Comparisons of Several Pareto Population Means

Transcription

1 Comparsons of Several Pareto Populaton Lahana Watthanacheewaul and Prachoom Suwattee Abstract Analyss of varance (ANOVA), the lelhood rato test (LRT) and the Krusal-Walls test for Pareto populatons are nvestgated. Snce Pareto data sets are non-normal dstrbutons, they must be transformed to normal dstrbutons wth a constant varance. The power of the ANOVA, lelhood rato and Krusal-Walls tests was compared n a number of dfferent stuatons and dfferent sample szes. It was found that the results depended on the locaton and shape parameters. They are set n three cases; the locaton parameters are the same and the shape parameters are dfferent, both the locaton and shape parameters are dfferent, and the shape parameters are the same but the locaton parameters are dfferent. It seems that the lelhood rato test was a good choce n almost every case, but t s dffcult to fnd a test based upon the lelhood rato. Index Terms Pareto populatons, Power, Shape parameters, Locaton parameters I. INTRODUCTION There are many methods for comparng of more than two populaton means such as the ANOVA and the LRT. If the data are non-normally dstrbuted, t s usual to apply the lelhood rato test to scale, locaton or shape parameters. owever, t s dffcult to fnd the exact dstrbuton of the generalzed lelhood rato statstc, Λ. or large samples the statstc lnλ s approxmately dstrbuted as ch-squared wth degrees of freedom, whereas for small samples the approxmated ch-square may be naccurate. Alternatvely, the Krusal-Walls non-parametrc test s used and, n case where a comparson of the means of a populaton wth one or more parameters, t s easer to apply the ANOVA. owever, n the latter case non-normal data must be transformed to normal dstrbutons so that the requred assumptons for the ANOVA ft. In the ANOVA, the usual basc assumptons are that the model s addtve and the errors are randomly, ndependently and normally dstrbuted wth zero mean and constant varance. When analyzng data that do not match the assumptons of a conventonal method of analyss, there are two choces; transform the data to ft the assumptons or develop some new robust methods of analyss []. If a satsfactory transformaton can be found, t wll almost always be easer to use t rather than to develop a new Manuscrpt receved August 4, 009. Lahana Watthanacheewaul s wth the aculty of Scence, Maejo Unversty, Chang Ma, Thaland (phone: ; fax: ; e-mal: lahanaw@yahoo.com, lahana@mju.ac.th). Prachoom Suwattee s wth the School of Appled Statstcs, Natonal Insttute of Development Admnstraton, Bango, Thaland (e-mal: prachoom@as.nda.ac.th). method of analyss. Transformatons are used for three purposes; stablzng response varance, mang the dstrbuton of the response varable closer to a normal dstrbuton and mprovng the ft of the model to the data []. Choosng an approprate transformaton depends on the probablty dstrbuton of the sample data, e.g., the square root transformaton s used for Posson data. urthermore, t s possble to transform the data usng a famly of transformatons already extensvely studed over a long perod of tme, e.g., [], [4]. A well-nown famly of transformatons often used n prevous studes was proposed by Box and Cox. owever, Box-Cox transformaton s not always applcable. The Box-Cox transformaton should be used wth cauton n some cases such as falure tme and survval data [5]. The Box-Cox transformaton was not satsfactory even when the best value of transformaton parameter had been chosen. rom studes of ncome dstrbuton, [6], t s well establshed that a hgh proporton of a populaton wll have low ncome. Pareto dstrbutons have an mportant role n these studes. It s a very rght long taled dstrbuton wth the shape parameter, whch determnes the concentraton of data and the locaton parameter, whch s the mnmum value of the varable [7]. Besdes the dstrbuton of ncome, the Pareto dstrbuton s ndeed common n varous felds, e.g., a queung system [8], moble communcaton networs [9]. As mentoned, the ANOVA requres certan basc assumptons. ence, the Pareto-dstrbuted data must be transformed before the ANOVA s appled. The ANOVA test, the lelhood rato test and the Krusal-Walls test are appled for comparsons of several Pareto populaton means. In addton, the power of the ANOVA, generalzed lelhood rato and Krusal- Walls tests s compared usng a smulaton study. II. TESTS OR COMPARISONS O SEVERAL MEANS A. The ANOVA Usually, a Box-Cox transformaton s used to transform data to normalty. or the ANOVA, the Box-Cox transformaton s n the form λ Xj, λ 0 Yj = λ () ln X j, λ= 0, for x j > 0, where Xj s a random varable n the j th tral from the th populaton, Y j the transformed varable of Xj and λ a transformaton parameter, but sometmes ths

2 transformaton when appled to the Pareto-dstrbuted data does not change the data suffcently so that all the requrements for the ANOVA are met. Moreover, the transformed data are sometmes almost the same value when the locaton parameter s large; they have a varance close to zero, so that the Kolmogorov-Smrnov (K-S) test for checng normalty and the Levene test for checng the homogenety of varances cannot be performed. In order to cope wth these problems, an alternatve transformaton proposed by Watthanacheewaul and Suwattee [0] s n the form λ λ X j (Xj 0.99 ), λ 0 λ Yj = () Xj ln, λ = 0 (Xj 0.99 ) where Xj s a random varable n the j th tral from the th Pareto dstrbuton, Y j the transformed varable of X j, the mnmum value of X j from the th Pareto dstrbuton, and λ a transformaton parameter. It s appled to transform any number of sets of Pareto data to normal wth equal varance. The null hypothess s 0 : μ =μ... =μ. It performs better than the Box-Cox transformaton for some specfc sets of data and so allows the normalty and homogenety of varances to be checed. B. The Lelhood Rato or Pareto data, testng the equalty of means s equvalent to testng the equalty of the shape parameters under the condton that the locaton parameters are equal. Theorem If X j s Par(, ), =,,, then, for large sample, the statstc of the lelhood rato test for testng the null hypothess 0 : = =... = and = =... = s x n = n ln n + n ln ln ln x n n n n + lnx n j lnxj x n j = j= = j= x j ln * ln = j= x j= x () n n + nln ln x n + () n = x j xj ln ln j= x () j= x () where n j * * n = j= x xj ln * = j= x * x s the mnmum value of all observatons and x s the mnmum value of each sample. () The test statstc has an approxmate ch-square dstrbuton wth degrees of freedom when 0 s true. C. The Krusal-Walls Ths s a non-parametrc analogue, based on ran, of one-way analyss of varance. The test statstc s n + = n R (4) n(n + ) = where R s the average ran of the member of the th sample () obtaned after ranng all of the n = n observatons. Krusal [] proves that f 0 s true, the statstc has a lmtng ch-square dstrbuton wth degrees of freedom as n smultaneously. The null hypothess s 0 : ξ =ξ =... =ξ where ξ s the medan of the th Pareto dstrbuton, ξ =. 0.5 III. EXAMPLE WIT REAL-LIE DATA Data on the major rce crop n the crop year 00/00 (Aprl st, 00 to March st, 00) from three Tambols, Nonghan, Nongyang and Nongjom, of Amphoe Sansa n the Chang Ma provnce, Thaland, were used as populatons. Each Tambol conssts of a set of 8, 8 and 94 farmer households, respectvely. Samples of szes, 8 and 0 were drawn from each Tambol. The sampled data are shown n Table I. = Table I The Major Rce Crop n Klograms from the three Tambols for the Crop Year 00/00 (Aprl st, 00 to March st, 00) Nonghan Nongyang Nongjom,00,440,600,800,00 5,000,000 5,400 7,500,400,800 7,800,000 4,00,600,800 7,000 4,000,000,700 4,000,50 6,000 4,800,00,50 4,900,700,500 4,00,800,000 4,500,500,400 4,00,500 5,000 6,800,600,600 4,000,00 8,000 7,000,900,50 4,500,00 8,500 8,00,500 4,500,800 6,000 4,50 4,000,00,500 5,800,850,000,70,700,600 6,000,900 5,000 5,400,000,600 4,000 4,500,600 4,0 4,70,600 5,800 4,800 4,00 4, They are Pareto data and they can be checed usng a Pareto probablty plot. The value of transformaton parameter = The ANOVA assumptons of the transformed data have been checed to be vald. or sgnfcance level α = 0.05,,78 =.. Snce = >

3 ., t s concluded that there s a sgnfcant dfference n at least one par among the three populaton means. Multple comparsons should now be appled to dentfy the dfference amongst the populaton means. The value of the statstc s and 0 s rejected. There s a sgnfcant dfference n at least one par among the three populaton means. The value of the statstc s.6 and 0 s rejected. It s concluded that there s a sgnfcant dfference n at least one par among the three populaton means. IV. SIMULATION STUDIES Snce the power functons of the three tests are n dfferent forms [] [4], they cannot be compared drectly. To draw a concluson, the power of the three tests s studed numercally for partcular cases by smulaton method. A. Smulaton Method A power comparson of several tests was suggested n two steps [5]. ) To fnd the crtcal value for reject on of the null hypothess, Pareto populatons of sze N = 4,000 are generated for = =... = and = =... =, =,,..,. rom each generated populaton,,000 random samples, each of sze n, =,,...,, are drawn. The test statstcs of the three tests are calculated from each of the,000 samples. or each test, the,000 values of the test statstcs are arranged n an ncreasng order and the 95 th percentle s dentfed. Ths gves the crtcal values at α= 0.05 for the three tests. ) Snce the proporton of rejectons of the null hypothess when the alternatve s true s needed, Pareto populatons of sze N = 4,000 are generated for varous parameter values and, =,,..,. Snce means of Pareto dstrbutons depend on both locaton parameter,, and shape parameter,, =,,.., are set for three cases: the locaton parameters are the same and the shape parameters are dfferent, both the locaton and shape parameters are dfferent, and the shape parameters are the same but the locaton parameters are dfferent. Snce there are many values of locaton parameter and shape parameter, the dfference of means of several Pareto dstrbutons s consdered. The dfference of means s measured by the coeffcent of varaton (C.V.), S.D.( μ, μ,..., μ) C.V.( μ ) =. (5) Mean( μ, μ,..., μ) To mae t clear, the examples of the calculaton of the coeffcent of varaton are llustrated n Appendx. rom each generated Par(, ) populaton, 0,000 random samples, each of sze n, are drawn. The test statstcs of the three tests are calculated. If the value of the test statstc s n the rejecton area, as defned by the crtcal values, then the null hypothess s rejected. The power of the test s the proporton of tmes that the null hypothess s rejected. Let be the power of the ANOVA test, the power of the lelhood rato test, and the power of the Krusal -Walls test.the values of parameters and the sgnfcant value are set as follows: ) = number of the populatons = ) n = sample szes from the th Pareto populaton between 0 and 0, for =,,..., ), the locaton parameter of the th Pareto populaton, s between,00 and,5 4), shape parameter of the th Pareto populaton, s between.7 and 50 and fxed by the locaton parameter and the coeffcent of varaton of populaton means 5) α = B. Results of the Power Comparson or a fxed null hypothess, the power of the three tests s obtaned as the proporton of rejecton when the alternatve hypothess s true wth dfferent values of the coeffcents of varaton of populaton means. When the coeffcent of varaton of populaton means s zero, the null hypothess s true. ence, the proporton of rejectng the null hypothess s equal to the level of sgnfcance α. When the coeffcent of varaton of populaton means s greater than zero, the null hypothess s false or the alternatve hypothess s true. Thus the proporton of rejecton of the null hypothess s the estmate of the power of the test. The results of the power of tests for dfferent sample szes and dfferent coeffcents of varaton of populaton means are compared when α s set equal to The results of comparson are dvded nto three cases. ) Same Locaton Parameters but Dfferent Shape Parameters rom the values of locaton parameters and shape parameters as Table A.I n Appendx, and samples of szes 0 to 0, the power of the three tests s shown n Table II-Table V. Table II Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton

4 Table III Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table IV Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table V Power of the s When the Populaton Vares wth Unequal Sample Szes of n = 0, n = 0, n = 0 Populaton In ths case, f the dfferences among the populaton means are small, the lelhood rato test has the hghest power for samples of szes 0 to 0. The power of all three tests s almost the same, f the dfferences among the populaton means are large. ) Dfferent Locaton Parameters and Dfferent Shape Parameters rom the values of locaton parameters and shape parameters as Table A.II n Appendx, and samples of szes 0 to 0, the power of the three tests s shown n Table VI- Table IX. Table VI Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table VII Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table VIII Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table IX Power of the s When the Populaton Vares wth Unequal Sample Szes of n = 0, n = 0, n = 0 Populaton

5 In ths case, f the dfferences among the populaton means are small, the lelhood rato test has the hghest power for the smulaton wth equal sample szes of n=0 and unequal sample szes but the power of ANOVA test s hgher than that of the lelhood rato test for the smulaton wth equal sample szes of n=0 and n=0. owever, all the three tests have almost the same power when the dfferences among the populaton means are large. ) Dfferent Locaton Parameters and Same Shape Parameters rom the values of locaton parameters and shape parameters as Table A.III n Appendx, and samples of szes 0 to 0, the power of the three tests s shown n Table X- Table XIII. Table X Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table XI Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table XII Power of the s When the Populaton Vares wth Equal Sample Szes of n = 0 Populaton Table XIII Power of the s When the Populaton Vares wth Unequal Sample Szes of n = 0, n = 0, n = 0 Populaton In ths case, f the dfferences among the populaton means are small, the Krusal-Walls test has the hghest power. If the dfferences among the populaton means are large, the lelhood rato test and the Krusal-Walls test have almost the same power and hgher power than ANOVA test. V. CONCLUSION To test the equalty of means of Pareto data, ANOVA test, lelhood rato test and Krusal-Walls test are nvestgated. The use of ANOVA test for testng equalty of means for several Pareto dstrbutons, transformed data to normal wth constant varances s needed. The data sets transformed by an alternatve transformaton meet the assumptons requred for the applcaton of ANOVA test. The power of ANOVA test s compared to those of the other two exstng tests, the lelhood rato and the Krusal-Walls tests. The power functons of the three tests are of dfferent forms and cannot be compared explctly. A numercal method s then used for comparson purposes. It s found that f the locaton parameters are the same and the shape parameters are dfferent, the lelhood rato test has the hghest power for samples of szes 0 to 0. The power of all three tests s almost the same f the dfferences among the populaton means are large. If the populatons have dfferent values of both locaton and shape parameters, the lelhood rato test has the hghest power for the smulatons wth equal samples of szes 0 and unequal sample szes but ANOVA test s hgher than the lelhood rato test a lttle for the smulatons wth equal samples of szes 0 and 0. owever, all the three tests have almost the same power when the dfferences among the populaton means are large. If the populatons have the same shape parameters but dfferent locaton parameters, the Krusal-Walls test has the hghest power. In ths case, as the dfferences among the populaton means are large, the lelhood rato test and Krusal-Walls test have the same power and hgher power than ANOVA test. To choose the approprate test, the locaton parameter and the shape parameter for each group should be checed. or the same locaton parameter but the dfferent shape parameter, sample szes do not affect on choosng the test. If the dfferences among the populaton means are small, the lelhood rato test should be selected. If the dfferences

6 among the populaton means are large, all three tests can be selected. or both the dfferent locaton parameter and shape parameter, samples of szes 0 and unequal sample szes, f the dfferences among the populaton means are small, the lelhood rato test should be selected. If the dfferences among the populaton means are large, anyone can be selected. or samples of szes 0 and 0, f the dfferences among the populaton means are small, the ANOVA test should be selected. If the dfferences among the populaton means are large, all three tests can be selected. or the dfferent locaton parameter and the same shape parameter, sample szes do not affect on choosng the test. If the dfferences among the populaton means are small, the Krusal-Walls test should be selected. If the dfferences among the populaton means are large, the Krusal-Walls test or the lelhood rato test can be selected. It seems that the lelhood rato test s a good choce n almost every case. owever, t s dffcult to fnd a test based on the lelhood rato. APPENDIX Table A.I Calculaton of the Populaton for the Same Locaton and Dfferent Shape Parameters wth = Locaton parameters Shape parameters Mean S.D. C.V Table A.II Calculaton of the Populaton for the Dfferent Locaton and Dfferent Shape Parameters wth = Locaton parameters Shape parameters Mean S.D. C.V Table A.III Calculaton of the Populaton for the Dfferent Locaton and Same Shape Parameters wth = Locaton parameters Shape parameters Mean S.D. C.V REERENCES [] W. Tuey, On the comparatve anatomy of transformatons, Annals of Mathematcal Statstcs, vol., 957, pp [] D. C. Montgomery, Desgn and Analyss of Experments, 5th ed. New Yor: Wley, 00, pp []. E. P. Box and D. R. Cox, An analyss of transformatons (wth dscusson), Journal of the Royal Statstcal Socety, Ser.B. vol. 6, 964, pp.-5. [4] J. A. John and N. R. Draper, An alternatve famly of transformatons, Appled Statstcs, vol. 9(), 980, pp [5] K. A. Dosum, and C. Wong, Statstcal tests based on transformed data, Journal of the Amercan Statstcal Assocaton, vol. 78, 98, pp [6] P. R. s, The graduaton of ncome dstrbutons, Econometrc, vol. 9(), 96, pp [7] N. L. Johnson, and S. Kotz, Contnuous Unvarate Dstrbutons. New Yor: Wley, 970. [8] C. M. arrs, The Pareto dstrbuton as a queue servce dscplne, Operatons Research, vol. 6, 968, pp [9] M. Chatterjee, S. K. Das, Optmal MAC state swtchng for CDMA 000 networs, Paper Research n Wreless Moblty and Networng, Unversty of Texas at Arlngton, 004. [0] L. Watthanacheewaul and P. Suwattee, Analyss of varance wth Pareto data, Proceedngs of the th Annual Conference of Asa Pacfc Decson Scences Insttute, ong Kong, 006, pp [] W.. Krusal, A Nonparametrc for the Several Sample Problem, The Annals of Mathematcal Statstcs, vol. (4), 95, pp [] P. B. Patna, The non-central χ and -dstrbutons and ther Applcatons, Bometra, vol. 6(), 949, pp. 0-. [] E. S. Pearson and. O. artley, Charts of the power functon for analyss of varance tests derved from the non-central -dstrbuton, Bometra, vol. 8(), 95, pp.-0. [4]. O. Posten and R.E. Bargemann, Power of the lelhood rato test of the general lnear hypothess n multvarate analyss, Bmometra, vol. 5(), 964, pp [5] D. Karls and E.Xeala, A smulaton comparson of several procedures for testng the Posson assumpton, The Statstcan, vol. 49, 000, pp