406654 0.767 0.44895 0.5079 0.569598.307.750973 0.83490 0.674840 3 0.69 7.80 3.690 0.83769 0.494577 0.8980 0.8474 0.5699.34646.83339.59543.349.070460