Estimation of the Parameters of the Exact Extreme Value Distribution

Transcription

1 IOSR Joural of ahemacs (IOSR-J) e-issn: , p-issn: 9-75X. Volume, Issue Ver. V (ar. - Apr. 07), PP Esmao of he Parameers of he Eac Ereme Value Dsrbuo Dr. Neama Qub, Dr. Asha Fayoum, Ohoud Al-Belad,, (Deparme Of Sascs, Faculy Of Scece / Kg Abdulazz Uersy, Saud Araba.) Absrac: Eac ereme alue dsrbuo s oe of he mos mpora compoud dsrbuos whch s based o he heory of he mamum of radom arable of radom umbers. Ths dsrbuo uses paral durao seres (PDS) daa o aalyze ereme hydrologcal. Ths dsrbuo s preseed wh s properes ad graphcal represeaos. omes (O), mamum lkelhood (L) ad Bayesa - based o o-formae ad formae pror- mehods are used o esmae he ukow parameers of he dsrbuo. arko Cha oe Carlo (CC) echque s used o compue he Bayesa esmaes. The smulao s performed o esgae ad compare bewee he esmaors wh dffere szes ad a se of he parameer's alues. I he sese of he mea squared error (SE), he resuls showed ha Bayesa -based o formae pror- mehod s he bes esmao mehod. Keywords: Eac ereme alue dsrbuo, mamum lkelhood, Bayesa esmao, CC. I. Iroduco Compoud dsrbuos ca be useful arous felds such as busess falures, relably ad lfe esg, rsk heory, queug heory, ecoomc heory, ereme alue heory, radar heory ad also for purely heorecal cosderaos. I may of hese cases he compoud dsrbuo or some specal cases of are good represeaos for he eplaao of he uderlyg sochasc pheomea. Also, from he fac ha some of he parameers of he smple dsrbuo could hemseles be radom arables, s que reasoable o assume ha he compoud dsrbuo would be suable may applcaos. Compoud dsrbuos whch s based o he heory of he mamum of radom arable of radom umbers back o 970 whe Todoroc preseed he heory of eac ereme alue. Ths compoud model uses paral durao seres PDF- Peak oer Threshold (POT)- whch coss of flood peaks eceedg some hreshold alue as a alerae aalycal ool for ama's aual dscharge mehod whch coss of mamum flow raes from each year wh he obserao perod o aalyze ereme flood. The heorecal bass for deelopme of he POT mehod was se by work of Todoroc [, ]. The model represes he sochasc aure of he umber ad magude of he eceedaces by Posso ad epoeal dsrbuos respecely.[] suded some properes of he compoud epoeal Posso dsrbuo hey also esmaed s parameer by mamum lkelhood mehod ad used he esmaors goodess of f es of he model o daa from Aswa sao o he Nle Rer. As a alerae o Posso dsrbuo for flood cou, egae bomal dsrbuo was proposed by [4]. [5] foud ha he Posso dsrbuo was beer ha he egae bomal dsrbuo cases whe he dfferece bewee he mea ad he arace of he aual umber of eceedeces was small. [] Compared bewee he PDS mehod ad he aual mamum(a) mehod for flood frequecy aalyss for daa from he Lja gaugg sao o he Saa Rer Sloea. The POT mehod gae bes resuls ha he A mehod. The POT model used he epoeal ad Pareo dsrbuos for modellg he magudes of eceedeces, ad he Posso, bomal ad egae bomal dsrbuos for he aual umber of ees aboe he hreshold. Four ess were used o check he adequacy of he Posso ad epoeal dsrbuos. For modellg he aual umber of eceedeces aboe he hreshold, he Posso dsrbuo gae bes resuls ha he bomal dsrbuo. The ma purpose of hs paper s o sudy he properes of he eac ereme alue model, ad esmae s parameers. Ths paper s orgazed as follows: Seco, roduco o Eac ereme alue dsrbuo. Seco roduces he dsrbuo of he eremes of radom arables of radom umbers. Seco prodes he heorecal cosderaos of he model. Seco 4 llusraes he sascal properes of he model. Seco 5 esmao of he ukow parameers of he dsrbuo usg O, L ad Bayesa mehods. A smulao sudy s coduced Seco. The coclusos are show Seco 7. II. Dsrbuo of Eremes of Radom Varables of Radom Numbers [] preseed a approach o aalyze ereme alues of radom umbers of occurrg pheomea. He defed a sequece of obseraos,,,... wh umber of occurrece, ad he probably fuco of, as p E p, He dered he dsrbuo fuco for gg a aeo o he wo arables: X ma,,,..., ad Z m,,,.... X ad Z deoed by F ad z () () F respecely, as k F p E k, () k0 0 z k F p E k. (4) k Suppose ha s a sequece of depede radom arables wh he commo dsrbuo fuco depede of he () ad (4) become: H ad DOI: / Page

2 Esmao of he Parameers of he Eac Ereme Value Dsrbuo F z k0 k H p E, (5) k Hz k p E. k F () III. Theorecal Cosderaos Accordg o eac ereme alue heory preseed Seco, [] deeloped a geeral sochasc model o descrbe ad predc behaor of floods. The heory of hs model s preseed as follows. Cosder a sream flow hydrograph, represeg he sa flood peaks a a ge sao wh a eral of me 0,. Le us cosder oly hose peaks Q V,,,..., ha eceed he base leel Q 0. Le he magude of flood eceedaces be defed as follows Accordg o he aure of flood pheomea, he umber of flood eceedace Q Q0, 0. a ge eral of me k (7), he magude of flood eceedaces ad he me of occurrece of flood eceedaces 0 are radom arables. Ulke he asympoc ereme alue model, he plays a mpora role he eac ereme alue model because he laer model cosders boh ad he smulaeously. The umber of eceedaces 0, s defed as sup, (8). By defo, 0,,,... for all 0 ad for 0 ad 0 ; hs meas, o-decreasg fuco of. I he ee ha here are eacly eceedaces occurrg 0,, le E. (9) I whch s a parcular umercal alue of he radom arable. Le sad for he epeced alue of p E. (0) I he case of flood aalyss, he radom arable of he larges eceedace wh a specfc me eral 0, plays a mpora role, X s defed as X sup,. () X X for 0 ad 0 X s a o-decreasg sochasc process. By defo, ; hs meas ha Deoe by F he dsrbuo fuco of X,.e []dered a epresso for P X as, s a, he X, amog he se of all eceedaces F PX, 0, 0. F, o he bass of mahemacal epecao of he codoal probably, () Equao () descrbes he mos geeral form of Todoroc s eac ereme alue sochasc flood model for he F P X 0,. Equao () s dsrbuo fuco of larges eceedaces a specfc me perod dffcul o be soled drecly uless oe deermes he probably P. Therefore, s ecessary o DOI: / Page 0 E

3 Esmao of he Parameers of he Eac Ereme Value Dsrbuo,, 0, are decal depede occurrg cosder a parcular form of equao () whch he,... radom arables, wh he commo dsrbuo fuco H ad he radom sequeces ad depede for all. Uder hese assumpos, equao () smplfes o where F P E0 P P E P E0 H P E H s he dsrbuo fuco of he eceedaces ad E 0 a ge me eral 0,. o eceedaces, I he flood coe, s a me depede Posso process. Suppose ha ad s probably fuco s ge by P E P ep ( α)α! are sochascally () P 0 s erpreed as he probably ha here wll be, 0,,,... s dsrbued as Posso wh parameer, (4) The epoeal dsrbuo has bee used frequely fg frequecy dsrbuos o magudes of eceedaces. Ths dsrbuo s applcable o magudes of eceedaces ealuaed for a rage of rucao leels. The probably desy fuco ( PDF ) ad he cumulae dsrbuo fuco ( CDF ) of epoeally dsrbued radom arables ca be epressed, respecely, as h ep, 0, 0. (5) H P - ep -. () By subsug equaos (5) ad (7) (4) we ge he esg form of he eac ereme alues model as F ep - p E0 ep! ep - ep ep -! (7) whch reduces o F ep ep. (8) Dffereag he CDF (9) of he process wh respec o he PDF s obaed as f ep - ep - ep -,- ;, 0. (9) Fgure : The PDF of he eac ereme alue model I s clear from he aboe Fg. ha he PDF of he eac ereme alue model ca ake dffere shapes DOI: / Page

4 Esmao of he Parameers of he Eac Ereme Value Dsrbuo Fgure : The CDF of he eac ereme alue model IV. Sascal Properes I hs seco we prese some properes of he eac ereme alue model. 4. The relably ad hazard rae fucos The relably fuco (RF) ad he hazard rae fuco(hrf) for he eac ereme alue model are ge respecely by R F ep ep. (0) ep ep ep f h. () R - ep - ep - From Fg. s clear ha he hazard rae s a creasg fuco 4. The cumulae hazard fuco The cumulae hazard fuco H s Fgure : The HRF of he eac ereme alue model H 4. The Quale fuco The quale fuco correspodg o he dsrbuo equao (9) s ge by q l l q, 0 q h d l ep ep The r h mome ad mome geerag fuco The mome geerag fuco of he eac ereme alue model s ge by where. deoes he gamma fuco. DOI: / Page () () Ee e f. d (4)

5 Esmao of he Parameers of he Eac Ereme Value Dsrbuo Fdg he r h mome by dffereag wh respec o ad pug = 0, oe obas he r h mome abou zero of he desy fuco (0) as r r (5) r E. 4.5 The mea, arace, mode ad meda The mea, arace, mode ad meda of he eac ereme alue model are ge respecely by X l. r E () X. V (7) ode l. (8) eda l 0.5 (9) mode 0.5 where s he Euler-aschero cosa 4. The skewess ad kuross coeffces The skewess ad kuross are ge respecely by Where.0 s A Apéry's cosa. Eac ereme alue dsrbuo s posely skewed dsrbuo Eac ereme alue dsrbuo s lepokurc dsrbuo (0) () V. Esmao mehods I hs seco, O, L ad Bayesa mehods s used o esmae he ukow parameers ad of he eac ereme alue model. Bayesa mehod s coduced usg o-formae pror dsrbuos ad also formae pror dsrbuos s used. The Bayes esmaes uder squared error loss fucos s obaed. 5. The mehod of momes ome esmaors of he shape parameer ad he scale parameer of he eac ereme alue dsrbuo are foud by equag he sample momes o he correspodg heorecal momes. I oher words, hey are he soluos of he followg equales X l ad S () ome esmaors of ad are he obaed as. S X ep ad. 5. amum lkelhood esmao Suppose ha X X, X,... X s a radom sample of sze from, eac ereme alue dsrbuo he he loglkelhood fuco s ge by log L log log ep. () (4) Dffereag (5) wh respec o ad respecely ad equag he resulg deraes o zero, we ge DOI: / Page

6 Esmao of he Parameers of he Eac Ereme Value Dsrbuo log L log L ep 0 ep 0 To oba he L esmaes of he shape parameer ad he scale parameer, umercal calculaos are requred o fd he soluo of he sysem of he olear equao (). 5.. Bayesa esmao usg o-formae pror Suppose we assume ha he ukow parameers ad hae he followg depede pror dsrbuos α~uform a, b, β~uform a, b. The, he jo poseror probably ca be wre as: where, ep ep A ep,, (5) () B A s he ormalzg cosa. Bayesa esmaors, s he jo pror dsrbuo of he parameers, ad uder squared error loss fuco ca be obaed by akg he mea of he equao (7). CC echque s used o fd he Bayesa esmaes of ad umercally. 5.. Bayesa esmao usg formae pror Le each of ad are depede radom arables ad follow oe parameer ered gamma k, respecely, he he pror dsrbuo for each parameers ca be wre as follows: k k The, he jo poseror probably ca be wre as: where k e, 0. k e, 0., ep ep A ep.,. dsrbuo, (7) (8) (9), s he jo pror dsrbuo of he parameers, ad uder squared error loss fuco ca be obaed by akg he mea of he equao (40). CC echque s used o fd he Bayesa esmaes of ad umercally. A s he ormalzg cosa. Bayesa esmaors VI. Smulao Sudy I hs seco, smulao sudy ad resuls are preseed. Smulao sudy hae bee performed o oba he esmaes preseed seco (5). Ths smulao was performed for dffere sample sze =0, 0, 00 ad a se of he parameer's. Each sample sze s repeaed 000 mes. For 000 replcaos, he esmaed SE of he esmaes are calculaed for each mehod. The resuls are show Table. Table : O, L ad Bayesa esmaes wh her (SE) uder eac ereme alue dsrbuo Parameers O L Bayesa o-formae formae (0.0) (0.0) (0.0) (0.008) (0.008) (0.008) 0.77 (0.005) (0.007) 0.70 (0.045) 0.77 (0.005) (0.007) 0.70 (0.045) (0.09) (0.09) (0.09) 0.55 (0.0574) 0.54 (0.0575) 0.54 (0.0575) 0.45 (0.004) 0.49 (0.05) 0.70 (0.004) 0.9 (0.008) 0.48 (0.05) (0.044) 0.75 (0.075) 0.75 (0.075) 0.74 (0.074) (0.0795) (0.0795) (0.079) 0.0 (0.0077) (0.008) (0.09) 0.88 (0.004) (0.057) (0.0579) 0.45 (0.057) 0.8 (0.00) 0.07 (0.009) 0.5 (0.08) (0.089) (0.098) (0.007) 0.57 (0.007) 0.78 (0.054) 0.54 (0.00) (0.000) 0.4 (0.057) DOI: / Page

7 (0.8) (0.8) (0.8) (0.0070) (0.0070) (0.0070) (0.057) (0.057) (0.057) (0.08) (0.08) (0.08) (0.00) (0.00) (0.00) (0.004) (0.004) (0.004) Esmao of he Parameers of he Eac Ereme Value Dsrbuo 0.77 (0.005) (0.007) 0.70 (0.045) (0.00) (0.0087) 0.77 (0.09) (0.00) (0.0087) 0.77 (0.09) (0.00) (0.0087) 0.77 (0.09) (0.000) (0.005) (0.000) (0.005) (0.000) (0.005) (0.) (0.7) (0.7) (0.00) (0.00) (0.00) (0.054) (0.054) (0.054) (0.077) (0.077) (0.077) (0.009) (0.009) (0.009) (0.008) 0.4 (0.054) (0.047) 0.47 (0.00) (0.005) 0.78 (0.08) 0.4 (0.00) (0.005) (0.09) 0.4 (0.00) (0.005) (0.09) (0.005) (0.005) (0.005) (0.887) 0.99 (0.890) 0.9 (0.87) 0.90 (0.0085) 0.90 (0.0085) (0.0084) (0.08) (0.08) (0.08) (0.07) (0.07) (0.07) 0.4 (0.00) 0.4 (0.00) 0.4 (0.00) (0.0047) 0.57 (0.0048) (0.0048) (0.0079) (0.0079) (0.0079) (0.0059) 0.55 (0.05) (0.05) 0.44 (0.00) (0.004) 0.79 (0.04) 0.0 (0.005) 0.5 (0.000) (0.05) (0.005) 0.57 (0.0058) (0.0) (0.00) (0.00) (0.00) 0.7 (0.005) (0.00) 0.70 (0.005) (0.080) (0.0850) (0.099) (0.0044) 0.75 (0.004) (0.07) (0.09) (0.0) (0.04) 0.75 (0.048) (0.055) 0.4 (0.008) 0.5 (0.00) (0.00) (0.0040) (0.004) (0.004) (0.007) (0.007) (0.007) 0.47 (0.00) 0.44 (0.0) 0.85 (0.0) 0. (0.000) (0.004) (0.0074) (0.000) (0.00) 0.47 (0.00) (0.0050) (0.00) (0.000) (0.00) (0.008) (0.000) (0.004) (0.00) (0.004) I s clear from he resuls ha he SE for he Bayesa esmaes of ad usg formae pror are smaller ha her correspodg SE of he L ad he O esmaes. Therefore, ca be cocluded ha he Bayesa mehod based o formae prors has proded beer esmaes of he parameers compared o he L ad O mehod. VII. Cocluso I hs paper, Eac ereme alue dsrbuo s preseed. Some of he sascal properes of he Eac ereme alue dsrbuo are suded. momes, mamum lkelhood ad Bayesa -based o o- formae ad formae pror dsrbuo -mehods are used o esmae he parameers of he dsrbuo. We fd ha he Bayesa esmaors based o formae pror dsrbuos are he bes esmaors. Refereces []. Todoroc, P., O some problems olg radom umber of radom arables, The Aals of ahemacal Sascs, 4(), 970, p []. Todoroc, P. ad Zelehasc, E., A sochasc model for flood aalyss, Waer Resources Research, () 970, p []. Kob, N.S.A., El-Gohary,.. ad El-Helbawy, A.T., Aalyss of Flood Frequecy Dsrbuo, Joural of Faculy of Commerce Al-Azhar Uersy Grls Brach, 9, 00, p [4]. Cuae, C., A oe o he Posso assumpo paral durao seres models,waer Resources Research, 5(), 979, p [5]. Bhuya, P.K., Sgh, R.D., Berdsso, R. ad Pada, S.N.,Flood aalyss usg geeralzed logsc models paral durao seres,. Joural of hydrology, 40, 0, p []. Bezak, N., Brlly,. ad Šraj,., Comparso bewee he peaks-oer-hreshold mehod ad he aual mamum mehod for flood frequecy aalyss, Hydrologcal Sceces Joural, 59(5), 04, p DOI: / Page