MAXIMUM LIKELIHOOD PARAMETER ESTIMATORS FOR THE TWO POPULATIONS GEV DISTRIBUTION

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1 MAXIMUM LIKELIHOOD PARAMETER ESTIMATORS FOR THE TWO POPULATIOS GEV DISTRIBUTIO Jose A. Raynal-Vllasenor Department of Cvl and Envronmental Engneerng Unversdad de las Amercas, Puebla 780 Cholula, Puebla, Meco E-mal: Abstract: The method of mamum lkelhood for estmatng the parameters of the two populatons general etreme value TPGEV probablty dstrbuton functon for the mama s presented for the case of flood frequency analyss. The proposed methodology s compared wth wdely used models, namely: two component etreme value TCEV, general etreme value GEV and Gumbel dstrbutons. The TPGEV dstrbuton behaved well for those selected sets of data n orthwestern Meco and the results of ths dstrbuton proved to be better than the TCEV model, when there are two populatons present n the flood sample of data. The paper contans several numercal eamples of the applcaton of the proposed methodology. Key words: probablty, flood frequency analyss, mamum lkelhood, parameter estmaton, med dstrbutons Introducton The method of mamum lkelhood has been acknowledged as one of the best methods for parameter estmaton of probablty dstrbuton functons. The propertes of ts estmators lke the nvarance property, Mood et al974, and the asymptotcally unbasedness, suffcency, for a partcular class of probablty dstrbuton functons, consstency and effcency, Haan, 977 and the remarkable sutablty when beng appled to cumbersome mathematcal epressons n ts lkelhood functons under some strct regularty condtons, have ganed t the well-known condton of prme choce for solvng problems of parameter estmaton of probablty dstrbuton functons. The use of the general etreme value GEV dstrbuton functon Jenknson, 955 and 969 for flood frequency analyss s wdespread enough and t s now a feasble opton, by the practcng engneers, for applcaton n the flood frequency analyss process ERC, 975; Prescott and Walden, 980; and Hoskng et al, 990. The use of a mture of probablty dstrbutons functons for modelng samples of data comng from two populatons have been proposed long tme ago Mood et al, 974. In the partcular case of etreme value dstrbutons, several optons have been proposed so far, the TCEV dstrbuton Gumbel,958; Todorovc and Rousselle, 97; Canfeld, 979; and Ross et al, 984 the med Gumbel dstrbuton, Gonzalez-Vllarreal, 970; Raynal-Vllasenor, 986; and Raynal-Vllasenor and Guevara-Mranda, 998, and the med general etreme value dstrbuton Raynal-Vllasenor and Santllan-Hernandez, 986; and Guterrez-Oeda and Raynal-Vllasenor, 988.

2 It s the purpose of ths paper to epand the knowledge of the two populatons general etreme value dstrbuton n the case of the mama, by provdng the procedure of applcaton of the method of mamum lkelhood for estmatng ts parameters. The General Etreme Value Dstrbuton for the Mama The probablty dstrbuton functon of the GEV dstrbuton for the mama s, ERC 975: F ep / where,, and are the scale, shape and locaton parameters. The probablty densty functon s gven by, ERC 975: / / f ep For etreme value type I Gumbel dstrbuton 0; =.396: - For etreme value type II Frechet dstrbuton < 0; >.396 : + / < For etreme value type III Webull dstrbuton > 0; <.396: - < + / where s the coeffcent of skewness. The Two Populatons General Etreme Value Dstrbuton Based n the general form for two populatons probablty dstrbutons functons Mood et al, 974: F m p F ; pf ; 6 where p s the proporton of the second populaton n the mture. The TPGEV dstrbuton can be constructed as: / F m pep p ep 7 /

3 and the correspondng probablty densty functon s: f m p / / ep p / / ep 8 The Method of Mamum Lkelhood The method of mamum lkelhood have been defned and appled to several probablty dstrbuton functons wth defned probablty densty functons pdf ERC, 975. Such method has sutable characterstcs lke the nvarance property Mood et al, 974, and the asymptotcally unbasedness, suffcency, consstency and effcency Haan, 977 n large sample estmaton and applcablty n estmatng the parameters of comple probablty densty functons. The lkelhood functon of ndependent random varables s defned to be the ont probablty densty functon of random varables and s vewed as a functon of the parameters. If X,..., X s a random sample of a unvarate probablty densty functon, the correspondng lkelhood functon for the observed X,..., X sample s Mood et al, 974: L, f 9 where denotes the parameter set and f. s the probablty densty functon. The logarthmc verson of eq. 8 s: Ln [ L, ] Ln[ f ] 0 and wll be used nstead of the former equaton because t s easer to handle n the computatonal procedure descrbed n the net secton. The set of parameters that mamze equaton 9, f they ests, wll be the mamum lkelhood estmators for the parameters of the probablty dstrbuton functon. Mamum Lkelhood Parameter Estmators for the Two Populatons GEV Dstrbuton for the Mama Based n the prncples contaned n the prevous secton, the log-lkelhood functon for the TPGEV dstrbuton for the mama s: 3

4 4 p Ln p L Ln /, ],,,,, ; [ / / / ep ep p and the correspondng frst order partal dervatves of such functon wth respect to each of the parameters are: DE F F C L Ln / / ; ; =, / ; F C L Ln DE f / / ; 3 =, DE Ln C L Ln / ep 4 =,

5 Ln L p f ; f ; DE 5 where: DE f 6 m / F ; ep 7 / F ; ep 8 C = -p ; C = p 9 The eact soluton provded by the system of equatons -4 s not known for the case of the of TPGEV dstrbuton, so the mamum lkelhood estmators of the parameters of the TPGEV dstrbuton may be obtaned by ether solvng numercally, e.g. by the method of ewton, the system of non-lnear equatons equatons -4, or by a drect mamzaton of the log-lkelhood functon, equaton, by a non-lnear optmzaton procedure, e.g. the multvarable constraned Rosenbrock method Kuester and Mze, 973. In ths study the former opton was the choce for estmatng the parameters of the TPGEV dstrbuton by the method of mamum lkelhood. The TCEV Dstrbuton The Two Component Etreme Value probablty dstrbuton has been defned Ross et al, 984 as: F ep ep ep 0 where s the locaton parameter and θ s the shape parameter of the TCEV dstrbuton. The mamum lkelhood parameters of the TCEV dstrbuton are obtaned by an teratve scheme usng the followng equatons Ross et al, 984: 5

6 ep ep ; =, ep ep ep ; =, where ψ. s the dgamma functon wth argument.. Results and Dscusson As eamples of applcaton, the annual flood dscharges of several gaugng statons, located n the states of Snaloa and Chhuahua, n orthwestern Meco, were processed and the sample mamum lkelhood estmators of the parameters of the TPGEV dstrbuton were computed. Those gaugng statons are located n an area that every year s affected by tropcal cyclones, durng summer and fall, and cold fronts, durng wnter, causng the presence of at least two populatons n the samples of flood data. The years of record, computed sample mean, standard devaton and coeffcent of skewness of the samples of flood data for the selected gaugng statons are shown n table. Table. Statstcal characterstcs of flood data of the selected gaugng statons Statstcal Characterscs Gaugng Staton Years of Record Mean Standard Devaton Coeffcent of Skewness El Oregano Santa Cruz Hutes El Zoplote Jana Ipalno Acattan San Bernardo Cho Tezocoma

7 The one populaton general etreme value and Gumbel dstrbutons computed parameters, were obtaned through the applcaton of user-frendly computer package FLODRO 4.0 Raynal-Vllasenor, 00 for the selected gaugng statons, and they are shown n tables and 3. The TPGEV and TCEV dstrbuton computed parameters for such gaugng statons were evaluated by usng computer code FLODRO 4.0 Raynal-Vllasenor, 00 and the results are contaned n tables 4 and 5. In order to compare the results provded by the TPGEV dstrbuton wth those produced by other wdely appled models, such the one populaton general etreme value GEV, Gumbel G and Two Component Etreme Value TCEV dstrbutons, n table 6 a complaton s presented of the desgn values for several return perods and ther standard errors of fttng, EE, produced by the methods mentoned above and the one proposed n the paper. The EE s defned as Kte, 988: / y EE 3 m where are the hstorcal values of the sample, y are the values produced by the dstrbuton functon correspondng to the same return perods of the hstorcal values, s the sample sze, and m s the number of parameters of the dstrbuton functon. Table. One populaton GEV and Gumbel EV-I dstrbutons parameters for the selected gaugng statons Gumbel Parameters GEV Parameters Gaugng Staton λ λ El Oregano Santa Cruz Hutes El Zoplote Jana Ipalno Acattan San Bernardo Cho Tezocoma The results of ths study provde the arguments to establsh the followng ponts: 7

8 The TPGEV dstrbuton functon behaved very well n the selected gaugng statons, ust n two out of ten t cannot reach convergence. The TCEV faled to attan convergence n three samples of flood data. In the case of the TPGVE, the lack of convergence was not solved by changng the ntal values n the optmzaton procedure, t seems that for a specfc sample of flood data the procedure ust wll have a lack of convergence, so n those cases such model smply won t work. The lack of convergence n the case of the TCEV t seems s assocated by the estmaton procedure tself, t won t converge n many nstances. Table 3. TPGEV dstrbuton parameters for the selected gaugng statons Staton λ λ p El Oregano Santa Cruz * * * * * * * Hutes El Zoplote * * * * * * * Jana Ipalno Acattan San Bernardo Cho Tezocoma * o convergence was attaned Table 4. TCEV dstrbuton parameters for the selected gaugng statons Staton p El Oregano Santa Cruz Hutes * * * * * El Zoplote Jana Ipalno Acattan San Bernardo * * * * * Cho * * * * * Tezocoma * o convergence was attaned The TPGEV dstrbuton functon has the least standard error of ft EE n fve gaugng statons and was very close to the least value n four addtonal gaugng statons. The GEV reached the least value of the EE n fve of the gaugng statons 3 one of the Gumbel etreme value type I nor the TCEV dstrbutons were even close to any of the least values of the EE n the ten selected gaugng statons 8

9 Table 5. Comparson of Desgn Values n m 3 /s and Standard Errors of Fttng n m 3 /s Between Several Models for One and Two Populatons Samples Model Q 5 Q 0 Q 0 Q 50 Q 00 EE El Oregano TCEV TPGEV GEV Gumbel Sta. Cruz TCEV TPGEV * * * * * * GEV Gumbel Hutes TCEV * * * * * * TPGEV GEV Gumbel Zoplote TCEV TPGEV * * * * * * GEV Gumbel Jana TCEV TPGEV GVE Gumbel Ipalno TCEV TPGEV GEV Gumbel Acattan TCEV TPGEV GEV Gumbel TCEV = Two Component Etreme Value Dstrbuton TPGEV = Two Populatons General Etreme Value Dstrbuton * o convergence was attaned n the estmaton of parameters process Bold numbers correspond to the dstrbuton wth best ft 9

10 Table 5. Comparson of Desgn Values n m 3 /s and Standard Errors of Fttng n m 3 /s Between Several Models for Two Populatons Samples cont d Model Q 5 Q 0 Q 0 Q 50 Q 00 EE San Bernardo TCEV * * * * * * TPGEV GEV Gumbel Cho TCEV * * * * * * TPGEV GEV Gumbel Tezocoma TCEV TPGEV GEV Gumbel TCEV = Two Component Etreme Value Dstrbuton TPGEV = Two Populatons General Etreme Value Dstrbuton * o convergence was attaned n the estmaton of parameters process Bold numbers correspond to the dstrbuton wth best ft 4 The TPGEV dstrbuton functon has the least standard error of ft EE n fve gaugng statons and was very close to the least value n three addtonal gaugng statons. The GEV reached the least value of the EE n fve of the gaugng statons 5 Wth regard to the desgn values, for those gaugng statons where the TPGEV dstrbuton produced the best ft, the produced values were much hgher than those for the GEV dstrbuton 6 The computaton of the parameters and desgn values for the TPGEV dstrbuton were made possble by the use of a personal computer. It wll be very dffcult, f not mpossble, to evaluate such parameters and desgn values wth a portable calculator or some other computng devce wth less capacty than a personal computer. Ths s a drawback that the proposed method has and there s no way to overcome t, gven the enormous number of calculatons that the optmzaton code has to perform n order to obtan the mamum lkelhood estmators of the parameters of the TPGVE dstrbuton Conclusons The procedure of fndng the estmators of the parameters of the TPGEV dstrbuton for the mama, usng the method of mamum lkelhood, has been presented. The TPGEV dstrbuton behaved well for those selected sets of flood data, ust n two out of the ten cases consdered for analyss, the TPGEV could not reach convergence n the 0

11 estmaton of parameters process. The lack of convergence was not solved by changng the ntal values n the optmzaton procedure, t seems that for a specfc sample of flood data the procedure ust wll have a lack of convergence, so n those cases such model smply won t work. In these cases another model of med dstrbutons should be used. The TCEV had three falures n the estmaton of the parameters process due to the lack of convergence. The lack of convergence n the case of the TCEV t seems s assocated by the estmaton procedure tself, t won t converge n many nstances. In fve cases the TPGEV dstrbuton produced the least standard error of ft and n other three cases was very close to the GEV dstrbuton whch has the least standard error of ft n such samples of flood data. It wll be wse to consder the presence of two populatons n the sample of flood data, n addton to the standard error of ft, to reach a decson on whch model to use for flood frequency analyss. Based n the results presented n the paper, the author recommend ths procedure to be ncluded n the standard methods for flood frequency analyss, as an addtonal model for the flood frequency analyss when there s the possblty that two populatons are present n the samples of flood data. Acknowledgements The author wsh to epress ther grattude to the Unversdad de las Amercas, Puebla for the support provded n the realzaton of ths paper. References Beran, M., Hoskng, J. R. M. and Arnell, Comment on Two Component Etreme Value Dstrbuton for Flood Frequency Analyss by Ross et al, Wat. Resour. Res.,, Canfeld, R. V The Dstrbuton of the Etremes of a Mture of Random Varable wth Applcatons to Hydrology, n Input for Rsk Analyss n Water Systems, E.A. McBean, K. W. Hpel and T. E. Unny, eds., Water Resources Publcatons, Gonzalez-Vllareal, F. J Contrbuton to the Frequency Analyss of the Etreme Values of the Floods n a Rver, Report # 77, Insttuto de Ingenera, Unversdad aconal Autonoma de Meco, Meco, D.F., Me. n Spansh Gumbel, E.J Statstcs of Etremes, Columba Unversty Press, ew York,. Y., 8. Guterrez-Oeda, C. and Raynal-Vllasenor, J. A Med Dstrbutons n Flood Frequency Analyss, X atonal Congress on Hydraulcs, Morela, Mch., Me., atonal Assocaton of Hydraulcs, 0-8. n Spansh Haan, C.T Statstcal Methods n Hydrology, The Iowa State Unversty Press, Ames, Iowa, 63.

12 Hoskng, J. R. M., 990, L-moments: Analyss and Estmaton of Dstrbuton usng Lnear Combnaton of Order Statstcs, J. R. Statst. Soc. B, 5, o., Jenknson, A. F The Frequency Dstrbuton of the Annual, Mamum or Mnmum Values of Meteorologcal Elements, Quart. J. Royal Met. Soc., 87, Jenknson, A. F Estmaton of Mamum Floods, Chapter 5, WMO, Techncal ote 98, Geneva, Swtzerland, Kte, G.W Frequency and Rsk Analyses n Hydrology, Water Resources Publcatons, Lttleton, Colorado, 87. Kuester, J. L. and Mze, J. H Optmzaton Technques wth FORTRA, Mc-Graw Hll Book Co., Mood, A. M., Graybll, F. and Boes, D. C Introducton to the Theory of Statstcs, McGraw-Hll Inc., Thrd Ed., ew York,. Y., atural Envronment Research Councl, ERC 975. Flood Studes Report, I, Hydrologc Studes, Whtefrars Press Ltd., London, 5. Prescott, P. and Walden, A. T Mamum Lkelhood Estmaton of the Parameters of the Generalzed Etreme Value Dstrbuton, Bometrka, 673, Raynal-Vllasenor, J.A Mamum Lkelhood Estmators of the Parameters of the Med Gumbel Dstrbuton, XII Congress of the atonal Academy of Engneerng, Saltllo, Coah., Me., n Spansh Raynal-Vllasenor, J. A., and Santllan-Hernandez, O. D., 986. Mamum Lkelhood Estmators of the Parameters of the Med General Etreme Value Dstrbuton, IX atonal Congress on Hydraulcs, Queretaro, Qro., Me., atonal Assocaton of Hydraulcs, In Spansh Raynal-Vllasenor, J. A. and Guevara-Mranda, J. L Mamum Lkelhood Estmators for the Two Populatons Gumbel Dstrbuton, Hydrologcal Scence and Technology J., Vol. 3, o. -4, pp Raynal-Vllasenor J. A., 00. Frequency Analyss of Hydrologc Etremes, Lulu.com, USA Ross, F., Florentno, M. and Versace, P., 984. Two Component Etreme Value Dstrbuton for Flood Frequency Analyss, Wat. Resour. Res., 07, Todorovc, P. and Rousselle, J., 97, Some Problems of Flood Analyss, Wat. Resour. Res., 75,