# Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask

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3 Table 2. Raks for the Learig Data Set of Table i R i S i oe has H * z,t =PZ z,t t =PX z,y t =H z, t = CF z,g t =CF * z,g * t 4 for all choices of z,tr. It follows at oce from the compariso of Eqs. 3 ad 4 that C * =C. Expressed i differet terms, the above developmet meas that the uique copula associated with a radom pair X,Y is ivariat by mootoe icreasig trasformatios of the margials. Sice the depedece betwee X ad Y is characterized by this copula, a faithful graphical represetatio of depedece should exhibit the same ivariace property. Amog fuctios of the data that meet this requiremet, it ca be see easily that the pairs of raks R,S,...,R,S associated with the sample are the statistics that retai the greatest amout of iformatio; see, e.g., Oakes 982. Here, R i stads for the rak of X i amog X,...,X, ad S i stads for the rak of Y i amog Y,...,Y. These raks are uambiguously defied, because ties occur with probability zero uder the assumptio of cotiuity for X ad Y. Pairs of raks correspodig to the learig data set are give i Table 2. Displayed i Fig. 2a is the scatter plot of the pairs R i,s i correspodig to these X i,y i. Fig. 2b shows the graph of the pairs R * i,s * i associated with the Z i,t i. The result is obviously the same. It is the most judicious represetatio of the copula C that oe could hope for. Upo rescalig of the axes by a factor of /+, oe gets a set of poits i the uit square 0, 2, which form the domai of the so-called empirical copula Deheuvels 979, formally defied by C u,v = v R i + u, S i + with A deotig the idicator fuctio of set A. For ay give pair u,v, it may be show that C u,v is a rak-based estimator of the ukow quatity Cu,v whose large-sample distributio is cetered at Cu,v ad ormal. Measurig Depedece It was argued above that the empirical copula C is the best sample-based represetatio of the copula C, which is itself a characterizatio of the depedece i a pair X,Y. It would make sese, therefore, to measure depedece, both empirically ad theoretically, usig C ad C, respectively. It will ow be explaied how this leads to two well-kow oparametric measures of depedece, amely Spearma s rho ad Kedall s tau. Fig. 2. Displayed i a is a scatter plot of the pairs R i,s i of raks derived from the learig data set X i,y i,i6. As for b, it shows a scatter plot of the pairs R i *,S i * of raks derived from the trasformed data Z i,t i =expx i,exp3y i, i6. For obvious reasos, the two graphs are actually idetical. Spearma s Rho Mimickig the familiar approach of Pearso to the measuremet of depedece, a atural idea is to compute the correlatio betwee the pairs R i,s i of raks, or equivaletly betwee the poits R i /+,S i /+ formig the support of C. This leads directly to Spearma s rho, viz. where = R i R S i S, R i R 2 S i S 2 R = R i = + = S i = S 2 This coefficiet, which may be expressed more coveietly i the form JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007 / 349

4 2 = + R i S i 3 + shares with Pearso s classical correlatio coefficiet, r, the property that its expectatio vaishes whe the variables are idepedet. However, is theoretically far superior to r, i that. E = ± occurs if ad oly if X ad Y are fuctioally depedet, i.e., wheever their uderlyig copula is oe of the two Fréchet Hoeffdig bouds, M or W; 2. I cotrast, Er = ± if ad oly if X ad Y are liear fuctios of oe aother, which is much more restrictive; ad 3. estimates a populatio parameter that is always well defied, whereas there are heavy-tailed distributios such as the Cauchy, for example for which a theoretical value of Pearso s correlatio does ot exist. For additioal discussio o these poits, see, e.g., Embrechts et al As it turs out, is a asymptotically ubiased estimator of =20, 2 uvdcu,v 3=20, 2 Cu,vdvdu 3 where the secod equality is a idetity origially prove by Hoeffdig 940 ad exteded by Quesada-Molia 992. To show this, oe may use the fact that uvdc u,v 3= 2 20, 2 R i S i + + 3= + ad that C C as. For more precise coditios uder which this result holds, see, e.g., Hoeffdig 948. Note i passig that uder the ull hypothesis H 0 :C= of idepedece betwee X ad Y, the distributio of is close to ormal with zero mea ad variace /, so that oe may reject H 0 at approximate level =5%, for istace, if z /2 =.96. Example For the observatios from the learig data set, a simple calculatio yields =/35=0.028, while r = Here, there is o reaso to reject the ull hypothesis of idepedece. For, if Z is a stadard ormal radom variable, the P-value of the test based o is 2PrZ 5/35=94.9%. Give a family C of copulas idexed by a real parameter, the theoretical value of is, typically, a mootoe icreasig fuctio of. A sufficiet coditio for this is that the copulas be ordered by positive quadrat depedece PQD, which meas that the implicatio C u,vc u,v is true for all u,v0,. The origial defiitio of PQD as a cocept of depedece goes back to Lehma 966; the same orderig, rediscovered by Dhaee ad Goovaerts 996 i a actuarial cotext, is ofte referred to as the correlatio or cocordace orderig i that field. I the Farlie Gumbel Morgester model, for example, oe has Fig. 3. Spearma s rho a ad Kedall s tau b as a fuctio of Pearso s correlatio i the bivariate ormal model 2 c u,v = uv C u,v =+ 2u 2v sice C is absolutely cotiuous i this case. A simple calculatio the yields 0 0 uvc u,vdvdu = ad, hece, =/3, as iitially show by Schucay et al As a secod example, if X,Y follows a bivariate ormal distributio with correlatio r, a somewhat itricate calculatio to be foud, e.g., i Kruskal 958, shows that arcsi FxGydHx,y 3= 6 r =2 2 where 0, 2 uvdc u,v =0 0 uvc u,vdvdu For those people accustomed to thikig i terms of r, the above formula may suggest that a serious effort would be required to thik of correlatio i terms of Spearma s rho i the traditioal bivariate ormal model. As show i Fig. 3a, however, the differece betwee ad r is miimal i this cotext. 350 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007

7 Table 4. Coordiates of Poits Displayed o the K-Plot Associated with the Learig Data Set of Table i W i: H i thus rely oly o the raks of the observatios, which are the best summary of the joit behavior of the radom pairs. Fig. 6. K-plot for the learig data set. Superimposed o the graph are a straight lie correspodig to the case of idepedece ad a smooth curve K 0 w associated with perfect positive depedece. Estimate Based o Kedall s Tau To fix ideas, suppose that the uderlyig depedece structure of a radom pair X,Y is appropriately modeled by the Farlie Gumbel Morgester family C defied i Eq. 2. I this case, is real ad as see i the Kedall s Tau subsectio there exists a immediate relatio i this model betwee the parameter ad the populatio value of Kedall s tau, amely = 2 9 K 0 w =PUV w PU w =0 vdv w w dv =0 +w v dv = w w logw ad k 0 =correspodig desity. The values of W :6,...,W 6:6 required to produce Fig. 6 ca be readily computed usig ay symbolic calculator, such as Maple. They are give i Table 4. The iterpretatio of K-plots is similar to that of QQ-plots: just as curvature is problematic, e.g., i a ormal QQ-plot, ay deviatio from the mai diagoal is a sig of depedece i K-plots. Positive or egative depedece may be suspected i the data, depedig whether the curve is located above or below the lie y=x. Roughly speakig, the further the distace, the greater the depedece. I this costructio, perfect egative depedece i.e., C=W would traslate ito a strig of data poits aliged o the x-axis. As for perfect positive depedece i.e., C=M, it would materialize ito data aliged o the curve K 0 w show o the graph. As for the chi-plot, the liearity or lack thereof i the K-plot displayed i Fig. 6 is hard to detect, give the extremely small size of the learig data set. However, see the Applicatio sectio ad Geest ad Boies 2003 for more compellig illustrartios of K-plots. Estimatio Now suppose that a parametric family C of copulas is beig cosidered as a model for the depedece betwee two radom variables X ad Y. Give a radom sample X,Y,...,X,Y from X,Y, how should be estimated? This sectio reviews differet oparametric strategies for tacklig this problem, depedig o whether is real or multidimesioal. Oly rak-based estimators are cosidered i the sequel. This methodological choice is justified by the fact, highlighted earlier, that the depedece structure captured by a copula has othig to do with the idividual behavior of the variables. A fortiori, ay iferece about the parameter idexig a family of copulas should Give a sample value of computed from Eq. 5 or 6, a simple ad ituitive approach to estimatig would the cosist of takig = 9 2 Sice is rak-based, this estimatio strategy may be costrued as a oparametric adaptatio of the celebrated method of momets. More geerally, if =g for some smooth fuctio g, the =g may be referred to as the Kedall-based estimator of. A small adaptatio of Propositio 3. of Geest ad Rivest 993 implies that where ad 4S N0, S 2 = W i + W i 2W 2 W i = I ji = j= # j:x i X j,y i Y j Therefore, a applicatio of Slutsky s theorem, also kow as the Delta method, implies that as 2 N, 4Sg Accordigly, a approximate 00 % cofidece iterval for is give by ± z /2 4Sg For a alterative cosistet estimator of the asymptotic variace of, see for istace, Samara ad Radles 988. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007 / 353

8 Table 5. Itermediate Values Required for the Computatio of the Stadard Error Associated with Kedall s Tau i W i W i Example (Cotiued) For the learig data set of Table, it was see earlier that =/5, hece =0.3. Usig the itermediate quatities summarized i Table 5, oe fids S 2 =0.043, hece a approximate 95% cofidece iterval for this estimatio is,, sice g9/2, ad hece,.964sg / =2.99. While the size of the stadard error may appear exceedigly coservative, this result is ot surprisig, cosiderig that the sample size is =6. The popularity of as a estimator of the depedece parameter stems i part from the fact that closed-form expressios for the populatio value of Kedall s tau are available for may commo parametric copula models. Such is the case, i particular, for several Archimedea families of copulas, e.g., those of Ali et al. 978, Clayto 978, Frak 979, Gumbel Hougaard Gumbel 960, etc. Specifically, a copula C is said to be Archimedea if there exists a covex, decreasig fuctio :0, 0, such that =0 ad Cu,v = u + v is valid for all u,v0,. As show by Geest ad MacKay 986 t =+40 t dt 7 Table 6 gives the geerator ad a expressio for for the three most commo Archimedea models. Algebraically closed formulas are available for various other depedece models, e.g., extreme-value or Archimax copulas. See, for example, Ghoudi et al. 998 or Capéraà et al Estimate Based o Spearma s Rho Whe the depedece parameter is real, a alterative rakbased estimator that remais i the spirit of the method of momets cosists of takig = h where =h represets the relatioship betwee the parameter ad the populatio value of Spearma s rho. I the cotext of the Farlie Gumbel Morgester family of copulas, for example, it was see earlier that =/3, so that =3 would be a alterative oparametric estimator to =9 /2. Now it follows from stadard covergece results about empirical processes to be foud, e.g., i Chapter 5 of Gaessler ad Stute 987, that N, 2 where the asymptotic variace 2 depeds o the uderlyig copula C i a way that has bee described i detail by Borkowf Arguig alog the same lies as i the Estimate Based o Table 6. Three Commo Families of Archimedea Copulas, Their Geerator, Their Parameter Space, ad a Expressio for the Populatio Value of Kedall s Tau Family Geerator Parameter Kedall s tau Clayto t / /+2 Frak Kedall s Tau subsectio, it ca the be see that uder suitable regularity coditios o h N, h 2 where 2 =suitable estimator of 2. A approximate 00 % cofidece iterval for is the give by ± z /2 h Substitutig C for C i the expressios reported by Borkowf 2002, a very atural, cosistet estimate for 2 is give by where ad 2 = 44 9A 2 + B +2C +2D +2E C = 3 j= k= R i S i A = + + B = D = 2 j= E = 2 j= log e t e R 4/+4D / Gumbel Hougaard logt / Note: Here, D = 0 x//e x dx is the first Debye fuctio. R i R i + S i 2 S i R k R i,s k S j + 4 A S i + max S R j i + +, R j + max R i R S j i + + +, S j + Example (Cotiued) For the learig data set of Table, it was see earlier that =/35, hece =3/ Burdesome but simple calculatios yield =7.77, hece a approximate 95% cofidece iterval for this estimatio is,, sice h3, ad hece,.96 h / =8.66. Here agai, the size of the stadard error is quite large, as might be expected give that =6. Maximum Pseudolikelihood Estimator I classical statistics, maximum likelihood estimatio is a wellkow alterative to the method of momets that is usually more / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007

11 Goodess-of-Fit Tests I typical modelig exercises, the user has a choice betwee several differet depedece structures for the data at had. To keep thigs simple, suppose that two copulas C ad D were fitted by some arbitrary method. It is the atural to ask which of the two models provides the best fit to the observatios. Both iformal ad formal ways of tacklig this questio will be discussed i tur. Graphical Diagostics Whe dealig with bivariate data, possibly the most atural way of checkig the adequacy of a copula model would be to compare a scatter plot of the pairs R i /+,S i /+ i.e., the support of the empirical copula C with a artificial data set of the same size geerated from C. To avoid arbitrariess iduced by samplig variability, however, a better strategy cosists of geeratig a large sample from C, which effectively amouts to portrayig the associated copula desity i two dimesios. Simple simulatio algorithms are available for most copula models; see, e.g., Devroye 986, Chap., or Whela 2004 for Archimedea copulas. I the bivariate case, a good strategy for geeratig a pair U,V from a copula C proceeds as follows: Step : Geerate U from a uiform distributio o the iterval 0,. Step 2: Give U = u, geerate V from the coditioal distributio Q u v =PV vu = u = u Cu,v by settig V=Q u U *, where U * =aother observatio from the uiform distributio o the iterval 0,. Whe a explicit formula does ot exist for Q u, the value v=q u u * ca be determied by trial ad error or more effectively usig the bisectio method; see Devroye 986, Chap. 2. Thus, for the Farlie Gumbel Morgester family of copulas, oe fids Q u v = v + v v 2u for all u,v0,, ad hece = u* if b = 2u =0 Q u u * b + b bu * if b = 2u 2b Fig. 8a displays 00 pairs U i,v i geerated with this algorithm, takig =ˆ = as deduced from the method of maximum pseudolikelihood. The six poits of the learig data set, represeted by crosses, are superimposed. Give the small size of the data set, it is hard to tell from this graph whether the selected model accurately reproduces the depedece structure revealed by the six observatios. To show the effectiveess of the procedure, the same exercise was repeated i Fig. 8b, usig a Clayto copula with =0. Here, the iappropriateess of the model is apparet, as might have bee expected from the fact that =5/6 for this copula, while =/5. Aother optio, which is related to K-plots, cosists of comparig the empirical distributio K of the variables W,...,W itroduced previously with K, i.e., the theoretical distributio of W=C U,V, where the pair U,V is draw from C. Oe possibility is to plot K ad K o the same graph to see Fig. 8. a Scatter plot of 00 pairs U i,v i simulated from the Farlie Gumbel Morgester with parameter = b Similar plot, geerated from the Clayto copula with =5/6. O both graphs, the six poits of the learig data set are idicated with a cross. how well they agree. Alteratively, a QQ-plot ca be derived from the order statistics W W by plottig the pairs W i:,w i for i,...,. I this case, however, W i: is the expected value of the ith order statistic from a radom sample of size from K, rather tha from K 0, as was the case i the K-plot. I other words W i: = wk wk w i K w i dw 9 i 0 where K w=pc U,Vw ad k =dk w/dw. These two graphs are preseted i Fig. 9 for the learig data set ad Clayto s copula with parameter =ˆ =0.449, obtaied by the method of maximum pseudolikelihood. As implied by the data i Table 5, K is a scale fuctio with steps of JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007 / 357

16 Fig. 3. Simulated radom sample of size 0,000 from six chose families C of copulas with parameter estimated by the method of maximum pseudolikelihood usig the peak volume Harricaa River data, whose pairs of raks are idicated by a X special case of the BB system whe 0, while settig 2 = actually yields the Kimeldorf Sampso family. Likewise, the Galambos ad Gumbel Hougaard distributios are special cases of Family BB5 correspodig, respectively, to = ad 2 0. Estimatio Table gives parameter values for each of the five models, based o maximum pseudolikelihood. For oe-parameter models, 95% cofidece itervals were computed as explaied above. For Fig. 4. Same data as i Fig. 3, upo trasformatio of the margial distributios as per the selected models for the peak ad the volume of the Harricaa River data, whose pairs of observatios are idicated by a X 362 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007

17 Table 0. Defiitio of the Five Chose Families of Copulas with Their Parameter Space Copula C u,v Parameters Gumbel Hougaard exp ũ +ṽ / Galambos uv expũ +ṽ / 0 Hüsler Reiss exp ũ + 2 log ũ ṽ ṽ + 2 log ṽ ũ 0 BB +u 2+v 2 / 2 / 0, 2 BB5 exp ũ +ṽ ũ 2 +ṽ 2 / 2 /, 2 0 Note: With ũ= logu, ṽ= logv ad stadig for the cumulative distributio fuctio of the stadard ormal. q-parameter models with q2, the determiatio of the cofidece regios relies o a estimatio of the limitig variace covariace matrix B B of the estimator ˆ =ˆ,...,ˆ q of =,..., q. Followig Geest et al. 995, the estimate of B is simply the empirical qq variace covariace matrix of the variables N,...,N q, for which a set of pseudo-observatios is available, amely i S i N pi = L pˆ, i,..., + +, where L p deotes the derivative of L,u,v=logc u,v with respect to p. Here, it is assumed that the origial data have bee relabeled so that X X. Likewise, =qq variace covariace matrix of the variables M,...,M q, for which the pseudo-observatios are M pi = N pi j L pˆ, j=i +, S j j uˆ, +L +, S j + j L pˆ, S j S i +, S j j vˆ, +L +, S j + for i,...,. A alterative, possibly more efficiet way of estimatig the iformatio matrix B is give by the Hessia matrix associated with L,u,v at ˆ, amely, the qq matrix whose p,r etry is give by i L p rˆ, +, S i + where L p r stads for the cross derivative of L,u,v with respect to both p ad r. I Table, the cofidece itervals for Models BB ad BB5 were derived usig the latter approach, as it produced somewhat arrower itervals. Goodess-of-Fit Testig As a secod step towards model selectio, oe should look at the geeralized K-plot correspodig to the five families uder cosideratio. The graphs correspodig to the BB are displayed i Fig. 5. For reasos give i the Graphical Diagostics subsectio, the graphs correspodig to the BB5, Gumbel Hougaard, Galambos, ad Hüsler Reiss copulas are idetical, sice they are all extreme-value depedece structures. The graphs appear i Fig. 6. The plots displayed i Figs. 5 ad 6 suggest that both the BB ad extreme-value copula structures are adequate for the data at had. A similar coclusio is draw from the formal goodess-of-fit tests based o S ad T, as idicated i Table 2. Agai, the geeralized K-plot ad the formal goodess-of-fit tests correspodig to the Galambos, Hüsler Reiss, ad BB5 extremevalue copula models yield exactly the same results, as evideced i Table 2. I a attempt to distiguish betwee the extreme-value copula structures, a cosistet goodess-of-fit test could be costructed from the process C C, as evoked but dismissed by Fermaia 2005, o accout of the uwieldy ature of its limit. However, this difficulty ca be overcome easily with the use of a parametric or double parametric bootstrap, whose validity i this cotext has recetly bee established by Geest ad Rémillard The bootstrap procedure is exactly the same as described i the Formal Tests of Goodess-of-Fit sectio, but with S replaced by the Cramér vo Mises statistic CM = C R i +, S i = W i C R i + C R i +, S i +2 +, S i +2 This bootstrap-based goodess-of-fit test was applied for each Table. Maximum Pseudolikelihood Parameter Estimates ad Correspodig 95% Cofidece Iterval for Five Families of Copulas, Based o the Harricaa River Data 95% cofidece Copula Estimates iterval CI Gumbel Hougaard ˆ =2.6 CI=.867,2.455 Galambos ˆ =.464 CI=.62,.766 Hüsler Reiss ˆ =2.027 CI=.778,2.275 BB ˆ =0.48, ˆ 2=.835 CI=0.022, ,2.25 BB5 ˆ =.034, ˆ 2=.244 CI=.000, ,.294 JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007 / 363

18 Fig. 5. a Graphs of K ad K for the BB copula with ˆ =0.48, ˆ =.835, based o the Harricaa River data. b Geeralized K-plot providig a visual check of the goodess-of-fit of the same model for these data. Fig. 6. a Graphs of K ad K for the Gumbel Hougaard copula with =ˆ =2.6, based o the Harricaa River data. b Geeralized K-plot providig a visual check of the goodess-of-fit of the same model for these data. of the five families of copulas still uder cosideratio. The results are summarized i Table 3. As it tured out, oe of the models could be rejected o this basis either. The Gumbel Hougaard ad Galambos copulas beig embedded i two-parameter models BB ad BB5, yet aother optio for choosig betwee them would be to call o a pseudolikelihood ratio test procedure recetly itroduced by Che ad Fa Their approach, ispired by a semiparametric adaptatio of the Akaike Iformatio Criterio, makes it possible to measure the trade-off betwee goodess-of-fit ad model parsimoy. Suppose it is desired to compare two ested copula models, say C=C, ad D=C,0. Let ˆ,ˆ represet the maximum pseudolikelihood estimator of,r 2 uder model C, ad write for the maximum pseudolikelihood estimator of R uder the submodel D. The test statistic proposed by Che ad Fa 2005 the rejects the ull hypothesis H 0 := 0 that model D is preferable to model C wheever CF =2 logc, 0 c ˆ,ˆ R i R i +, S i + +, S i + is sufficietly small. To determie a P-value for this test, oe must resort to a oparametric bootstrap procedure, which proceeds as follows. For some large iteger N ad each k,...,n, do the followig: Step : Draw a bootstrap radom sample X,Y,...,X,Y with replacemet from X,Y,...,X,Y. Step 2: Use the method of maximum pseudolikelihood to determie estimators ˆ,ˆ ad of, ad, 0 uder models C ad D, respectively. Step 3: Compute the Hessia matrices B ad B 2 associated with logc, u,v ad logc,0 u,v at ˆ,ˆ ad, respectively. Step 4: Determie the value of 364 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007

19 Table 2. Results of the Bootstrap Based o the Cramér vo Mises Statistic S ad Kolmogorov Smirov Statistic T : Observed Statistic, Critical Value q Correspodig to =5% ad Approximate P-Value, Based o N=0,000 Parametric Bootstrap Samples Copula S q95% P-value T q95% P-value Gumbel Hougaard Galambos Hüsler Reiss BB BB CF,k = B 2 ˆ ˆ,ˆ ˆ B ˆ ˆ,ˆ ˆ T The P-value associated with the test of Che ad Fa 2005 is the give by N CF,k CF N k= The coclusios draw from this aalysis ot reported here are cosistet with Table, which idicate that the iterval estimates for the BB5 family are compatible with the Galambos model sice = is a possible value but ot with the Gumbel Hougaard because 2 =0 is excluded from its 95% cofidece iterval. Likewise, the parameter itervals for Family BB suggest that either the Gumbel Hougaard or the Kimeldorf Sampso families are adequate for the data at had. Additioal tools that may help to distiguish betwee bivariate extremevalue models will be preseted i the ext sectio. Graphical Diagostics for Bivariate Extreme-Value Copulas I the bivariate case, extreme-value copulas are characterized by the depedece fuctio A, asieq.. Whe the margial distributios F ad G of H are kow, a cosistet estimator A of A has bee proposed by Capéraà et al It is give by where t H z z A t = exp0 z z dz, t 0, H t = Z i t is the empirical distributio fuctio of the radom sample Z,...,Z with Z i =logfx i /logfx i GY i for i,..., Table 3. Results of the Bootstrap Based o the Cramér vo Mises Statistics CM : Observed Statistic, Critical Value q Correspodig to =5% ad Approximate P-Value, Based o N=0,000 Parametric Bootstrap Samples Copula CM q95% P-value Gumbel Hougaard Galambos Hüsler Reiss BB BB These authors showed that if Z Z are the associated ordered statistics, the A ca be writte i closed form as pt if 0 t Z A t = tq t i/ t i/ Q pt Q i if Z i t Z i+ tq pt if Z t i terms of a weight fuctio p so that p0= p= ad quatities Q i = i k= Z k / Z k /, i,..., The asymptotic behavior of the process loga loga is give by Capéraà et al. 997 uder mild regularity coditios, ad could be used to perform a goodess-of-fit test, say, usig the Cramér vo Mises statistic 0 loga t/a t 2 dt Whe the margis are ukow, however, as is most ofte the case i practice, it would seem reasoable to use a variat Â of the same estimator, with Z i replaced by the pseudo-observatio Ẑ i = log R i +/log R i + S i i +, Before a proper test ca be developed, it will be ecessary to examie the asymptotic behavior of the process logâ t loga t This may be the object of future work. For additioal discussio o this geeral theme, refer to Abdous ad Ghoudi For the time beig, a useful graphical diagostic tool for extreme-value copulas may still cosist of drawig Â ad A o the same plot. Fig. 7 shows such a plot for the four families of extreme-value copulas retaied for this study. Here, the weight fuctio used was pt= t. The reaso for which o goodessof-fit test could distiguish betwee these models is obvious from the graph: the geerators of the four families are ot oly fairly close to A, they are practically idetical. Coclusio Usig both a learig data set ad 85 aual records of volume ad peak from the Harricaa watershed, this paper has illustrated the various issues ivolved i characterizig, measurig, ad modelig depedece through copulas. The mai emphasis was JOURNAL OF HYDROLOGIC ENGINEERING ASCE / JULY/AUGUST 2007 / 365

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