ANALYSIS OF VARIANCE WITH PARETO DATA

Transcription

1 Proceedgs of the th Aual Coferece of Asa Pacfc Decso Sceces Isttute Hog Kog, Jue -8, 006, pp ANALYSIS OF VARIANCE WITH PARETO DATA Lakhaa Watthaacheewakul Departmet of Mathematcs ad Statstcs, Maejo Uversty, Chag Ma, 5090, Thalad Prachoom Suwattee School of Appled Statstcs, Natoal Isttute of Developmet Admstrato, Bagkok, 00, Thalad ABSTRACT I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad ormally dstrbuted about zero mea ad costat varace. For aalyzg data whch do ot match the assumptos of the covetoal method of aalyss, we have two choces. We may trasform the data to ft the assumptos, or we may develop ew methods of aalyss wth assumptos whch ft the orgal data. If we ca fd a satsfactory trasformato, t wll almost always be easer to use t rather tha to develop a ew method of aalyss. I aalyss of varace wth Pareto data, the data should frst be trasformed to ft all the assumptos requred. Although the best trasformato s chose, the basc assumptos are sometmes ot satsfed. The value of trasformato parameter s obtaed whe the slope of curvature of the log lkelhood fucto s early zero. For certa staces, the well-kow Box-Cox trasformato ca be used to get the ormalty but caot be used to get the homogeety of varaces. Furthermore, some cases the data trasformed by the metoed trasformato are almost of the same values, so they caot be checked ether for ormalty of the data or the homogeety of varaces. To solve these problems, a alteratve trasformato s proposed. Whe the trasformed data have met the requred assumptos of ormalty ad homogeety of varaces, we the ca apply the aalyss of varace to test the equalty of the populato meas or the treatmet effects of the orgal Pareto populatos. Moreover, umercal studes of the powers of the tests obtaed from ANOVA of the trasformed data are also gve. The power of the aalyss of varace test creases as creases. Furthermore, whe the dffereces amog the meas are larger, hgher powers of the tests are obtaed. Keywords: Pareto dstrbuto, The Box-Cox trasformato, A alteratve trasformato INTRODUCTION I the aalyss of varace (ANOVA) the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad ormally dstrbuted about zero mea ad equal varaces. Wth some specfc sets of data, the basc assumptos are ot satsfed so aalyss of varace caot be appled approprately. Tukey [8] suggested that aalyzg data whch do ot match the assumptos of the covetoal method of aalyss, we have two alteratve ways to go about. We may trasform the data to ft the assumptos, or we may develop some ew methods of aalyss wth assumptos fttg the orgal data. If we ca fd a satsfactory trasformato, t wll almost always be easer to use the covetoal method of aalyss rather tha to develop a ew oe. Motgomery [5] suggested that trasformatos are used for three purposes, stablzg respose varace, makg the dstrbuto of the respose varable closer to the ormal dstrbuto, ad mprovg the ft of model to the data. Choosg a approprate trasformato depeds o the probablty dstrbuto of the sample data. For example, the square root trasformato s used for Posso data ad the logarthmc trasformato s used for logormal data. Moreover, we ca use the relatoshp betwee the stadard devato ad the mea for stablzg varace. Furthermore, we ca trasform the data by usg a famly of trasformatos studed for a log tme. May authors have studed the trasformatos of the data to meet the requremets of the aalyss of varace [] [] [6]. The Box-Cox trasformato s ofte used to trasform the data to fulfll the requremets but t mght ot be satsfactory some cases. Doksum ad Wag [8] dcated that the Box-Cox trasformato should be used wth cauto some cases such as falure tme ad survval

2 600 LAKHANA WATTHANACHEEWAKUL, PRACHOOM SUWATTEE data. Joh ad Draper [] showed that the Box-Cox trasformato was ot satsfactory eve whe the best value of trasformato parameter have bee chose. The purpose of ths paper s to study the applcato of aalyss of varace for skewed dstrbutos foud, for example, come data. Oe such a dstrbuto s the Pareto dstrbuto. It s a very rght log taled dstrbuto wth mea where s the locato parameter ad the shape parameter []. Besdes persoal come, ths dstrbuto has bee used extesvely the sze dstrbuto of urba areas [] [7], the servce tme dstrbuto [9] [] ad modelg data traffc the teret data [6] [9]. THE BOX-COX TRANSFORMATION Box ad Cox [] gave a modfed the smple form of the power trasformato as Y X, 0 = l X, = 0 () where X s a radom varable the j th tral from the th dstrbuto Y the trasformed varable of X, ad a trasformato parameter. Trasformato () ca be appled to all x > 0 whe x are the values of X. Ths trasformato has bee used extesvely practce. Sce the trasformato parameter s usually ukow, Box ad Cox [] appled orthodox large sample maxmum lkelhood theory to obta the approprate values of trasformato parameter. May statstcas studed the methods of estmato of the parameter of trasformato,, appearg the Box-Cox trasformatos [] [] [5] [0] []. However, the reowed Box-Cox trasformato sometmes s ot a approprate trasformato of some specfc sets of data eve wth sutable value of the trasformato parameter. As metoed above, the ANOVA some basc assumptos are requred. To aalyze Pareto data by ANOVA, the exstg sets of data should frst be trasformed to satsfy the requred assumptos. Ths meas the trasformed Pareto data should have ormal dstrbuto wth homogeety of varaces. However, the Box-Cox trasformato whe appled to the Pareto data sometmes does ot gve the data that meet all the requremets of ANOVA. Moreover, the trasformed data are sometmes almost of the same values whe the locato parameter s large. They have a varace closed to zero so the Kolmogorov-Smrov (K-S) test for checkg ormalty ad the Levee test for checkg the homogeety of varaces caot be performed. To correct these problems, we propose the alteratve trasformato that performs better tha the Box-Cox trasformato for specfc sets of data so that the ormalty ad homogeety of varaces ca be checked ad ANOVA ca be approprately appled. AN ALTERNATIVE TRANSFORMATION A trasformato for ay sets of Pareto data to ormalty wth equal varaces proposed here s the form X ( X 0.99 ), 0 Y = X l, = 0 ( 0.99 ) X () where X s a radom varable the j th tral from the th Pareto dstrbuto Y the trasformed varable of X, the mmum value of X from the th Pareto dstrbuto, ad a trasformato parameter. As the Box-Cox trasformato, The lkelhood fucto relato to the Pareto observatos s gve by

3 ANALYSIS OF VARIANCE WITH PARETO DATA 60 where J( y; x) = k ( 0.99 ) (,, ) = exp. ( ; ) x x L μ σ x μ J y x, () ( πσ ) σ = j= k y x = j= k ( 0.99 ) = x x = j= x k = x ( x 0.99 ). = j= We also apply the orthodox large sample maxmum lkelhood theory to (), the same approach as for the Box-Cox trasformato, to obta the maxmum lkelhood estmate (MLE) for the trasformato parameter. For a fxed, the MLE s for μ ad σ are ad ˆ μ = j = x ( x 0.99 ) k ( 0.99 ) ( 0.99 ) x x x x ˆ σ = = j= j= (). (5) Substtute the values of ˆ μ ad ˆ σ to the lkelhood equato (). Thus for fxed, except for a costat, the maxmzed log lkelhood s ( 0.99 ) ( 0.99 ) k x x x x f( ) = l L( x ) = l = j= j= k + l x ( x 0.99 ). (6) = j= Sce appears o the expoet of the observatos, t s cosdered to be too complcated for solvg the maxmzed log lkelhood fucto. The maxmzed log lkelhood fucto s a umodal fucto so the value of the trasformato parameter s obtaed whe the slope of the curvature of the maxmzed log lkelhood fucto s early zero. Hece we ca also use the alteratve method for fdg the sutable value of. The proposed method s as follows: Step Set,, ad ε. Step If f ( ) < ε the let =. If f ( ) ε, the cosder f ( ). If f ( ) < ε the let =. If f ( ) ε ad f ( ) ε, the proceed to Step. Step If f ( ) ε ad f ( ) < 0, let m =. The set =, ad = m = m. Set =. Repeat ths process utl f ( ) > 0. If f ( ) ε ad f ( ) > 0, let m =. The set =, ad = + m = + m. Set =. Repeat ths process utl f ( ) < 0. Step Calculate m =. If f ( ) < f ( ), let = + m. Otherwse, let = m.

4 60 LAKHANA WATTHANACHEEWAKUL, PRACHOOM SUWATTEE Step 5 If f ( ) < ε, stop the terato. If f ( ) ε ad f ( ) > 0, let = ad go back to Step. If f ( ) ε ad f ( ) < 0, let = ad go back to Step. After the Pareto data are trasformed by the alteratve trasformato, the ormalty of the trasformed data ca be checked by the Kolmogorov-Smrov goodess-of-ft test ad the homogeety of varaces of the trasformed data ca be checked by the Levee test. A NUMERICAL STUDY For a umercal study, order to atta the most effectve use of the proposed trasformato, we set the values of parameters ad the sgfcat value as follows: ) k = umber of the populatos =, ) = sample sze from the th Pareto populato =0, 0, 0, 50, ) = locato parameter of the th Pareto populato s betwee 0 ad 50, ) = shape parameter of the th Pareto populato s betwee ad 8, 5) α = The Box-Cox ad the Alteratve Trasformatos As a umercal study, Pareto populatos of sze N =,000 ( =,,) are geerated for dfferet values of parameters,. From a Par(, ), 0,000 radom samples, each of sze, are draw. The we trasform each set of the sample data to ormalty by the Box-Cox trasformato ad the proposed trasformato. The results of the goodessof-ft tests ad the tests of homogeety of varaces wth 0,000 replcated samples of varous szes are show Table to Table 8. TABLE of Normalty, ad of the p-values for Usg Data Trasformed by the Box-Cox Trasformato wth =0,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, * () * * * (),,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * * 00,5,50,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * *

5 ANALYSIS OF VARIANCE WITH PARETO DATA 60 Note that () * meas the dstrbuto has zero varace. The Kolmogorov-Smrov test caot be performed. The Kolmogorov statstc, D = sup F ( x) F0 ( x ), where F ( x ) s the emprcal dstrbuto fucto ad x x μ F0 ( x) s the hypotheszed dstrbuto fucto. For test of ormalty F0 ( x) =Φ. σ () * meas the dstrbuto has zero varace. The Levee test caot be performed. The Levee statstc, F * = MSTR k MSE where MSTR = (...) k d d, MSE = ( d.) k d, d. = d, k k d.. = ( d ) = j= = d = Y Y, wherey = meda of group. = j= j= TABLE of Normalty, ad of the p-values for Usg Data Trasformed by the Box-Cox Trasformato wth =0 TABLE of Normalty, ad of the p-values for Usg Data Trasformed by the Box-Cox Trasformato wth =50,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * * 00,5,50,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * *,,,, 0,0,0,, ,,

6 60 LAKHANA WATTHANACHEEWAKUL, PRACHOOM SUWATTEE TABLE (CONTINUE),,,,,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * * 00,5,50,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * * TABLE of Normalty, ad of the p-values for Usg Data Trasformed by the Box-Cox Trasformato wth =0, 0, 0,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, * () * * * (),,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * *

7 ANALYSIS OF VARIANCE WITH PARETO DATA 605 TABLE (CONTINUE),,,, 00,5,50,, * * * *,,5 * * * *,5,7 * * * *,6, * * * *,9,8 * * * * TABLE 5 of Normalty, ad of the p-values for Usg Data Trasformed by the Alteratve Trasformato wth =0 TABLE 6 of Normalty, ad of the p-values for Usg Data Trasformed by the Alteratve Trasformato wth =0,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, ,, ,5, ,6, ,9, ,5,50,, ,, ,5, ,6, ,9, ,,,, 0,0,0,, ,,

8 606 LAKHANA WATTHANACHEEWAKUL, PRACHOOM SUWATTEE TABLE 6 (CONTINUE),,,,,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, ,, ,5, ,6, ,9, ,5,50,, ,, ,5, ,6, ,9, TABLE 7 of Normalty, ad of the p-values for Usg Data Trasformed by the Alteratve Trasformato wth =50,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, ,, ,5, ,6, ,9,

9 ANALYSIS OF VARIANCE WITH PARETO DATA 607 TABLE 7 (CONTINUE),,,, 00,5,50,, ,, ,5, ,6, ,9, TABLE 8 of Normalty, ad of the p-values for Usg Data Trasformed by the Alteratve Trasformato wth =0, 0, 0,,,, 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, ,, ,5, ,6, ,9, ,5,50,, ,, ,5, ,6, ,9, We have see that, for the same sets of Pareto data, all sets of the data trasformed by the alteratve trasformato ca be checked by the K-S test ad for homogeety of varaces by the Levee test (Table 5 to Table 8), whereas specfc sets of the data trasformed by the Box-Cox trasformato caot do so (Table to Table ). Furthermore, the data trasformed by the alteratve trasformato always meet all the requred assumptos for ANOVA. Power of the ANOVA Test We trasform each set of the sample data to ormalty ad homogeety of varaces by our proposed alteratve trasformato. The the trasformed data sets are used to test the equalty of the populato meas by ANOVA. The power of the F-test as obtaed from ANOVA gve by Patak [7, p.0] s

10 608 LAKHANA WATTHANACHEEWAKUL, PRACHOOM SUWATTEE β( μ,..., μ ) = ( ) k pf df Fα k ( μ μ) t k ( ) σ μ μ = ( k ) + t ( ) t ( ) = σ ( k ) k + t ( k ) e =. F + F df t = 0 ( ) ( )! ( ), ( ) F k k tb k t k α + (7) where μ = y, j= μ k = = j= y, ad σ μ. k = ( y ) = j= TABLE 9 Powers of the ANOVA Tests of Equalty of Meas Usg Trasformed Data,,,, Power of the ANOVA Test = = = 0 = = = 0 = = = 50 = 0, = 0, = 0 0,0,0,, ,, ,5, ,6, ,9, ,5,0,, ,, ,5, ,6, ,9, ,00,00,, ,, ,5, ,6, ,9, ,5,50,, ,, ,5, ,6, ,9, We see that the power of the ANOVA test creases as creases. Furthermore, whe the dffereces amog the populato meas are larger, hgher powers of the tests are obtaed. The above results stll hold for a larger umber of populatos. CONCLUSION The alteratve trasformato as proposed ths paper performs better tha the well-kow Box-Cox trasformato for data sets obtaed from Pareto dstrbutos terms of the ormalty of the data ad the homogeety of varaces whch ca be checked by the Kolmogorov-Smrov test ad the Levee test, respectvely. I some cases, the data trasformed by the Box-Cox trasformato are almost of the same values. I such staces, the data would have varace ear zero so they caot be checked ether for ormalty by the K-S test or the homogeety of varace by the Levee test. The results of the smulato dcated that the data sets trasformed by the alteratve trasformato always meet the assumptos requred for the applcato of ANOVA. The power of the test depeds o the sample szes, ad also o the shape ad locato parameters of the populatos.

11 ANALYSIS OF VARIANCE WITH PARETO DATA 609 REFERENCES [] Alperovch, G. & Deutsch, J. The sze dstrbuto of urba areas: testg for the approprateess of the Pareto dstrbuto usg a geeralzed Box-Cox trasformato fucto, Joural of Regoal Scece, 995, 5(): [] Adrews, D. F. A ote o the selecto of data trasformatos, Bometrka, 97, 58: 9-5. [] Atkso, A. C. Regresso dagostcs, trasformatos ad costructed varables (wth dscusso), Joural of the Royal Statstcal Socety, Ser.B, 98, : -6. [] Box, G.E.P. & Cox, D. R. A aalyss of trasformatos (wth dscusso), Joural of the Royal Statstcal Socety, Ser.B, 96, 6: -5. [5] Carroll, R. J. A robust method for testg trasformatos to acheve approxmate ormalty, Joural of the Royal Statstcal Socety, Ser.B, 980, : [6] Chatterjee, M. & Das, S. K. Optmal MAC state swtchg for cdma 000 etworks, Paper Research Wreless Moblty ad Networkg. Uversty of Texas at Arlgto, 00. (Ole) Avalable URL: [7] Cordoba, J. O the dstrbuto of cty szes, Rce Uversty, 00. (Ole) Avalable URL: [8] Doksum, K. A. & Wag, C. Statstcal tests based o trasformed data, Joural of the Amerca Statstcal Assocato, 98, 78: -7. [9] Fscher, M. J. & et al. Usg quatle estmates smulatg teret queue wth Pareto servce tmes, Proceedg of the 00 Wter Smulato Coferece, 00: [0] Ha, A. K. A o-parametrc aalyss of trasformatos, Joural of Ecoometrcs, 987, 5: [] Harrs, C. M. The Pareto dstrbuto as a queue servce dscple, Operatos Research, 968, 6: 07-. [] Hkley, D. V. O power trasformatos to symmetry, Bometrka, 975, 6(): 0-. [] Joh, J. A. & Draper, N. R. A alteratve famly of trasformatos, Appled Statstcs, 980, 9(): [] Johso, N. L. & Kotz, S. Cotuous uvarate dstrbutos, Wley, 970. [5] Motgomery, D. C., Desg ad Aalyss of Expermets, Wley & Sos, 00. [6] Ntyasuddh, D. A test for more tha two Posso populato meas, Ph.D. Dssertato, Natoal Isttute of Developmet Admstrato., 00. [7] Patak, P.B. The o-cetral χ ad F- dstrbuto ad ther applcatos, Bometrka, 99, 6: 0-. [8] Tukey, W. O the comparatve aatomy of trasformatos, Aals of Mathematcal Statstcs, 957, : [9] Yag, X. Desgg traffc profles for bursty teret traffc, MIT Laboratory for Computer Scece, Cambrdge, 00. (Ole) Avalable URL: