Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with peak-over-threshold data

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1 Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with peak-over-threshold data B. Renard, M. Lang, P. Bois To cite this version: B. Renard, M. Lang, P. Bois. Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with peak-over-threshold data. Stochastic Environmental Research and Risk Assessment, Springer Verlag (Germany, 2006, 2 (2, p p. 2. <0.007/s >. <hal > HAL Id: hal Submitted on Feb 200 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Author-produced version of the article published in Stochastic environmental research and risk assessment, 2006, 2 (2, The original publication is available at doi : 0.007/s

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18 !!,8" M!&0! %9!!,&2,!&,,4!70,& 5$,&6!#%0&,&,"4707 0& $42%/!!%%"%605," 0/! 7!0!7, $! &%/," /%2!"!<070!2 M%8&," %+ 42%"%64'(!!,8" M!&0! %9!!,&2,!& 9,4!70,& 5$,&6!#%0&,&,"4707 0& $42%/!!%%"%605," 0/! 7!0!7, %/#,07%& %+ 5$,&6!#%0& /%2!"7,&2 +%!5,7 0&6M%8&,"%+42%"%64'(F!!,8",&2 0& 0-8!,&2 022!&,;%< /%2!"7 +% #!,; +"%:,&,"4707!00X/!7!&! 0!&7285!&!M,5?8!7, M!&0! %9!!,&2 "0<0 ;4 5! %7#!5 0<! /8" 0<,0,!,4!70,& 5$,&6!#%0&,&,"4707' 70/8",&!%87 70&6"! 5$,&6! 0& $! /!,&%+7!<!,"$42%"%605,"7!?8!&5!7 %5$,7 05&<0%&/!&,"!7!,5$,&207; 77!77/!& '(F 05;,&27 M D., 07 05," 0&+!!&5! 870&6!-!/! %2! 7, &&,"7 %+, 'D 82$%//! DD. %2X"!7 74& $Q0?8!7 2!7 5%&&,077,&5!7!& $42%"%60! &0<!70 Q %& #!""0!@4%&(#%& #!""0!,4E,&27,4,4!70,&/! $%27+%5$,&6!#%0& 2!!5 0%&0&"%&6,&6! 2!#!&2!& #%5!77!7M%8&,"%+0/!!0!7&,"4707'F*.!!2J DDD@"%%27 0/, 0%&,&29%%;>%"8/!'<!<0!:%42%"%64*# J,""0&6+%2!07,&2 M!20&6!.,4!70,& +"%%2 +!?8!&54,&,"4707 :0 $ $07 %05,"0&+%/, 0%&M%8&,"%+42%"%64'DF 0!,&2,&&!DD@,50"0, 0&6 $!0997,/#"! $!0997 %##!,&2 $! ,/#"! M%8&," %+ $! /!05,&, 07 05," 77%50, 0%& '*F *F* %9! 42!&,&2 0!0&6%&,4!70,&0&+!!&5!0&$022!&,;%< /%2!"7 $%86$ $!!<!709"!I8/#,;%<5$,0&%&!,"%/! $%2M%8&,"%+ $! %4,", 07 05,"%50! 4!0!7, 07 05,"! $%2%"%64'.. %79I!6,&2,27!&(2<,&5!2,##%,5$!70&R/%2!""0&6%+!-!/! $42%"%605,"!<!& 7 42%"%64'50!&5!,5 05!+% $!7!& 84%&2%& %50! 4 8#5!:7;0 J,&2 Z E,5/,!; %&7, 0%&,4,##%,5$ %, 70! +"%%2 +!?8!&54 /%2!""0&6 J!06$!2 "!,7 7?8,!7!7 0/, 0%& M%8&," %+ 42%"%64 '(. 8#5!:7;0J %&7, 0%&,4,##%,5$ %, 70! +"%%2 +!?8!&54 /%2!""0&6,-0/8/ "0;!"0$%%2!7 0/, 0%& M%8&," %+ 42%"%64 '(,&&! DDF%%"7+%, 07 05,"&+!!&5!#0&6!>!",6*#!:%;

19 Annex. Griddy Gibbs sampling $!,0/%+ $07/! $%207 %70/8",!,7,/#"!+%/ $!#%7!0% %&2 ; 3$! 09977,/#"0&6,"6%0 $/5,&9!:0!&,7+%"%:' $%%7!,7,0&6<,"8! ( θ 0/!7' o?[?ka,&27!?[ o,/#"!? \ ( L?? θ θ θ θ o,/#"!? ( \ L??? θ θ θ θ θ o ] o,/#"!? \ ( L???? θ θ θ θ θ θ o ]]??? o,/#"! \ ( L θ θ θ θ +!&2 $!<!5 %77!0!7 35%&<!6!7 % $!,6! #%7!0% %&,7?!&27 %0&+0&0 4 &%2! % 2!5!,7! $! 0&+"8!&5!%+ $! 7,0&6#%0& $! +07 0!, 0%&7,! 878,"4 2!"!!2,&2 0&+!!&5! 07/,2!:0 $ 0!, 0%&7!&70 0<0 4,&,"4707/,4 9! &!5!77,4 % 2!!/0&!,55!#,9"! <,"8!7 +%,&2@ 878,"4, "!,7, +!: $%87,&27%+ 0!, 0%&7,!87!2+%,&2@%!6!&!,"4 $07/! $%2#!7!& 7 $!7,/!2,:9,5;7,7 % $! 0!, 0<! 70/8", 0%&!5$&0?8!7 %&<!6!&5! $87 $,7 % 9!/%&0 %!2 94 5$%%70&6 7!<!,"7,0&6#%0& 7%5%/#8 0&65%&<!6!&5!0&205!7!"/,&!,"DD.%,&&! DDF#%<02!7%/!6802!"0&!7+%0/#%<0&6 $!&8/!05,"70/8", 0%&7! +050!&54 &+% 8&,!"4 $! ,/#"0&6,"6%0 $/ 5,& 878,"4&% 9! 87!2 0& $07,: <!70%& 9!5,87! 0 0&<%"<!7 9!0&6,9"! % 7,/#"! +%/ $! +8" 5%&20 0%&," 2!&70 0!7 (???? L θ θ θ θ + θ 0!,&2,&&! DD#%#%7!2 $! 87!%+, 2075!!,##%-0/, 0%&%+ $!58/8", 0<!2!&70 4+8&5 0%&752+%+ $!7! %&7' $%%7!,602%+#%0& 7 / : <,"8,!2G ;G /: G / G E/: G 3%& $07602 %%9,0&H / :H %/#8! $!58/8", 0<!78/7%+H / :H %%9,0&,&,##%-0/, 0%&%+ $!@,/#"!+%/,8&0+%/ %&%&UV,&7+%/94 $!0&<!7!%+ $!,##%-0/,!@ $07,"6%0 $//87 9!,22!2,!,5$ 7!#%+ $! !, 0%& + $! +8" 5%&20 0%&," 2!&70 0!7,!;&%:&%&"48# %#%#% 0%&,"0 4 (???? θ L θ θ θ θ 9!!#",5!294 (???? θ θ θ θ + 5,& + $!#%285 %+#0%,&2"0;!"0$%%2,&2 $!58/8", 0<!78/7%+H / :H /87 $!&9!20<02!294 $! %,"78/ %#%<02! $!@,##%-0/, 0%&&<!70%&%+ $!@5,&9!/,2!:0 $,"0&!,0&!#%", 0%&9! :!!& :% 602#%0& 7$!6025$%05!07 $!/%7 0/#%,& 0778!%+ $07!5$&0?8!'0 $,7 %9!9%,2!&%86$ % 5%<! $!,&6!%+ $! %&7,&2 +0&!!&%86$ %!&78!, !&,558,54;!!#0&60&/0&2 $, $0778##"!/!&,47!#075%/#8 0&6 0/!!&70<!%/! 0/#%<!/!& 7%+ $!/! $%2,!2!7509!20&0!,&2,&&!DD

20 Annex 2. Chib method $!,0/%+ $07/! $%2 07 % 5%/#8! $!/,60&," %&%+ $!%97!<, 0%&7 :$05$07 $!&%/,"00&65%&7,& %+ $!,4!7 $!%!/' π L L< L L ^ ^ ^ $07!", 0%&7$0# 9!0&6 8! +%,&4 <!5 % "! 87 5%&702!,#, 058", θ θ !5 "45%/#8,9"!:$05$07&% $!5,7!%+ $!2!&%/0&, %%&702! $!+%"%:0&6!", 0%&7$0#' ^ ^ ^ ^ ^ ^ ^ ^ ^ θ θ L θ L θlθ θ Lθ θ θ ^ ^ ^ θ Lθ θ $! +07!/ 5,& 9!!<,"8,!2 $,&;7 % $! 7,/#"!%+ $! +07 /,60&," %& &6,,8770,&;!&!",7,&!-,/#"!$!",7!/5,&9!5%/#8!294&8/!05," 0&!6, 0%&' ^ ^ ^ ^ θ Lθ θ ^ ^ θ θ θ θ $! ,/#"0&6 5,& 9! 87!2 % 5%/#8! 0&!/!20,4!/7 ^ ^ ^ ^ θ Lθ θ θ 07 0&2!!2 $! +07 /,60&," %+ $! %& ^ ^ ^ θ θ + θ Lθ θ θ!<,"8,!2, θ $! ,"6%0 $/5,& $879!,##"0!2 % $!&%&&%/,"0!2#%7!0%2!&70 4:0 $ $!+07 5%/#%&!& 79!0&6+0-!2 $, ^ ^ ^ 07 θ θ θ θ θ + θ " $%86$ $07,##%,5$ 07 $!%! 05,"4 <,"02 +%,&4 <,"8! :0 $&%&!%#%7!0% #%9,90"0 4 $09!5%//!&27 $! 87!%+, $06$2!&70 4#%0& % 0&5!,7! $!/! $%2,558,54 ^

21 List of tables,9"!'%9,90"0 42!&70 4+8&5 0%&787!20&!-!/!<,"8!7,&,"4707,9"!'%2!"7%+5$,&6!7%& $! %&7#,,/!!7,9"!,'8,& 0"!75,"58", 0%&,&2M,5%90,&/, 0-!7+%#0%7#!50+05, 0%&7'-#%&!& 0,",!&!,"0!2,! %9,&28/9!" %&7,9"! 9' 8,& 0"!7 5,"58", 0%&,&2 M,5%90,&/, 0-!7 +%#0% 7#!50+05, 0%&7' > %&,9"!('0%7#!50+05, 0%&+% $!P/!0<!, 85!&0%07,9"!.'%7!0%#%9,90"0 0!7%+/%2!"7,&2!", 0<!7,4!7+,5 %7 List 7+% $!P/!0<!, 85!&0%07,,&2,77%50,!2#!,;79,&2 $!0&!,0<,"7#%5!77,7,+8&5 0%&%+ 0/!+%/%2!"7 <,&2 0!7%+ $!0&!,0<,"7#%5!77#,,/!!7+%/%2!"7 < 9,&2 < 5 < 8,& 0"!7 58<!7:0 $/%2!"7 <,&2 <!7# 7%"02,&2 2,7$!2"0&! F' 0% 2%!2"0&!,&2#%7!0% 9,7/,60&,"2!&70 0!7%+ $!5$,&6!#%0& 0& /%2!" ' %7!0%/,60&," 2!&70 0!7%+ $! 75,"!#,,/!!7 0&/%2!"7 <,&2 8,& 0"!7/!,& 58<!7 7%"02"0&!7,&2 DT 5%&+02!&5! 0&!<,"7 2%!2"0&!7, D' <%"8 0%&%+ $! 4!,! 8&+"%%27%"02"0&!:0 $ 0/!,&2DT5%&+02!&5! 0&!<,"2%!2"0&!7

22 2,, "%5;,-0/8/2,, #,,/!! %&7 #,,/!! %&7 -#%&!& 0," % σ L % σ σ;&%:& $!7$%"2<,"8! >!&!,"0!2,! % % σ % L σ σ;&%:& $!7$%"2<,"8! > 0 % σ >,9"! 8/9!" % µ % µ L % µ µ > >!&!,"0!2-!/!>,"8!> % µ % µ % L µ µ > > % µ >

23 Sampling method POT Distribution Exponential Generalized Pareto Model stationary step-change linear trend exp M 0 exp M exp M 2 (t 0+ τ + 0+ > τ M 0 gp (t M gp 0+ τ 0+ > τ M 2 gp + Block Maximum Gumbel GEV M 0 gu (t M 0 gev (t M gu µ 0+ τ µ µ 0+ > τ gev M µ 0+ τ µ µ 0+ > τ,9"! M 2 gu µ µ + µ + gev M 2 µ µ + µ +

24 (a (b Model stationary step-change linear trend M 0 exp σ "%6 Specification on I "%6 M 0 gp σ + Specification on ( "%6 ( "%6 "%6 exp M σ "%6 Specification on "%6 "%6 M gp σ + Specification on ( "%6 ( "%6 "%6 ( "%6 exp M 2 σ + "%6 Specification on + "%6 "%6 + "%6 "%6 M 2 gp + σ + Specification on + ( "% ( "%6 "%6 ( "%6 + + (c M 0 gu µ "%6"%6 Specification on "%6"%6 "%6 "%6 "%6 gu M µ "%6"%6 "%6"%6 "%6 "%6 "%6 "%6"%6 "%6 "%6 "%6 Specification on gu M 2 µ + µ + "%6"%6 Specification on + µ µ + "%6"%6 "%6"%6 "%6 "%6 + "%6 "%6 "%6 "%6 + µ µ + "%6"%6 "%6"%6 "%6 "%6 + "%6 "%6 "%6 "%6,9"!,

25 stationary Step change linear trend M 0 gev ( µ + "%6 Specification on "%6 ( "%6 ( "%6 "%6 ( "%6 "%6 M gev µ + ( "%6 Specification on ( "%6 ( "%6 "%6 ( "%6 "%6 ( "%6 ( "%6 "%6 M 2 gev ( + µ + µ + "%6 Specification on ( "% µ µ + ( + "%6 "%6 + + ( "%6 "%6 ( "% µ µ + ( + "%6 "%6,9"!9

26 !,;7 &!,0<,"7 $,&6! $,&6! -#%&!& 0,"!&!,"0!2,! % -#%&!& 0," #%0& #%0&,,/!!7 S S S F. F/ 7 (/ 7 &%#0% 0&+%/, 0%& τ S S τ S &%#0% 0&+%/, 0%&!&2 5%!+050!& S &%#0% 0&+%/, 0%& 4#! #,,/!!7 B0 L [. [( 0 L [F [FRF < < 0 L [F [FR(,9"!( 8&0+%/%& U] V < < < < B0L [. [. &%&7, 0%&,4 7, 0%&,4 8&0+%/%& U] V!&!,"0!2,! % %&!& 0," &!,0<,"7 F F R R R.( R!,;7. ( *. L [ [.

27 &/ 50 (a Q(m 3.s Q(m 3.s (b rank of event Time (years Interarrivals time (years & (c rank of event 5 Interarrivals time (year Year &

28 (a Density Prior Post. (b Density Density 2 Post. Post. 2 (c Density τ Prior 0 Post Density Prior Post &. &, Frequency τ &"

29 posterior, model M 0 gp posterior 2, model M gp prob &J &- 90 0year return flood (m 3.s years &#