Variance Estimation Using Linear Combination of Tri-mean and Quartile Average

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1 Amercan Journal o Bologcal and Envronmental tatstcs 07; 3(): do: 0.648/j.ajbes IN: (Prnt); IN: (Onlne) Varance Estmaton Usng Lnear Combnaton o Tr-mean and Quartle Average howkat Maqbool, hakeel Javad Dvson o Agrcultural tatstcs and Economcs, FOA, Wadura, kuast, Kashmr, Inda Department o tatstcs & O.R., A.M.U., Algarh, Inda Emal address: showkatmaq@gmal.com (. Maqbool), shakeel.jd@operamal.com (. Javad) To cte ths artcle: howkat Maqbool, hakeel Javad. Varance Estmaton Usng Lnear Combnaton o Tr-mean and Quartle Average. Amercan Journal o Bologcal and Envronmental tatstcs. Vol. 3, No., 07, pp do: 0.648/j.ajbes Receved: October, 06; Accepted: January, 07; Publshed: February 9, 07 Abstract: In ths paper, we have proposed a class o moded rato type varance estmator or estmaton o populaton varance o the study varable, when Tr Mean and Quartle average o the aulary varable are known. The bas and mean square error (ME) o the proposed estmator are obtaned. From the numercal study t s observed that the proposed estmator perorms better than the estng estmators n the lterature. Keywords: mple Random amplng, Bas, Mean quare Error, Tr-mean, Quartle Average, Aulary Varable. Introducton The use o an aulary varable n the estmaton o the nte populaton total or mean o a characterstc y s a common occurrence n practce. The theory o basc sample survey as avalable n standard tet books on samplng deals wth the case whch comprses lnear estmators such as mean, total, proporton and smple sample desgn usng smple random samplng wth replacement. It contnues to supply new and mproved procedures or estmaton o varances assumng ndependence o observatons. Rato, derence and regresson estmators utlze an aulary varable or more ecent estmaton o the parameter n queston. uch estmators take advantage o the correlaton between the varable and the characterstc y. In a smlar manner, then, t seems reasonable that under sutable condtons ecent estmaton o the varance o the estmator o the nte populaton total or mean o the characterstc y s also possble usng such estmaton technques. Wth the ncreasng growth n the number and dverse uses o sample surveys worldwde, t s oten desred to analyze and nterpret the resultng volumnous data by swter methods (Cochran (970)). A basc requrement o a good survey s that a measure o precson s provded or each estmate computed rom survey data collected on the bass o the survey desgn. An mportant queston s how to choose an approprate varance estmator. The choce n general s very dcult. Factors lke accuracy o the varance estmator, tmelness, cost, smplcty and other admnstratve convenences must be consdered. Here, we consder a nte populaton U { U, U,..., UN } o N dstnct and dentable unts. Let Y be a real varable wth value Y measured on U,,,3,,N gvng a vector Y { y, y,..., yn }. The goal s to estmate the populaton meansy N y N y y N or ts varance ( ) Y N on the bass o random sample selected rom the populaton U. ometmes n sample surveys along wth the study varable Y, normaton on aulary varable, whch s postvely correlated wth Y, s also avalable. The normaton on aulary varable, may be utlzed to obtan a more ecent estmator o the populaton. In ths paper, our am s to estmate the populaton varance on the bass o a random sample o sze n selected rom the populaton U. Ogus and Clark (97) proposed the use o rato or derence estmators o the varance under a passon samplng desgn (a desgn n whch each samplng unt s gven an ndependent chance o beng selected nto the sample wthout replacement) or the purpose o reducng the eect o the random sample sze on the varance estmator. Under a sample desgn n whch one sample unt s selected n each stratum wth probablty proportonal to sze (PP). hapro and Bateman (978) consdered reducng the bas o the

2 6 howkat Maqbool and hakeel Javad: Varance Estmaton Usng Lnear Combnaton o Tr-mean and Quartle Average estmator o the varance n a one-per stratum desgn by usng a varance estmator as the Yates Grundy varance estmator or a two sample unts per stratum desgn wth jont ncluson probabltes. The problem o constructng ecent estmators or the populaton varance has been wdely dscussed by varous authors such as Isak (983), who proposed rato and regresson estmators. Prasad and ngh (990) consdered a rato type estmator or estmatng populaton varance by mprovng Isak s (983) estmator. Gupta and habr (008) proposed a new hybrd class o estmators and showed that n some cases ther ecency s better than the tradtonal rato estmators, where as Kadlar & Cng (006,007) proposed the moded estmators usng coecent o varaton (C.V.) and ther lnear combnatons. ubraman & Kumarapandyan (05) mproved the already estng estmators by ntroducng moded estmators wth the use o known parameters lke C.V., Kurtoss, Medan, Quartles and Decles. Recently, Maqbool et al. (06) proposed a moded rato estmator usng non-conventonal locaton parameters because these parameters take care o outlers n the data. Outlers can be generated by rom a smple operatonal mstake to ncludng small sample rom a derent populaton, and they make serous eects o statstcal nerence. Even one outlyng observaton can destroy least squares estmaton, resultng n parameter estmates that do not provde useul normaton or the majorty o the data. ME ( ˆ ) Bas ( ) R ˆ R ˆ R () s y ( ) 4 ( ) γ β( ) ( λ γ y β( y ) + ( β( ) ( λ Here γ s the samplng racton and λ s covarance between the study and aulary varable. 4. Kadlar and Cng (006) Estmators Kadlar and Cng (006) suggested our rato type varance estmators usng known values o C.V. and coecent o kurtoss o an aulary varable ˆ + C kc s + C kc γ ya A ( β( ) ) λ ( ME ( ˆ kc) 4 γ y ( β( y) ) + A ( β( ) A ( λ (). Notatons Let N populaton sze,n sample sze, γ, Y study n varable, aulary varable., Y populaton means,, y sample means. y, Y, populaton varances, s s sample varances. C, C coecent o varaton, ρ correlaton coecent, β ( ) aulary varable, β ( ) β ( y) y skewness o the kurtoss o the aulary varable, kurtoss o the study varable, M d medan o the aulary varable, B(.)bas o the estmator, ME(.) Mean square error, rato type varance estmator, ˆ, ˆ ˆR Kc estng moded rato estmators, Q a (Q 3 +Q )/ s populaton sem-quartle average o the aulary varable, TM(Q +Q +Q 3 )/4s a Tr-Mean, j proposed estmator by howkat and Javad. 3. Rato Type Varance Estmator Proposed by Isak (983) Isak (983) suggested a rato type varance estmator or the populaton varance when the populaton varance Y o the aulary varable s known. Its bas and mean square error are gven by ˆ + β( ) kc s + β( ) kc γ ya A ( β( ) ) λ ( ME ( ˆ kc) 4 γ y ( β( y) ) + A ( β( ) A ( λ ˆ β( ) + C kc3 s β( ) + C kc3 γ ya 3 A3 ( β( ) ) λ ( ME ( ˆ kc3) 4 γ y ( β( y) ) + A3 ( β( ) A3 ( λ ˆ C + β( ) kc4 sc + β( ) kc4 γ ya 4 A4 ( β( ) ) λ ( ME ( ˆ kc4) 4 γ y ( β( y) ) + A4 ( β( ) A4 ( λ (3) (4) (5)

3 Amercan Journal o Bologcal and Envronmental tatstcs 07; 3(): Here β ( ) C,, 3, 4 s + c s + β( ) s β( ) + c sc + β( ) A A A A 5. Recent Developments ubraman and Kumarapandyan (05) proposed a generalzed moded rato type estmator or estmatng populaton varance usng the known parameters o the aulary varable and ther estmator s gven as ˆ + αw s + αw The bas and ME o the estmator n (6) as suggested by the ubraman and kumarapandyan (05) s gven as Bas ( ) ˆ ya A ( ) ( ) γ β( ) ( λ ME ( ˆ ) 4 γ y β( y) + A ( β( ) A ( λ A s + αω When the study varable Y and the aulary varable are negatvely correlated and the populaton parameters o the aulary varable are known, ubraman & Kumarapandyan (05) proposed the ollowng generalzed moded product type varance estmator as ˆ + τw,,,. s + τw The notatons n the estmator (6) ˆ are eplaned n detal n ubraman & Kumarapandyan (05). When α 0 n Equaton (6), reduces to Isak (983) ˆ estmator. When α n Equaton (6), ˆ reduces to Kadlar and Cng (006) estmator. 6. Proposed Estmator The accuracy o the varance estmator s an mportant crteron or the choce o a varance estmator and t can be assessed by mean square error (ME) o the varance estmator that has the best statstcal propertes or the proposed analyss and nterpretaton o data. Compromses may have to be made because derent analyss o the same data may suggest derent varance estmator. In short, t s, thereore seen that the process o evaluatng alternatve varance estmator and selectng a specc estmator or use n partcular comple survey n a dcult and knotty problem, nvolvng both subjectve and objectve elements. Keen judgment n makng ntellgent compromses between the consderaton o accuracy, cost, tmelness and smplcty s requred. The perormance o the estmator o the study varable can be mproved by usng known populaton parameters o an aulary varable, whch are postvely correlated wth a study varable. We have proposed a new moded rato type varance estmator o the aulary varable by usng lnear combnaton o Tr-mean and populaton sem nter quartle average o the aulary varable. It s hghly senstve to outlers as ts desgn structure s based on only etreme values o the data (or more detals see Ferrell (953)). ˆ ) j s We have derved here the bas and mean square error o the proposed estmator ˆJ to rst order o appromaton as gven below. The Taylor seres lnearzaton approach derves a lnear appromaton to the survey estmator and then obtans varance estmate or ths lnear combnaton takng account o the samplng desgn. Lete y y y 0 y and s ( + e ) and e0 and e we obtan: 0 s e (7). Further we can wrte s ( + e ) and rom the denton o E[ e0 ] E[ e ] 0, E[ e0 ] ( β( y) ), n E[ e ] ( β( ) ), E[ e0e ] ( λ ). n n The proposed estmator 0 ˆJ s gven below: ˆ ) j s H + s + e + H ˆ + J ( e0 ) ˆ J y ( + e0 ), where A J ( + A e ) J, where H TM + Qa ˆ ( )( ) J y + e0 + AJe + ( TM + Q ) ˆ ( + )( ) 3 3 J y e0 AJe AJe AJe Epandng and neglectng the terms more than the 3 rd order, we get ˆJ y + y e0 y A J e y A J e0e + y A J e a

4 8 howkat Maqbool and hakeel Javad: Varance Estmaton Usng Lnear Combnaton o Tr-mean and Quartle Average ˆJ y y e0 y A J e y A J e0e y A J e + (8) By takng epectaton on both sdes o (8), we get E( ˆ ) ( ) ( ) ( ) ( ) J y ye e0 yaje e yaje e0e + yaje e Bas( ˆ ) ( ) ( ) J yaje e yaje e0e Bas( ˆ ) [ ( ( )] (9) J γ yaj AJ β( ) λ quarng both sdes o (8) and (9), neglectng the terms more than nd order and takng epectaton, we get E( ˆ ) E( e ) A E( e ) A E( e e ) J y y 0 + y J y J 0 The bas and mean square error o the above estmator n (7) ater smplcaton s gven as Bas ( ) ME ( ˆ ) J ya J AJ ( ) ˆ J where A J γ β( ) ( λ ( ) 4 γ y β( y) + AJ ( β( ) AJ ( λ. + ( TM + Q ) 7. Numercal Illustraton a The perormance o the proposed estmator s assessed wth that o smple random samplng wthout replacement (RWOR) sample varance and estng estmators. We use the data o Murthy (967) page 8 n whch ed captal s denoted by (aulary varable) and output o 80 actores are denoted by Y(study varable). We apply the proposed and estng estmators to ths data set and the data statstcs are gven below: N 80, 8.454,A 0.965, n0, C , A ,.64, β ( ).8664,A , Y 5.864, β ( y).667, A , ρ 0.943, β ( ).05, λ.09, y , Md7.5750, Q 9.38, C y 0.354, A , Q ,Q ,Q D 5.95,Q a.065,q R.8, TM9.38, Tr-mean (TM)9.38 The results obtaned are shown n Table. Table. Bas and Mean quare Error o the estng and the proposed estmators. Estmators Bas Mean quare Error Isak (983) Kadlar&Cng(006) Kadlar&Cng(006) Kadlar&Cng(006) Kadlar&Cng(006) ubraman&kumarapandyan(05) Proposed Estmator (Maqbool & hakeel) Concluson The paper proposes a rato type varance estmator usng known values o an aulary varable. From Table we see that the bas and ME o the estng estmators ranges rom 6. to0.87 and 380 to 395 respectvely, whle as the proposed estmator suggested by Maqbool & hakeel has a bas o 3.0 and ME whch show that the bas and mean square error are less than the already estng estmators n the lterature and the percentage relatve ecency comes out be 39.8%. Hence the proposed estmator may be preerred over estng estmators or use n practcal applcatons. In uture work, we hope to adopt the estmator presented here to strated random samplng and hope to develop a varance estmator usng two aulary varables. Acknowledgement The authors apprecate the eedback provded by the reerees that helped mprove the presentaton o the paper. The econd author s thankul to UGC or provdng the unds under UGC-BR Research tart-up Grant, (No. F.30-90/05(BR), FD Dary No. 767, dated: Reerences [] Cochran, W. G. (94). amplng Technques. Thrd Edton, Wley Eastern lmted. [] Gupta, at and habbr, Javd (008). Varance estmaton n smple random samplng usng aulary normaton. Hacettepe journal o Mathematcs and tatstcs, 37 (), [3] Ferrell, E. B. (953). Control charts usng Md-ranges and Medans. Industral Qualty control, 9(5), [4] Isak, C. T. (983). Varance estmaton usng aulary normaton. Journal o the Amercan tatstcal Assocaton, 78,7-3. [5] Kadlar, C. & Cng, H. (006). Rato estmators or populaton varance n smple and strated samplng. Appled mathematcs and Computaton, 73, [6] Kadlar, C. & Cng, H. (007). Improvement n Varance estmaton n smple random samplng. Communcatons n tatstcs: Theory and methods, 36, [7] Murthy, M. N. (967). amplng theory and methods. Calcutta tatstcal Publshng House, Inda. [8] Ogus, J. L., and Clark, D. F. (97). The annual survey o manuactures: A report on methodology. U bureau o the census techncal paper 4, U Government prntng oce, Washngton DC. [9] Prasad, B., and ngh, H. P. (990). ome mproved rato type estmators o nte populaton varance n sample surveys. Communcaton n tatstcs: Theory and methods, 9, [0] Maqbool,., Raja, T. A., and hakeel Javad (06). Generalzed moded rato estmator usng non-conventonal locaton parameter, Int. J. Agrcult. tat. c, (),

5 Amercan Journal o Bologcal and Envronmental tatstcs 07; 3(): [] hapro, G. M., and Bateman, D. V. (978). A better alternatve to the collapsed stratum varance estmate. Proceedngs o the socal statstcs secton, Amercan tatstcal Assocaton, [] umraman, J. and Kumarapandyan, G. (05). Generalzed moded rato type estmator or estmaton o populaton varance. r-lankan journal o appled tatstcs,vol6-, [3] Wolter, K. M. (985). Introducton to varance estmaton. prnger- Verlag.