Estimation of Co-efficient of Variation in PPS sampling.

Transcription

1 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.409 Estmato of Co-effcet of Varato PPS samplg. Archaa. V ( st Author) Departmet of Statstcs, Magalore Uverst Magalagagotr, Koaje Karataka, Ida. Emal: archaa_stkl@ahoo.com Arua Rao K ( d Author) Departmet of Statstcs, Magalore Uverst Magalagagotr, Koaje Karataka, Ida. Emal: aruaraomu@ahoo.com. Itoducto. Iferece cocerg the populato coeffcet of varato (C.V) has attracted the atteto of several emet researchers the past. It dates back to the works of Mcka(9) ad Pearso(9). At that tme the prmar cocer was for the estmato ad cofdece terval for the C.V of the ormal dstrbuto. Subsequet works cocetrated o the mproved estmato of the cofdece terval for the ormal C.V. The clude the works of Mahmoudvad ad Hassa(007) ad Pachktkosotkul(009). Subsequetl the focus was to develop tests for equalt of the populato C.V s of two or more groups whe the uderlg dstrbuto s ormal. Lkelhood rato, wald ad score test prcples are used to develop the tests. The research papers ths drecto cludes Ahmed (00), Nar ad Rao(00) ad Jafar ad Behbooda(00) ad the referece cted there. For a log tme ferece regardg C.V dd ot draw the atteto of surve statstcas. The frst reported work s due to Das ad Trpath (980) who proposed estmato of C.V whe the desg s smple radom samplg wthout replacemet (SRSWOR). Subsequetl several papers have addressed the estmato of fte populato C.V. Oe of the poeerg work ths drecto s due to Trpath et al., (00) where the proposed a class of estmators for the populato C.V. Ths class cludes the rato ad regresso estmators of populato C.V usg sample C.V ad certa hbrd estmators where regresso estmators are used to propose a ew rato estmator. Patel ad Shah (007) examed the small sample mea square error (MSE) of several estmators cludg some estmators belogg to the hbrd class. All these estmators refers to the case of smple radom samplg. Rajaguru ad Gupta (006) proposed three classes of estmators of populato C.V uder stratfed radom samplg. Probablt proportoal to sze samplg (PPS) s wdel used sample surves. Thus t becomes mportat to propose estmators of C.V uder PPS samplg. I ths paper we have proposed a estmator of C.V uder probablt proportoal to sze wth replacemet samplg(ppswr). The mea square error of the estmator s derved to the order of O( - ). The performace of the proposed estmator s compared wth the estmator uder smple radom samplg usg smulatos. It s commo

2 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.400 practce the appled research s to draw the samples usg a partcular samplg desg but to use ormal theor estmators for the aalss of the data. Thus the performace of the proposed estmator s also compared for the sample C.V whe the samples are draw usg PPSWR scheme. The sample C.V uder PPSWR scheme turs out to be the best estmator terms of the MSE ad justfes the practce followed b the appled researchers. The rest of the paper s orgazed as follows: I secto, estmators of C.V uder PPSWR scheme s proposed. Small sample comparso of the performace of estmators of C.V s preseted secto. The results ad dscussos are gve secto 4 ad the paper cocludes secto 5.. Estmators of C.V uder PPSWR samplg. I theor of samplg t s customar to deote the stud varable b ad the auxlar varable b x. Let (X, ), =,.., N be the values of the varables x ad for the th ut the populato. Let ad σ deote the populato mea ad populato varace for the stud varable where, N N = ad σ = ( ). N = N = σ The prmar focus of terest s to estmate θ =. Uder PPSWR scheme a ubased estmator of σ ad s gve b, = = (.) N p v( = ) N ( ) = p (.) where = p Here x p =, X = X.e. probablt of selectg th ut the sample. X Now we ca defe the estmator of C.V uder PPSWR as, where ( σ ) ( σ ) = (.) = + ( v ) N p N N Aother possble estmate of C.V uder PPSWR s the sample C.V ad s gve b, ( ) s = (.4) where s the sample mea usg the PPS sample ad s s the sample varace usg the PPS sample.

3 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.40 The sample C.V uder smple radom samplg scheme s defed as, s = (.5) where s the sample mea ad s s the sample varace ad are gve b, ( ) = =. = = ad s The expressos for the bas ad MSE of the estmators, ad θ θ are derved b the authors to the order of O( - ). The dervatos of MSE s of the above estmators uder PPSWR scheme s algebracall tedous, but sce t s ot used awhere the paper to save space these expressos are ot reported here.. Small sample comparso of the estmators of C.V... Smulato Expermet. A smulato expermet s carred out to compare the MSE s of the varous estmators. The expressos for the MSE of the estmators derved the prevous secto were to the order of O(-) ad thus are the asmptotc MSE s. I order to estmate the exact MSE, t s ecessar to go for a smulato expermet. The estmato of mea uder PPSWR samplg scheme has got a smaller MSE compared to the estmator uder SRSWR/WOR f the sze varable s hghl correlated wth the stud varable. Thus t s ecessar to corporate the correlato coeffcet the smulato expermet. For ths purpose 000 samples were geerated from a bvarate ormal dstrbuto. Care s take to esure that all the values of the sze (auxlar) varable s postve. From ths populato a sample of sze s selected usg PPSWR ad SRSWR schemes. From the PPS sample the estmators of populato C.V as gve b equato (.) s computed alog wth the sample C.V. Usg SRS samples, sample C.V s also computed. The MSE of these three estmators are estmated usg 0,000 smulatos. The cofguratos used the smulatos are the followg. The values of the C.V of the stud varable used the smulatos were 0., 0., 0.5, 0.8,.0 ad.0. For each fxed value of the stud varable a set of four values of C.V of the auxlar varable are cosdered. The are 0.5,.0,.5 ad tmes the C.V of the stud varable. The correlato coeffcets r used the smulato stud are -0.9, -0.7,-0.5,-0.,-0., 0, 0., 0., 0.5, 0.7, 0.9. The sample szes cosdered are =00, 00. The total umber of cofguratos works out to be=6*4**=58.. Small sample MSE. Table (.) represets the rato of the MSE of the sample C.V uder PPS sample ad SRS sample compared to the estmator of C.V uder PPS sample. For bravt the latter estmator s referred as the PPS estmator of C. V. Geerall whe comparg the MSE of the two estmators, the MSE of the estmator havg smaller value s take the deomator. However sce our motvato s compare the MSE of the sample C.V s to the MSE of the PPS estmator, the MSE of

4 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.40 the latter s take the deomator. Thus a lttle cauto s eeded to terpret the results. The result s preseted ol for =00 ad the patter remas the same for the other sample szes. Before preparg ths table, the MSE of the estmate of C.V s carefull examed for the 4 values of the C.V of the auxlar varable for each of the estmators. The MSE almost remas the same rrespectve of the C.V of the auxlar varable ad the table (.) the rato s refers to the rato of the average MSE of the estmator the umerator to the average MSE of the estmator the deomator; the average s take over the four values of the MSE correspodg to the values of C.V of the auxlar varable. Table.: Relatve MSE (%) of the sample C.V uder PPS ad SRS sample compared to the estmator of C.V uder PPSWR. C. V of the stud varable Correlato co-effcet r (=00) θ From the table t s clear that the estmator whch has got the smallest MSE correspods to the sample C.V uder PPS samplg followed b the MSE of the sample C.V uder SRS scheme. Ths s true for varous values of the correlato coeffcet. For moderate values of C.V, the relatve effcec (.e., verse of the values reported here) of these estmators s ver hgh 4

5 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.40 compared to the PPS estmator of C.V. As the value of C.V of the stud varable creases, although the effcec s hgher for these estmators the ga sample sze s reduced. For eg: whe θ =0., r=-0.9 the relatve MSE of these estmators are respectvel 4.76 ad.8. Ths dcates that whe the PPS estmator requres a sample of sze 00 to atta the same MSE, the sample C.V requres the sample sze of 5 ad 4 uder PPS ad SRS scheme. Whe θ = 0.8, r=- 0.9 the relatve MSE for these estmators are 7.8 ad 0.4 respectvel dcatg that a sample szes of 8 ad 0 are requred to atta the same MSE of PPS estmator whch requres a sample sze 00. We have examed the MSE of the PPS estmator across the correlato coeffcets for each value of the C.V of the stud varable. The results are ot full reported here to save space. Table (.) presets the MSE of the PPS estmator for 0., 0.5 ad.0 for selected values ofθ. The results dcate that the MSE of the PPS estmator s relatvel hgher whe the correlato co-effcet s low. Table.: The MSE s of the proposed three estmators for selected value of C.V of the stud varable. C. V of the stud varable θ Correlato co-effcet r (=00) Dscussos: I ths paper a estmator of populato C.V s derved whe the samplg desg s PPSWR. Geerall t s expected that ths estmator performs well compared to the estmator of C.V amel sample C.V uder SRS scheme. The umercal results dcate that for the estmato of C.V, SRS s more effcet compared to the PPSWR scheme. At the tme of umercal computato t was observed that the estmator of σ usg PPS scheme turs out to be egatve several cases. The percetage s approxmatel 0%. Thus the PPS estmator s ot defed such cases. A larger questo that eeds to be addressed s Whether SRS samplg s more effcet for the estmato of compoud parameters lke C.V, skewess ad kurtoss compared to the PPS scheme. The results dcate that the appled researchers are justfed usg sample C.V as a 5

6 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.404 estmator of populato C.V whe the samples are draw usg PPRWR desg. If ths estmator s take as the estmator of C.V uder PPS desg, the the PPS desg turs out to be more effcet compared to the SRS desg. 5. Cocluso: Two estmators are proposed for populato C.V whe the samples are selected usg PPSWR scheme. The frst estmator s the PPS estmator obtaed b ubasedl estmatg σ (populato varace) ad (populato mea) usg PPS scheme. The secod estmator s the sample C.V. The sample C.V uder PPS samplg turs out to be the best estmator compared to the PPS estmator ad sample C.V uder SRSWR. We advocate that for greater effcec t s advsable to select the sample usg PPS scheme ad use the sample C.V as a estmator of the populato C.V. Further research s eeded to shed lght o the more geeral questo amel Whether usg the estmator derved uder SRS scheme turs out to be more effcet uder PPS scheme for complex parameters lke skewess, kurtoss ad g dex. Dervg estmators for these uder PPS scheme s algebracall tedous whle ther estmators SRS scheme tur out to be smple. SELECTED REFERENCES. Ahmed, S.E (00): Smultaeous estmato of Co-effcet of Varato. Joural of Statstcal Plag ad Iferece. 04, -5.. Das, A.K. ad Trpath, T.P. (98 a): Samplg strateges for coeffcet of varato usg kowledge of the mea usg a auxlar character. Tech. Rep. Stat. Math. 5/8. ISI, Calcutta.. Das, A.K. ad Trpath, T.P. (98 b): A class of estmators for co-effcet of varato usg kowledge o coeffcet of varato of a auxlar character. Paper preseted at aual coferece of Id. Soc. Agrcultural Statstcs. Held at New Delh, Ida. 4. Mahmoudvad, R. ad Hassa, H.(007): Two ew cofdece tervals for the C.V a ormal dstrbuto. Joural of Appled Statstcs, 6: 4, Murth M.N. (967): Samplg Theor ad Methods, Statstcal Publshg Socet, Calcutta. 6. Nar, K.S. ad Rao, K.A. (00): Tests of coeffcet of varato ormal populato. Commucatos Statstcs: Smulatos ad Computato (), Patel, P. A. ad Shah Ra. (009): A Mote Carlo comparso of some suggested estmators of Co-effcet of varato fte populato. Joural of Statstcs sceces, (), Pachktkosolkul, W. (009): Improved cofdece tervals for a Co-effcet of varato of a ormal dstrbuto. Thalad statstca, 7(), Rajaguru, A. ad Gupta, P. (006): O the estmato of the co-effcet of varato from fte populato-ii, Model Asssted Statstcs ad applcato, (), Rajaguru, A. ad Gupta, P. (00): O the estmato of the co-effcet of varato from fte populato-i, Model Asssted Statstcs ad applcato, 6(), Trpath, T.P. Sgh, H.P. ad Upadhaa, L.N. (00): A geeral method of estmato ad ts applcato to the estmato of co-effcet of varato. Statstcs Trasto, 5(6),

7 It. Statstcal Ist.: Proc. 58th World Statstcal Cogress, 0, Dubl (Sesso CPS00) p.405 Abstract: Co-effcet of varato (C.V) s a utless measure of dsperso ad s wdel used rather tha stadard devato. I the recet ears estmato of C.V fte populatos has draw the atteto of several researchers, however ma of these works o C.V cofed to the case of smple radom samplg. I ths paper a estmator of C.V s proposed PPS samplg wth replacemet alog wth ts asmptotc mea square error. The expressos for the desg effcec are also derved. The proposed estmator s compared wth the estmator of C.V smple radom samplg wth replacemet usg several real lfe data settgs. B treatg the fte populato as a sample from a fte ormal populato, the effcec of the estmators PPS wth replacemet samplg s compared wth smple radom samplg wth replacemet (SRSWR) usg extesve smulato. The results dcate that for the estmato of C.V, PPSWR samplg s more effcet compared to SRSWR. Kewords: Co-effcet of varato, PPS samplg, Smple radom samplg, Desg effcec, Modelbased comparso. 7