The generalized Pareto sum

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1 HYDROLOGICAL PROCESSES Hydrol. Proce. 22, ) Publihed online 28 Noveber 27 in Wiley InterScience The generalized Pareto u Saralee Nadarajah 1 *SauelKotz 2 1 School of Matheatic, Univerity of Mancheter, Mancheter M6 1QD, UK 2 Departent of Engineering Manageent Syte Engineering, George Wahington Univerity, Wahington, DC 252, USA Abtract: The generalized Pareto ditribution ha received uch popularity a odel for extree event in hydrological cience. In thi note, the iportant proble of the u of two independent generalized Pareto ro variable i conidered. Exact analytical expreion for the probability ditribution of the u are derived a detailed application to drought data fro Nebraa i provided. Copyright 27 John Wiley & Son, Ltd. KEY WORDS generalized Pareto ditribution; hydrological cience; drought data; probability ditribution Received 31 May 26; Accepted 17 October 26 INTRODUCTION Su of ro variable arie ot frequently in hydrological cience. Soe iportant exaple with repect to extree hydrological event are: 1) Strea flow fro two river, or river bain, ay cobine to provide water upply to a ingle region. In thi cae, the occurrence of iultaneou low flow i of outot iportance given the regional water upply reiliance on cobined available trea flow. 2) Confluence of two river whoe cobined flow create flood hazard. In thi cae, the occurrence of iultaneou high flow i of utot iportance. 3) Inter-arrival tie of drought event i the u of the drought the ucceive non-drought. In thi cae, the occurrence of long period of drought non-drought will be of interet ee the Application ection). See Loaiciga Leipni 1999) for ore exaple.) In thi note, the iportant proble of coputing the u of two independent generalized Pareto GP) ro variable are conidered. Why chooe the GP ditribution? Thi ditribution i widely recoended a the odel for extree event in hydrology the theoretical otivation for thi choice can be found in Kotz Nadarajah 2)). Dargahi- Noubary 1989) advocated the ue of GP for fitting a ditribution of annual axia of the wind peed that of axiu flood of the Feather river. Soe recent hydrological application of the GP ditribution include Maden Robjerg 1997a, 1997b), Maden et al. 1997a) Maden et al. 1997b), Lab 1999), Rauen 21), Martin Stedinger 21a, 21b, 22), * Correpondence to: Saralee Nadarajah, School of Matheatic, Univerity of Mancheter, Mancheter M6 1QD, UK. E-ail: aralee.nadarajah@ancheter.ac.u Kuar et al. 23), Catellarin et al. 24), Lab Kay 24). Suppoe X Y are independent GP ro variable. The probability denity function pdf) of X Y can be pecified by: f X x D 1 cx ) 1/c 1 1 f Y y D 1 dy, 2 repectively, for x>ifc), <x</c if c> ), y> if d ), <y</d if d>), >, >, 1 <c<1 1 <d<1. In thi note, exact expreion for the pdf of the u S D X C Y are derived a detailed application to drought data fro Nebraa ee the following two ection) i provided. The analytical calculation involve the Appell function of the firt ind defined by: F 1 a 1,a 2,a 3 ; a 4 ; x 1,x 2 D 1 1 id jd a 1 icj a 2 i a 3 j x i 1 xj 2 a 4 icj i!j! the Gau hypergeoetric function defined by: 2F 1 a 1,a 2 ; a 3 ; x D 1 id a 1 i a 2 i a 3 i x i i!, where q i D q q C 1 ÐÐÐ q C i 1 denote the acending factorial. The propertie of the above pecial function can be found in Prudniov et al. 1986) Gradhteyn Ryzhi 2). EXACT DISTRIBUTION OF S D X C Y Variou repreentation are derived for the pdf of S D X C Y. Theore 1 expree the pdf explicitly in ter of the Appell Gau hypergeoetric function. Copyright 27 John Wiley & Son, Ltd.

2 THE GENERALIZED PARETO SUM 289 Theore 1 Suppoe X Y are independent GP ro variable with pdf 1) 2), repectively. The pdf of S D X C Y can be expreed a one of the following: 1. if either c d orc, d></d or c>, d></dthen d F 1 1, 1 c, 1 c, 2; d, ) d. 3 d 2. if either c, d> ½ /d or c>, d> /d < < /c then 1 c ) 1/c 1 2F 1 1, 1 c ;1C 1 d ; c d c 3. if c>, d> ½ /c then c c C d 1 ) 1/c 1 d ) 1/cC1/d 1 B ). 4 1 c, 1 ). 5 d Proof The joint pdf of S, W D X C Y, X/S can be written a: f, w D cw ) 1/c 1 d w. 6 The pdf S follow by integrating Equation 6) with repect to all poible value of w. If either c d orc, d>>/dor c>, d> </dthen note that the following expreion can be written: 1 cw ) 1/c 1 dw. 7 d w If either c, d> ½ /d or c>, d> /d < < /c then 1 1 / d d w cw ) 1/c 1 dw. 8 If c>, d> ½ /c then / c 1 / d d w cw ) 1/c 1 dw. 9 The reult in Equation 3) follow by applying Lea 1 fro the Appendix to calculate the integral in Equation 7). The reult in Equation 8) follow by applying Lea 2 fro the Appendix to calculate the integral in Equation 8). Finally, note that the integral in Equation 9) reduce to a beta integral on applying the tranforation x Dfw 1 C / d g/f/ d C / c 1g. Uing pecial propertie of the Appell Gau hypergeoetric function, one can reduce Equation 3) 4) to eleentary for when 1/c 1/d tae integer value. Thi i illutrated in the corollarie below. Corollary 1 If 1/c ½ 1 1/d ½ 1 are integer then Equation 3) can be reduced to: d I 1 c, 1 ), d where I a, b atifie the recurrence relation d I a, b D b 1 fd C c d g { c ) ) } 1 a 1 b d a C b 2 d I a, b 1 b 1 fd C c d g for a>1b>1 with the initial value { } d I 1, 1 D d C c d log, d c { I a, D c ) } 1 a 1 c a 1 I,b D { ) d 1 b 1}. d b d Corollary 2 If 1/c ½ 1 1/d ½ 1 are integer then Equation 4) can be reduced to: c d 1/c 1 ) i I i, 1 c id ), ) 1/d 1 i Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )

3 29 S. NADARAJAH AND S. KOTZ where I a, b atifie the recurrence relation a d b a 1 c b I a, b D c a b C 1 fd C c d g b 1 a c I a 1,b c a b C 1 for a>1b>1 with the initial value [ {d } c C c d 1 b I,b D 1]. b c d c APPLICATION Here, the drought proble dicued in the Introduction ection i returned to an application of the reult of Theore 1 i provided. The drought data i ued fro the State of Nebraa, freely downloadable fro the webite: xgrg3.htl. The data coprie of the onthly odified Paler Drought Severity Index PDSI) fro the period fro January 1895 to Deceber 24. A drought i aid to have happened when PDSI i below i defined by the theory of run Yevjevich, 1967). The State of Nebraa i divided into eight cliate diviion nubered1,2,3,5,6,7,89 thereinocliatediviion 4 for Nebraa ee Figure 1 for a geographical ap of the eight diviion). Soe tatitic of the oberved drought for the eight cliatic diviion are uarized in Table I. The real drought data et for the 83 drought event in cliate diviion 1 i illutrated in Table II. Uing the PDSI data, data on drought non-drought were obtained for each cliate diviion. The interet i in deterining the ditribution of the inter-arrival tie of drought S D drought X C non-drought Y). It i aued that the obervation of X Y are independent, an auption confired by the tet of aociation reult in Table III. The ditribution of S wa deterined by fitting the odel given by Equation 1) 2) to the oberved value of X Y, repectively, uing the Equation 3) 5) to copute the fitted pdf. For coparion, the ditribution Cliate diviion Table I. Baic drought tatitic for Nebraa PDSI data Nuber of drought Drought frequency nuber per year) Mean drought onth) Stard deviation of drought onth) 1 83 Ð75 6Ð 8Ð 2 66 Ð6 8Ð6 12Ð 3 89 Ð81 6Ð3 9Ð Ð74 6Ð3 1Ð5 6 9 Ð82 6Ð3 1Ð Ð74 6Ð1 9Ð Ð69 6Ð5 13Ð Ð67 7Ð5 1Ð9 Cae Table II. Drought data for Nebraa cliate diviion 1 Drought onth) Non-drought onth) Cae Drought onth) Non-drought onth) Figure 1. The eight cliatic diviion of the State of Nebraa of S for four other odel coonly ued in hydrology were alo coputed. Thee are: Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )

4 THE GENERALIZED PARETO SUM 291 Table III. p-value of the tet of aociation between drought non-drought Cliate diviion p-value 1 Ð447 2 Ð623 3 Ð2 5 Ð597 6 Ð64 7 Ð321 8 Ð393 9 Ð2 1. Gaa ditribution odel with the pdf of X Y taen to be: f X x D xc 1 exp x/ c 1 c f Y y D yd 1 exp y/ d, 11 d repectively, for x>, y>, >, c>, > d>. 2. Weibull ditribution odel with the pdf of X Y taen to be: f X x D cxc 1 expf x/ c g c 12 f Y y D dyd 1 expf y/ d g d, 13 repectively, for x>, y>, >, c>, > d>. 3. Log noral ditribution odel with the pdf of X Y taentobe: f X x D expf log x 2 / 2 2 g p 2x 14 f Y y D expf log y 2 / 2 2 g p 2y, 15 repectively, for x>, y>, >>. 4. Gubel ditribution odel with the pdf of X Y taen to be: f X x D 1 exp x ) { exp exp x )} 16 f Y y D 1 exp x ) { exp exp x )}, 17 repectively, for 1 <x<1, 1 <y<1, > >. Note that odel with hape cale paraeter have been choen. The lat two odel however only have the cale paraeter: the log noral the Gubel ditribution do not have hape paraeter. The fitting of each of the five odel GP, gaa, Weibull, log noral, Gubel) wa perfored by the ethod of axiu lielihood. The axiu lielihood etiate of, c, d their tard error deterined by inverting the oberved inforation atrix) are given in Table IV VIII for the eight cliate diviion. The lat colun of Table IV VIII Table IV. Paraeter etiate tard error for the GP odel given by Equation 1) 2) Cliate diviion O tard error) Oc tard error) O tard error) Od tard error) NLLH 1 4Ð64 Ð73) Ð327 Ð139) 3Ð84 Ð761) Ð739 Ð186) 487Ð Ð265 1Ð68) Ð394 Ð169) 5Ð72 1Ð294) Ð566 Ð23) 426Ð51 3 3Ð958 Ð669) Ð384 Ð138) 4Ð546 Ð793) Ð481 Ð148) 512Ð Ð349 Ð684) Ð283 Ð116) 5Ð313 1Ð39) Ð513 Ð171) 486Ð Ð391 Ð663) Ð284 Ð112) 5Ð423 1Ð15) Ð386 Ð16) 525Ð Ð917 Ð652) Ð346 Ð13) 5Ð565 1Ð88) Ð495 Ð17) 485Ð Ð283 Ð694) Ð297 Ð119) 8Ð12 1Ð674) Ð273 Ð175) 464Ð Ð269 Ð937) Ð32 Ð139) 6Ð353 1Ð46) Ð434 Ð195) 462Ð213 Table V. Paraeter etiate tard error for the gaa odel given by Equation 1) 11) Cliate diviion O tard error) Oc tard error) O tard error) Od tard error) NLLH 1 6Ð318 1Ð112) Ð94 Ð127) 15Ð5 2Ð934) Ð632 Ð82) 52Ð Ð4 2Ð7) Ð84 Ð126) 16Ð139 3Ð346) Ð72 Ð13) 435Ð34 3 7Ð344 1Ð272) Ð863 Ð112) 1Ð778 1Ð92) Ð777 Ð1) 528Ð74 5 6Ð387 1Ð131) Ð978 Ð134) 13Ð293 2Ð464) Ð743 Ð99) 499Ð37 6 6Ð4 1Ð81) Ð981 Ð129) 1Ð21 1Ð735) Ð838 Ð18) 535Ð Ð68 1Ð182) Ð915 Ð125) 13Ð334 2Ð464) Ð755 Ð11) 498Ð Ð149 1Ð33) Ð97 Ð129) 12Ð198 2Ð275) Ð892 Ð126) 474Ð Ð17 1Ð522) Ð93 Ð134) 13Ð93 2Ð527) Ð788 Ð112) 468Ð58 Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )

5 292 S. NADARAJAH AND S. KOTZ Table VI. Paraeter etiate tard error for the Weibull odel given by Equation 12) 13) Cliate diviion O tard error) Oc tard error) O tard error) Od tard error) NLLH 1 5Ð567Ð721) Ð896 Ð7) 7Ð496 1Ð221) Ð713 Ð57) 497Ð Ð559 1Ð162) Ð845 Ð75) 9Ð43 1Ð596) Ð765 Ð69) 432Ð Ð699 Ð757) Ð849 Ð64) 7Ð142 1Ð7) Ð8 Ð6) 523Ð Ð818 Ð764) Ð895 Ð67) 8Ð45 1Ð251) Ð789 Ð64) 495Ð Ð874 Ð733) Ð899 Ð65) 7Ð638 1Ð3) Ð852 Ð67) 533Ð44 7 5Ð522 Ð747) Ð869 Ð67) 8Ð654 1Ð275) Ð797 Ð64) 494Ð Ð798 Ð828) Ð854 Ð65) 1Ð238 1Ð395) Ð893 Ð77) 472Ð Ð37 Ð976) Ð891 Ð74) 9Ð22 1Ð373) Ð827 Ð73) 466Ð538 provide the negative logarith of the axiized lielihood NLLH). Although the five odel are not neted, the tard lielihood ratio tet LRT) can be ued to copare thoe with the ae nuber of paraeter. Aong the odel with four paraeter, it i clear that the GP provide the bet fit. The odel with two paraeter, log noral Gubel, have uch larger NLLH value that even the criteria uch a AIC or BIC will not find their fit ignificant. Thu, one can conclude baed on the NLLH value that the GP provide the ot reaonable fit aong the five odel fitted. Table VII. Paraeter etiate tard error for the log noral odel given by Equation 14) 15) Cliate diviion O tard error) O tard error) NLLH 1 1Ð569 Ð121) 1Ð867 Ð144) 536Ð Ð832 Ð158) 2Ð43 Ð177) 479Ð16 3 1Ð597 Ð12) 1Ð779 Ð133) 57Ð32 5 1Ð587 Ð124) 1Ð938 Ð151) 548Ð Ð596 Ð119) 1Ð851 Ð138) 593Ð Ð552 Ð121) 1Ð958 Ð153) 543Ð Ð59 Ð129) 2Ð9 Ð17) 531Ð Ð764 Ð145) 2Ð17 Ð166) 523Ð881 Cliate Diviion 1 Cliate Diviion Cliate Diviion 3 Cliate Diviion Cliate Diviion 6 Cliate Diviion Cliate Diviion 8 Cliate Diviion Figure 2. Probability plot for the for the fit of Equation 3) 5) for data on inter-arrival tie of drought fro the eight cliate diviion of Nebraa Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )

6 THE GENERALIZED PARETO SUM Cliate Diviion 1 Cliate Diviion Inter Arrival Tie Inter Arrival Tie Cliate Diviion 3 Cliate Diviion Inter Arrival Tie Inter Arrival Tie Cliate Diviion 6 Cliate Diviion Inter Arrival Tie Inter Arrival Tie Cliate Diviion 8 Cliate Diviion Inter Arrival Tie Inter Arrival Tie Figure 3. Fitted value of Equation 3) 5) a well a the oberved hitogra for data on inter-arrival tie of drought fro the eight cliate diviion of Nebraa Table VIII. Paraeter etiate tard error for the Gubel odel given by Equation 16) 17) Cliate diviion O tard error) O tard error) NLLH 1 62Ð65 6Ð983) 712Ð885 77Ð867) 167Ð Ð1 57Ð87) 673Ð636 82Ð413) 983Ð Ð2 8Ð378) 628Ð479 66Ð687) 1138Ð Ð632 7Ð715) 75Ð358 77Ð972) 148Ð Ð278 8Ð356) 639Ð196 67Ð429) 1153Ð Ð711 7Ð172) 721Ð68 79Ð775) 144Ð Ð937 21Ð256) 732Ð555 84Ð13) 149Ð Ð84 53Ð435) 668Ð435 77Ð781) 182Ð981 To ae the goodne-of-fit of the GP odel two ethod were purued. Firtly, the probability plot of the fit for the eight cliate diviion were drawn, ee Figure 2. A probability plot i where the oberved probability i plotted againt the probability predicted by the fitted odel. Thu, F S i veru i Ð375 / n C Ð25 wa plotted, a recoended by Blo 1958) Chaber et al. 1983), where F S Ð i the cuulative ditribution function cdf) correponding to Equation 3) 5) i are the orted value, in the acending order, of the oberved inter-arrival tie. Alo the fitted denitie f S of S were copared with the epirical verion, ee Figure 3. Both Figure 2 3 ugget that the fit of the GP odel i reaonable. Finally, it wa exained to ee whether the bet fitting GP odel with four paraeter for each of the eight cliate diviion) can be reduced to a ipler odel. Let i, c i, i d i denote the value of, c, d, repectively, for the ith cliate diviion. The following odel were fitted: Model 1: i D i c i D d i two paraeter per cliate diviion). Model 2: i D i three paraeter per cliate diviion). Model 3: c i D d i three paraeter per cliate diviion). Model 4: i D i D c 1 c i D d i D c 2 for all i two paraeter in total). The NLLH value for the firt three odel are hown in Table IX. It follow by the LRT that odel 1 provide a good a fit a the full odel with four paraeter per cliate diviion) for at leat ix of the eight cliate diviion. The fit of odel 4 reulted in NLLH D 3871Ð with Oc 1 D 4Ð795 Oc 2 D Ð438. Coparion of thi NLLH value with the u of the econd colun of Table IX ugget no evidence againt hoogeneity of the inter-arrival tie ditribution acro the eight cliate diviion. Thi i conitent with the auption ade Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )

7 294 S. NADARAJAH AND S. KOTZ Table IX. NLLH value for Model 1 3 Cliate diviion Model 1 Model 2 Model Ð53 487Ð Ð Ð59 426Ð Ð Ð85 512Ð36 512Ð Ð81 487Ð44 487Ð Ð89 526Ð34 525Ð Ð Ð Ð Ð Ð72 464Ð Ð Ð43 462Ð369 Lea2Equation2Ð2Ð6Ð15), Prudniov et al., 1986, volue 1) For >ˇ>, a x 1 a x ˇ 1 x C z dx D a Cˇ 1 z B, ˇ 2 F 1, ; C ˇ; a ). z REFERENCES Inter Arrival Tie Figure 4. Fitted value of Equation 3) 5) a well a the oberved hitogra for the pooled data on inter-arrival tie of drought fro all the eight cliate diviion of Nebraa in regional frequency analyi that the ditribution of flood i baically the ae in a region. Hence, one can conclude that the inter-arrival ditribution of drought event for the State of Nebraa i ditributed according to Equation 3) 5) with D D 4Ð795 c D d D Ð438. The fitted denity of thi ditribution i hown in Figure 4. ACKNOWLEDGEMENTS The author would lie to than the two referee the Editor-in-Chief for carefully reading of the paper for their great help in iproving the paper. APPENDIX The proof of Theore 1 require the following technical lea. Lea1Equation2Ð2Ð8Ð5), Prudniov et al., 1986, volue 1) For a>, >ˇ>, a a 1 a x ˇ 1 ux vx dx D a Cˇ 1 B, ˇ F 1,,, C ˇ; ua, va. Blo G Statitical Etiate Tranfored Beta-variable. John Wiley Son: New Yor. Catellarin A, Vogel RM, Brath A. 24. A tochatic index flow odel of flow curve. Water Reource Reearch 4: Art No. W314. Chaber J, Clevel W, Kleiner B, Tuey P Graphical Method for Data Analyi. Chapan Hall: Boca Raton, FL. Daragahi-Noubary GR On tail etiation: an iproved ethod. Matheatical Geology 21: Gradhteyn IS, Ryzhi IM. 2. Table of Integral, Serie, Product ixth edition). Acadeic Pre: San Diego, CA. Kotz S, Nadarajah S. 2. Extree Value Ditribution: Theory Application. Iperial College Pre: London. Kuar R, Chatterjee C, Kuar S, et al. 23. Developent of regional flood frequency relationhip uing L-oent for Middle Ganga Plain Subzone 1f) of India. Water Reource Manageent 17: Lab R Calibration of a conceptual rainfall-runoff odel for flood frequency etiation by continuou iulation. Water Reource Reearch 35: Lab R, Kay AL. 24. Confidence interval for a patially generalized, continuou iulation flood frequency odel for Great Britain. Water Reource Reearch 4: Art No. W751. Loaiciga HA, Leipni RB Analyi of extree hydrologic event with Gubel ditribution: arginal additive cae. Stochatic Environental Reearch Ri Aeent 13: Maden H, Robjerg D. 1997a. Generalized leat quare epirical Baye etiation in regional partial erie index-flood odeling. Water Reource Reearch 33: Maden H, Robjerg D. 1997b. The partial erie ethod in regional index-flood odeling. Water Reource Reearch 33: Maden H, Pearon CP, Robjerg D. 1997a. Coparion of annual axiu erie partial erie ethod for odeling extree hydrologic event. 2. Regional odeling. Water Reource Reearch 33: Maden H, Rauen PF, Robjerg D. 1997b. Coparion of annual axiu erie partial erie ethod for odeling extree hydrologic event. 1. At-ite odeling. Water Reource Reearch 33: Martin ES, Stedinger JR. 21a. Generalized axiu lielihood Pareto-Poion etiator for partial erie. Water Reource Reearch 37: Martin ES, Stedinger JR. 21b. Hitorical inforation in a generalized axiu lielihood fraewor with partial annual axiu erie. Water Reource Reearch 37: Martin ES, Stedinger JR. 22. Cro correlation aong etiator of hape. Water Reource Reearch 38: Art. No Prudniov AP, Brychov YA, Marichev OI Integral Serie volue 1, 2 3). Gordon Breach Science Publiher: Aterda. Rauen PF. 21. Generalized probability weighted oent: application to the generalized Pareto ditribution. Water Reource Reearch 37: Yevjevich V An Objective Approach to Definition Invetigation of Continental Hydrologic Drought, Hydrologic Paper No. 23. Colorado State Univerity: Fort Collin, CO. Copyright 27 John Wiley & Son, Ltd. Hydrol. Proce. 22, )