Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying...
Transcript of Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying...
Time-Varying Models for Extreme Values
Gabriel Huerta
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, U.S.A.��������������� ���������� ��������� , ����� ��� ���������� ��������������������������Bruno Sanso
Department of Applied Mathematics and Statistics, University of California, Santa Cruz, U.S.A.� ����������� �!���#"���"!�$����� , ����� �%�� �!���#"���"!�$��������� � �������Summary. We propose a new approach for modeling extreme values that are measured in time and space. Firstwe assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location,scale or shape parameters define the space-time structure. The temporal component is defined through a Dy-namic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of thedata in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a processconvolution form. We show how to produce temporal and spatial estimates of our model via customized MarkovChain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels pro-duced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coastof Venezuela.Some key words: Spatio-temporal process, Extreme values, GEV distribution, Process convolutions, MCMC,ozone levels.
1. Introduction&('�)+*-,/. 0 1325460725,98:0;*50 1�,<0�*>=61�0�*50 )?250�.@,<)BAC,9D�EF150HGJIK46,/LM4NLO'#151�0�*5=+'#)F.6*P2�'H.6Q�,<R9ST8:Q�U ,<8VQV'�WX'�Y '#)607L '�)FL 0 )?251MQ�2�,<'#)F*Z ,9)[=F=68]\7'�^#0 1725460@Q�1�0�QN'�W`_a0�1�L 0�.b,9)c_d0OU ,9L 'e&(,<2fSeWg'�1h254F0H=!0 1�,<' .iG�j�j#k�lfm�k#k#m ndoP4F,9*h=6R9'�2V*-EFD�D�0�*f2M*J2�4FQ�2h254F01MQ�)F. '#8qp+EFLr2�EFQ�2�,<'#)F*sLO'�EFR9.at!0:*-E6=!0 1�,98h=!'#*50�.dIK,u2�4BQv.60�LO1�0�Q#*-,9)6D:2�150�)F.�nsoP460hw?EFQ�)?25,<x+L Q�25,9'�)a'�W�*-EFLM4BQV251�0 )+.a,9*y 0�SzWg'�1X25460K^�Q�R9,/.6Q�2�,<'#)h'�W!,98:=+'#1-2MQ�)?2{0 )|^|,<1�'�)F8h0�)?2�Q�R =!'�R9,/LOS;.60�LO,/*5,<'#)F* n}oP4603Q�)FQ�R9S *-,/*}'�W�Q~25,98h0>*50 1�,90�*�'�W�8VQ�U ,987E68^�Q�R9E60�*s*5EFLM4eQ�*`25460:'#)60;,9)eAC,9D�EF150TG;I�'�EFR9.Bt+0:)FQ�25E61MQ�R9R9S@L�Q�1�15,90�.N'�E 2st|SNLO'#)F*5,9. 0�15,9)6DH)6'�) l�*-2�Q�2�,<'#)FQ�1�S@=61�'|L 0�*�*-0�*IK,<254T25,98:0z^�Q�15S|,9)6D:=FQ�1MQ�8:0 250 1M*�n}oP46,/*K,/*�25460sWg' L EF*K'�W}254F,9*K=+Q�=!0 1�n{� E6=6=!'#*50s254FQ�2��6,9)aQ�.6. ,<25,9'�)H25':2�460J.6Q�2�Q:Q�2K'�)F0*5=+0�LO,<x+LJR9' L Q�2�,<'#)��!I(0J4+Q�^�07.6Q�2�Q]Q�^�Q�,<R/Q�t6R90JQ�2~)60 ,9D�4|t!'�1�,<)FDVR<' L Q�25,9'�)F*�n3oP460;,<)6Wg'�1�8:Q�25,9'�)@Wg1�'�8�25460�*-0h.6Q�2MQ]L Q�)dt!0=!'?'#R<0�.a2�'�D�0 25460�1~25'T'�t 2MQ�,9)BQ]D#R<'#tFQ�R{. 0�*5L 15,9= 2�,<'#)d'�WX2�460:t+0�4FQ�^|,<'#1`'�W�8VQ�U ,98:QH'�^�0�1`QH150�D�,9'�)�nz��)N2546,/*s=FQ�=+0�1���I�0IK,9R<RX*546'�I�4F'�I�25460:2�,<8:0 l$^�Q�1�S|,<)FDH8:' . 0 R/*z. 0�^�0 R9'�=!0�.dWg'�1z'#)60:R<' L Q�25,9'�)�L Q�)�t!0h)+Q�25EF1�Q�R<R9Sd0OU|2�0 )F. 0�.d2�'@,<)FL R<E+. 0VQ*5=FQ�2�,9Q�R�LO'�8:=!'�)60�)?2�noP460�251MQ�. ,<25,9'�)FQ�R?Q�=6=61�'#Q#LM4sWg'�1C25460PQ�)FQ�R<S *5,9*�'�WF0 U|251�0 8:0�^�Q�R9E60�*�,/*}tFQ�*50�.z'#)z254F0(EF*50�'�W6254F0P�~0�)60 1MQ�R9,<Y�0�.J�XU|251�0 8:0� Q�R9E60 Z �s� � \P.6,9*-251�,<tFE 25,9'�)�nXoP4F,9*P,/*KD�,9^�0�)Ht|S� Z�� \X��0 U =V�3���$G(���V� � ���� �K�����5�5 ¡ ¢ Z G�\
IK460�150:�£,9*JQTR9' L Q�2�,<'#)B=+Q�1MQ�8:0O2�0 1�� � ,/*JQ@*�L Q�R90:=FQ�1MQ�8:0 250 1JQ�)F.B�H,/*JQ@*-4+Q�=!0V=FQ�1MQ�8:0O2�0 1�nVoP460H�¤*5,<D#)�.60 )6'�250�*254F07=!'#*5,u2�,<^#0z=FQ�1523'�W�2�460;Q�15D#E68:0 )?2�nKoP46,/*3. ,/*-251�,<t6E625,9'�)a0 )+LO'�8:=FQ#*5*50�*K25460hA61�¥0�LM460 2 Z �v¦§k?\r�!¨B0�,<tFE6R<R Z �v©§k#\KQ�)F.�~E68;t+0�R Z ��ª«k?\sW�Q�8:,9R<,90�*�nN¬�LOR90�Q�1hQ�L�LO'�EF)#2;'�WP25460T*f2MQ�2�,9*-25,/L Q�R�8:' . 0 R9,<)FDdWg'�1;0OU|251�0 8:0]^�Q�R<E60�*7E+*-,9)6Da2�460T�s� �. ,/*f2�15,9t6E 2�,<'#)H,/*�=61�0�*50 )?250�.],9)@&('�R90�* Z m�k�kFG�\On}3)F. 0 1�2�460zQ�*�*-E68:= 2�,<'#)v'�WC,9)F. 0 =!0 )+. 0 )FL 0��6L '�)F. ,<25,9'�)+Q�R�'�)H�{® � Q�)F.]� �254F0`R9, y 0 R9,<4F'?' .v'�WCQh*5Q�8:=6R<0`'�W�8VQ�U|,98VQ6� *�Q�S � � ® ¯ ¯�¯O® ��° ,/*(1�0�Q#. ,9R<SVQ�^�Q�,9R9Q�t6R90>Wg1�'�8 Z G�\rnXoP4|EF*��|Q;±PQ�S�0�*5,/Q�)]Q�)+Q�R9S|*5,/*L Q�)bt+0TL Q�151�,<0�.�'�E 2ht?S�,98h=!'#*5,9)6DNQa=61�,9'�17. ,/*-251�,<t6E625,9'�)�'#)b�{® � Q�)+.e�dQ�*J,9)c&('�R90�*;Q�)F.bo}Q�IK) Z G�j�j#²#\rnN�sQ�0O2MQ�)Q�)+.@�~1�,9D�'�R90O2525' Z m�k�k�³?\�=6150�*-0�)?2KQ:8:' . 0�R�IK,u2�4d. S|)FQ�8:,/L Q�R<R9S]^�Q�1�S|,<)6Dh=FQ�1�Q�8h0 250�1�*�Wg'#1P25460JQ�)FQ�R9S *-,/*P'�W�0OU|251�0 8:0�*P'�WQ]EF)6,<^�Q�15,/Q�2�0z25,98:0h*50 1�,<0�* n~oP460 ,91~8h' . 0�RC,9)FL R<EF.60�*~. S|)FQ�8:,/L7LM4FQ�)6D�0�*KWg'�1s*�L Q�R90�´�*-4+Q�=!0z=FQ�1MQ�8:0O2�0 1M*3Q�)F.d*50�w?E60 )?2�,9Q�RE6=�.6Q�2�,<)FDVIK,u2�4dµ>¶�·r¸$¹�ºO»<¼z½�¹¾»¿¸�¼O·MÀ�t6E 2>2�460 1�0s,9*P)F']*-=FQ#LO0s'�1K*5=FQ�L 0Ol$25,98:0s*-251�EFLO25E61�0�n¬>)@Q�R<250�15)+Q�25,9^�0~Q�=6=F15'?Q�LM4V2�'h8:' . 0�R<,9)6Dh0OU|251�0 8:0z.6Q�2MQ7,/*(254FQ�2>'�WCL '�)F*5,/. 0 1�,<)FD72�460z. ,9*-251�,9t6E 25,9'�)T'�WC0OU6LO0�0�.6Q�)FLO0�*'�^�0�1{Qs46,<D#4h25461�0�*546'�R/.�n�ÁX,/L y Q�)F.6* Z G�j|Â�Ã�\C'#t 2�Q�,<)60�.;254603�~0�)60 1MQ�R9,<Y�0�.hÁ{Q�1�0O2�'~Ä`,9*-251�,9t6E 25,9'�)��#IK46,9R/*f2�Á�,9L y Q�)+.6* Z G�j?Â|G�\
2 G. Huerta and B. Sanso
Time
Ozo
ne
1990 1992 1994 1996 1998 2000 2002
0.0
0.1
0.2
0.3
Fig. 1. Daily maxima of ozone concentration at station Merced in Mexico City from 1990 to 2002.
,9)#2�15' . E+LO0�.BQv=!'�,9)?2z=615' L 0�*�*~Q�=6=61�'#Q#LM4�nJoP460�*50h,9. 0�Q�*~4FQ�^#07t!0 0 )�0OU|2�0 )F. 0�.BQ�)F.NQ�=6=6R9,<0�.N,9)b�|8:,u2�4B0 2zQ�R�n Z G�j�j|Â�\O��|8:,<254 Z G�j � j#\O�?Ä~Q�^|,9*5'�)]Q�)F.v�|8:,<254 Z G�j#j�k?\{Q�)F.v8:'�1�0>150�LO0�)#2�R<SV¬`*5*5E6)��L��Q�'J0 2PQ�R�n Z m�k�k�³|\rn��K0 D?Q�1M. ,9)6Ds25460~8:' . 0 R9,<)FD'�W}.6Q�2MQ7Q�2P. ,��!0�150�)#2PR9'|L�Q�2�,<'#)F*�,9)H*5=FQ#LO0��6&�Q�*�*5'�)vQ�)F.H&('�R90�* Z G�j�j#j#\XL '�)F*5,/. 0 1�25460s*-=+Q�25,/Q�R�. ,/*f2�15,9t6E 2�,<'#)v'�W�0OU|251�0 8:0�*EF*5,<)FD:Qh2�461�0�*546'�R/.v8:' . 0�R�tFQ�*50�.H'�)T2�460z=+'#,<)?2>=61�' LO0�*5*PQ�=F=615'?Q�LM4�nX�~,9R<R90 R/Q�)+.H0 23Q�R$n Z m�k�k�³|\�. 0 ^#0 R9'�=@Q;4F,<0�1�Q�1�LM46,/L Q�R8:' . 0 R tFQ#*-0�.J'#)72�460>�~0 )60�1�Q�R<,9Y 0�.7Á{Q�1�0O2�'`Ä`,9*-251�,9t6E 25,9'�)�n}oP460P=61�0�*50 )?2{=FQ�=+0�1�*54FQ�150�*�IK,<254h254F0�R/Q�*-2�2fI('~=FQ�=+0�1�*�LO,<250�.254F0VW�Q�LO2z254FQ�2J25460V8:' . 0�R<,9)6Da,/*ztFQ�*50�.�'#)�4F,<0�1�Q�1�LM46,/L Q�R{*f2�15EFLO25E61�0�*z254FQ�27Q�*�*-E68:0:2�460]0OU ,/*f2�0 )FL 0:'�W>Q�)eE6)F.60 1�R<S|,9)6D*5=FQ�2�,9Q�R+8:' . 0 R$n{¬>)TQ�R<250 1�)FQ�25,9^�03I�'�EFR9.Vt!0`25';L '�)F*5,/. 0 1�8:'|.60 R/*�tFQ#*-0�.V'�)v87E6R<25,9^�Q�1�,9Q�250`.6,9*-251�,<tFE 25,9'�)F*(Wg'#1(0 U?2�150�8:0�*�nAFQ�8:,<R9,<0�*s'�WP*-EFLM4e. ,/*f2�15,9t6E 2�,<'#)F*z4FQ�^#0;t!0 0�)�.60 ^�0�R<'#=+0�.���,9)�0�. Wg'�1M.BQ�)F.Bo}Q�IK) Z G�j#j�²F�CG�j�j?Â�\3Q�)F.��>0 ��0 1�)FQ�)�Q�)F.o}Q�IK) Z m�k�k�³|\Pt|S@Wg' LOEF*5,<)FDv'�)N'�t 2MQ�,9)6,<)FD]pF0 U|,9t6R90hQ#*-S|8:= 2�'�25,/L7.60 =!0 )F. 0�)FLO0h=61�'�=!0 1525,90�*�n~Ä`E6=6EF,9* Z m�k�k?Ã�\KL '�)F*5,9.60 1M*Q�)eQ�=F=615'?Q�LM4dt+Q�*50�.d'#)�L '�=6EFR9Q#* n;� 'HW�Q�1���8:' . 0�R9*stFQ#*-0�.N'#)N8;E6Ru2�,<^�Q�15,/Q�2�0:.6,9*-251�,<tFE 25,9'�)F*`Wg'#1z0OU|251�0 8:0�*~4FQ�^#0;t!0 0�)Q�=F=6R<,90�.T'#)6R<S]25'VQ;Wg0 I *-=FQ�25,/Q�R�R9' L Q�2�,<'#)F*�n��)b2�46,9*;=FQ�=+0�1;I�0]Wg' LOE+*7'#)b25460TEF*50v'�WK25460a�s� � . ,9*-251�,9t6E 25,9'�)cQ�)F.�Q�R<R9'�I 254F0v=FQ�1�Q�8h0 250�1�*J25'N^�Q�15S�,<)[25,98:0Q�)+.£*5=FQ#LO0�n ��)§�|0�LO25,9'�)im�I�0a=615'#=+'?*-0a*5'�8:0dQ�=F=615'?Q�LM460�*;Wg'�1V2�,<8:0N*50 1�,<0�*:'�Wz0OU|2�150�8h0�*:2�Q y 0�)£Q�2v'�)60aR9' L Q�25,9'�),9)e*5=FQ#LO0�nv¨B0]*-25EF.6Sd254F0V25,98:0V^�Q�15S|,9)6D@t+0�4FQ�^|,<'#1z'�W�25460V=+Q�1MQ�8:0O2�0 1M*32�4FQ�2;D�'�^�0�15)N25460].6,9*-251�,<tFE 25,9'�)e'�W�254F0v.6Q�2MQEF*5,<)FDTÄ`S|)FQ�8:,/L��,<)60�Q�1z_a' . 0�R9*sQ�*~=61�0�*50 )?2�0�.a,9)B¨�0�*-2JQ�)F.��3Q�151�,9*5'�) Z G�j#j?Â�\rns¨B0:,9R<R9EF*-251MQ�2�072�460:=+'?*5*5,<tF,<R9,u2�,<0�*~'�WL Q�= 25EF15,9)6Dh25,98:0z^�Q�15S|,9)6Dh251�0 )F.F* nXoP460sEF*50z'�W{Q:. S|)FQ�8:,/L~8h' . 0�R�Q�R9R9'�I>*�Wg'#1K254F0J. 0O2�0�Lr2�,<'#)@'�WCt!'�2�4a*-46'#1-2>Q�)F.TR<'#)6D1MQ�)6D#0JLM4FQ�)6D#0�*�n�¨�07Q�R/*5']LO'�)+*-,/. 0 1>25460J=F15'#t6R<0�8q'�W�1�0 R/Q�2�,<)FD:251�0 )F.6*>25'V2�,<8:0J^�Q�15S|,9)6D]LO'�^�Q�1�,/Q�250�* n(±(S@. '#,<)6D]*5'VI�0Q�1�0;Q�t6R90J25'@. 0�*5L 15,9t+0h46'�I 2�460h*-251�0 )FD�254B'�WX2�460hR<,9)60�Q�1`1�0 R/Q�2�,<'#)F*546,<=BLM4FQ�)6D#0�*`IK,u2�4d2�,<8:0�nz��)�*-0�Lr2�,<'#)��]I�0h0OU|250�)F.254F0`8:' . 0 R!2�'hQ�R<R9'�I�Wg'�1P*-=+Q�25,/Q�R!^�Q�1�,9Q�25,9'�)]'�W�25460s=FQ�1�Q�8:0O250�1�*�n��`EF1�*5=FQ�2�,<'�l�2�0 8:=!'�1MQ�R!8h' . 0�RFWg'#1P0OU|251�0 8:0�*(,/*(t+Q�*50�.'�)T=61�' LO0�*�*�LO'�)|^#'�R9E 25,9'�)F*(EF*5,<)FDh8:0 2546' .6*P=61�0�*50 )?2�0�.��|Wg'#1�0 U6Q�8:=6R90�� ,9)��3,<D?. '�) Z m�k#k#m#\rn{oP46,/*P=615'�^|,/. 0�*�Q;150�. EFLr2�,<'#)'�W�2�4603=+Q�1MQ�8:0O2�0 1�*5=FQ�L 0K254+Q�2(,/*X0��!0�Lr2�,<^#03,9)V4FQ�)F.6R<,9)6D7R/Q�1�D�0K)|E68;t+0�1�'�W�R<' L Q�25,9'�)F*�n}��)V254603xF)FQ�R+*50�Lr2�,<'#)VI(0`=61�0�*50 )?2Q:. ,/*5L EF*5*5,9'�)T'�W�'�E61P1�0�*5E6Ru2M* n2. GEV distribution with parameters varying in time&('�R90�* Z m�k�k6G�\~=61�0�*50 )?2�*JQ�)�Q�=6=F15'?Q�LM4�25'd8:' . 0�R�LM4FQ�)6D�0�*JQ�L 15'?*5*~2�,<8:0H,<)�8VQ�U|,98VQaEF*5,<)FDa25460@�s� � .6,9*-251�,<tFE 25,9'�)�noP460�*-03LM4+Q�)6D#0�*�Q�150`. 0OxF)F0�.:2�4615'#E6D�4].60O250�158:,9)6,/*f2�,9L>WgE6)FLr2�,<'#)F*�Q�)F.:8VQ�Sh150 Wg0 1X2�'J2�150�)F.V'�1�*-0�Q�*5'�)FQ�R<,<2fSJ,9)V254F0~.6Q�2�QFn
Time-Varying Models for Extreme Values 3&('�)+*-,/. 0 1TQ�*50�w?E60 )+LO0N'�W7LO'#)F. ,<25,9'�)FQ�R<R9Sc,9)F. 0�=+0�)F. 0�)#2@'�t+*-0�15^�Q�2�,<'#)F* � � ® � � ® ¯�¯ ¯ ® ��° *5EFLM4�254+Q�2H0�Q�LM4§'#tF*50 1�^�Q�25,9'�)Wg'�R9R9'�I>*XQJ�s� � .6,9*-251�,<tFE 25,9'�)VIK,<254vQ~2�,<8:0`^�Q�15S|,9)6DzR9'|L�Q�2�,<'#)h=+Q�1MQ�8:0O2�0 1���,�n 0�n9� ���������HZ � � ® � ®-��\rnXoC'J'#t 2�Q�,<) ��� I�0x6UdQ:2�,<8:07R90 )FD�254a'#13t6R9' L y '�WX.6Q�2MQ]Q�)F.NL Q�R/LOEFR9Q�250z2�46078VQ�U ,987E68 '�W�2�4607'#tF*-0�15^�Q�25,9'�)F*K'#t 2�Q�,<)F0�.a. E61�,9)6DV254F0 $l�2�4=!0 1�,<' .V'�W�25,98h0~'�1(t6R9'|L y n{oC'78:' . 0 R�LM4FQ�)FD�0�*�,<)]25,98h0#�F&('�R90�* Z m�k#k6G�\�8:0 )?25,9'�)F*�*50 ^�0�1�Q�R6=!'#*�*-,9t6,9R<,<25,90�*�n{A6'�1(0OU6Q�8:=6R90��� � �����{��� � � � � � �����{��� � � ��� � � � '�1P*-'#8h0`'�2�460 1P46,9D�460�1-l%'�1M. 0�1X=+'#R<S|)6'#8:,9Q�R+'#) � n{¬>R/*-'+�|460~*5E6D#D�0�*-2�*�254F0`EF*50~'�WLO'�^�Q�15,/Q�2�0�*{IK,<254TQ�)]0 U|=F150�*5*5,<'#):'�W�254F03Wg'#158 � � �����}��� � � � n��3'�) l�*-2�Q�2�,<'#)FQ�1�S78:' . 0�R9*�L Q�)vQ�R/*-'7t+03Wg'�1�87E6R/Q�2�0�.:Wg'#1254F0J*-4FQ�=+0zQ�)F.+´�'#1K*5L�Q�R90~=+Q�1MQ�8:0O2�0 1M* n�A6'#1K0OU6Q�8:=6R90�� � � �§0OU = Z ������� � � \��6� � ��������� � � '�1P� � �����(��� � � ��� � � � n¬>R<250�15)+Q�25,9^�0�R<S#� I(0s=61�'�=!'#*50`25460zEF*50s'�W�Ä3S|)FQ�8h,/L �,9)60�Q�1K_a' . 0 R/* Z Ä C_d*�\O�FQh^�0�15SVD#0 )60�1�Q�R�L R9Q#*5*P'�W�2�,<8:0J*50 1�,<0�*8:' . 0 R/* �|Q�*�,<)v¨B0�*f2PQ�)F. �`Q�1�15,/*-'#) Z G�j#j?Â�\C25'J8:' . 0 R!=FQ�1�Q�8h0 250�1�LM4FQ�)FD�0�*�Q�LO1�'#*�*}25,98:0>Wg'#1�R9'|L�Q�2�,<'#)��?*�L Q�R<03'�1�*54FQ�=!0�nA6'#1}Q325,98:0Ol%^�Q�1�S?,9)6D`R<' L Q�25,9'�)J8:' . 0�R���I�0�Q#*5*5E68:0X2�4FQ�2}254F0P.FQ�2�Q � � ® � � ® ¯ ¯�¯O® ��° . 0 xF)60�*�Q`LO'�)+. ,u2�,<'#)FQ�R9R9S~,9)F. 0�=+0�)F. 0�)#2*50�w?E60 )+LO0zQ�)F. ���������]Z � � ® � ®5��\K*-';254F0JLOE68;E6R9Q�25,9^�0z. ,/*f2�15,9t6E 2�,<'#)TWgE6)FLr2�,<'#)@'�W�0�Q�LM4 ��� ,/*� �rZ���� \ � 0OU = � � � G(�b� � � � �B� �� �>� ���5�5 ¡ ¢ � � � G�®�¯ ¯�¯O®! �¯
oP46,/*PQ�*�*-E68:= 2�,<'#)],/*�tFQ#*-0�.]'#)vQ�*5S|8h=625'�2�,9L3150�*-EFRu2M*(Q�)F.v*5'7,<2PL�Q�)]t!0sLO'�)+*-,/. 0 1�0�.vQ�*�Q�)vQ�=6=61�'�U|,98VQ�2�,<'#)�nC��)+*f2�0�Q�.]'�WQ:. 0O2�0 1�8:,<)6,/*-25,/L~WgE6)FLr2�,<'#)@'�)T� � � I(0JQ#*5*5E68:0~254FQ�2� � � "�#��$ � �&% � �'% �(��)�Z kF® \$ � � �*� $ � ��� �,+ � ��+ �(��)�Z kF®.-�\IK460�150 $ � ,/*K*f2MQ�250s^#0�Lr2�'�1P'�W�. ,98h0�)F*5,<'#)F*0/21NG3�4" � ,/*KQ65-1�0 D#150�*5*5'�187s'�W�. ,<8:0�)F*-,9'�)+*0/91BG3� �*� ,/*KQ:/21;/v0 ^�'#R<E625,9'�)8VQ�2�15,<U<� ,/*P254607'�tF*50 1�^�Q�25,9'�)+Q�R�^�Q�1�,9Q�)FLO0zQ�)+.=- ,/*>Q9/;1>/T0 ^�'#R<E625,9'�)@LO'�^�Q�1�,9Q�)FLO0~8VQ�2�15,<U�n(oP46,/*>. 0 xF)60�*`QVÄ C_Z " � ® ® �*� ®8-�\rn:±(SNEF*-,9)6DaQTÄ C_ I�0h'�t 2MQ�,9)B8:'�1�0;pF0OU ,9t6,<R9,<2fSd,<)B254+Q�2s251�0 )F.F*zQ�1�0;)6'�2JLO'�)+*f2�1�Q�,<)60�.a2�'T4+Q�^?,9)6DTQ*5=+0�LO,<x+L>=FQ�1�Q�8:0O251�,/LPWg'�1�8@nC¨�0~L�Q�)]Q#*5*50�*�*{2�460`*5,<D#)6,<x+L Q�)FLO0>'�W�*-4F'�152X250�158¤LM4FQ�)6D�0�*{25'#D�0 25460�1�IK,<254]R9'�)6D7250�158 '�)F0�*�n¬>R/*5'F� ,<2>,9*K=!'#*�*5,<t6R90~25']w?EFQ�)#2�,uWgSv46'�I§2�460z0 ��0�LO2�*>. EF0s25'VLO'�^�Q�1�,/Q�250�*PLM4FQ�)6D#0`IK,<254T2�,<8:0�nÁ�'#*-250�15,9'�1{,9) Wg0�150�)FLO0~Q�)F.V*-251�EFLr2�E61�03E6)F.60 1�'#E61�)60 Ii�s� � . ,/*f2�15,9t6E 2�,<'#)VWg'�R9R<'�I>*(*f2MQ�)F.FQ�1M.V_dQ�1 y '�^hLM4FQ�,9)]_a'�)?2�0&�Q�1�R9' Z _�&�_B&P\>8:0 2546' .6* Z *-0�0��!Wg'�1s0OU6Q�8h=FR<0#���sQ�8h0�158VQ�)��CG�j�j|Â�\rns¨�0;tF15,90OpFSd.60�*�LO1�,<t!0J2�460:150�R<0�^�Q�)?23WgE6R9RXLO'#)F. ,<l25,9'�)+Q�RC.6,9*-251�,<tFE 25,9'�)F*�n>Ä`0OxF)60725460;^�0�Lr25'#1�*@? � Z�� � ® � � ®�¯ ¯ ¯�® � ° \A�<Bi� Z � � ®-� � ® ¯ ¯�¯O®-� ° \>Q�)F.DCN� Z $ � ® $ � ® ¯ ¯�¯ ® $ ° \rn¬�2>0�Q#LM4T*-250�=H'�W}'#E61>_B&�_B&iQ�R9D�'#15,<254F8T�|I�0z. 1�Q�Iz�
E ��Q�LM4@^�Q�R9E60;'�W{� � Wg15'#8GF Z � �AH ��� ® $ � ® ®8-�\�� � ��G�®�¯ ¯ ¯�®! ^?,/Q]_a0 251�'�=!'�R9,9*-l �3Q�*-25,9)6D?*>*f2�0 =F*�n`�~,9^�0�)a25460hLO'#)F. ,<l25,9'�)FQ�R�,<)F.60 =!0 )F. 0�)FLO0J*-251�EFLO25E61�0s'�W}25460z8:' . 0 R$�|254F0JLO'�)+. ,u2�,<'#)6,9)6DV'�)D? 1�0�. E+LO0�*�2�' ��� nE oP460b*5L�Q�R90�Q�)F. *54FQ�=+0e=FQ�1�Q�8h0 250�1�*TQ�1�0�*5,987E6R/Q�2�0�.iWg15'#8IF Z � H ?T®.B3®5��\@Q�)F.JF Z � H ?H®.B~® � \HIK,u2�4 ,<)F.6,<^|,/. EFQ�R_a0O2�15'#=+'#R<,/*fl �3Q#*f2�,<)FD#*:*f2�0 =F*�n�oP460d*�L Q�R90dQ�)+. *-4FQ�=+0a=+Q�1MQ�8:0O2�0 1M*V'�W~25460��s� � . ,/*-251�,<t6E625,9'�)§Q�1�0dQ�*�*5E68:0�.LO'�)+*f2MQ�)?2�,9)T25,98:0�nE oP460s'�tF*50 1�^�Q�25,9'�)+Q�R�^�Q�1�,9Q�)FLO0 L Q�)Ht!0J*5Q�8:=6R<0�.]Wg1�'�8 Q�)T��)|^�0 1M*50s�sQ�8h8VQ:. ,/*f2�15,9t6E 2�,<'#)�nE oP460:0 ^�'#R<E625,9'�)BL '�^�Q�1�,9Q�)FLO0;8:Q�251�,uUK- L Q�)Bt!0V8:' . 0 R90�.BIK,u2�4b. ,9*�LO'#E6)?2`W�Q#Lr2�'�1M*s'�1zQ�*`WgE6)FLO25,9'�)F*J'�W�2�460V'�t6l*-0�15^�Q�2�,<'#)FQ�R�^�Q�15,/Q�)+LO0 n3��)a2�460h*50�L '�)F.aL�Q�*50��+25460h0 ^#'�R9E 25,9'�)dL�Q�)dt!0;*�Q�8:=6R90�.dIK,u2�4NQ�)d��)|^#0 1M*-0 l��sQ�8h8VQ:'#1��)?^#0 1M*-0 l$¨ ,/*54FQ�152�. ,/*-251�,<t6E625,9'�)�nE &('�)F. ,<25,9'�)+Q�RK'�)�B~� Q�)F.�-��PI(0d*�Q�8:=6R90@254F0d*f2MQ�250a^#0�LO25'�1LC§EF*5,<)6De2�460B½NM�·�O�¶�·.P�½�¹¾»¿¸�¼O·r¹RQTS6U`¶�º8VWO�¶�·XPY+¹RZ\[ »<¶�¸$¹]M^Q Z A�A�±K�6\�Q�R9D�'#15,<25468 Q#*�=F150�*-0�)#2�0�.v,<)d&�Q�15250�1PQ�)F.L_s'#46) Z G�j#j�³?\X'�1KA61W`E64|IK,91-2�4 l��6LM46)FQ�25250�1 Z G�j�j�³?\OnA6'�1�IPQ�1M.v,9)a25,98:0���I�07Q�=6=FR<ST254F071�0�LOEF1�*5,<^#0sxFR<250 1�,9)6D]0�w?EFQ�25,9'�)F* Z ¨�0�*-2~Q�)F. �3Q�151�,/*-'#)���G�j#j? �!&(4FQ�=6250 1`³?\O�F2�''�t 2MQ�,9)aF Z $ �AH b\� ® ®.-�\A� � � G�®Mm ®�¯ ¯ ¯ ®! ���25460K=!'#*-250�15,9'�1{. ,/*-251�,<t6E625,9'�)h'�W $ � D�,9^�0 )hQ�R9R|25460K,9) Wg'#158VQ�2�,<'#);Q�^�Q�,9R/Q�t6R90E6=z25'325,98:0 � Z]b\� \rn�±PQ�L y IPQ�1M.6*�,<)J25,98h0#��I(0�xF1M*f2�*-,987E6R/Q�2�0cF Z $ °\H b:° ® ®8-�\�Q�)+.s25460�)���1�0�LOEF1�*5,<^#0 R9S3I�0�*�Q�8:=6R9025460z'�25460�1 $ � ^�Q�R<EF0�*PWg1�'�8dF Z $ �AH $ � ¡ � ® b:° ® ®.-�\(Wg'�1 � �� �cG#® ¯�¯ ¯O®�G�n¬>)N,<8:=6R90 8:0 )?2MQ�25,9'�)N'�W{254F,9*~Q�R9D�'#15,<25468@�+25'#D�0O2�460 1`IK,u2�4a254F0;.6Q�2�QvQ�)FQ�R9S|Y 0�.@,<)a2�46,/*3=FQ�=+0�1~Q�150;Q�^�Q�,<R/Q�tFR<0sWg1�'�8 254F0 � 6efeTehg8i^jlkmjngpoTqWrsg!tfumo4v �swfx o4tmymj4klv6n¬3*`Q�)@,9R<R9EF*-251MQ�25,9'�)a'�W�25460J=F15'#=+'?*-0�.H8:' . 0�RCI�0z150 25E61�)@25']254607.FQ�2�Q],<)dA},9D�E61�0VG�nKoC']0OU =6R9'�1�0s2546070OU ,9*-250�)FLO07'�W=!'#*�*-,9t6R9032�150�)F.6*�I(0zL '�)F*5,9.60 1PQ7=FQ�1-2�,9L E6R/Q�1PL Q�*50~'�W�2�,<8:0Ol%^�Q�1�S|,<)6DV�s� � n oP46,9*P,/*�D�,9^�0�)vt|S]QJx+1�*-2K'�1M. 0 1�=+'#R<S|)6'#8:,9Q�RÄ �_ IK4F0 1�0a" � � �*� ��G�®{z � � ,9*(E6) y )6'�IK)HQ�)F.;- � ��| ®{z � �f|T¦�k6�?IK,u2�4HQJ'#)60`.6,<8:0 )+*-,9'�)FQ�R!*-2�Q�250 l$=FQ�1�Q�8:0O250�1
4 G. Huerta and B. Sanso
Time
ozon
e
1990 1992 1994 1996 1998 2000 2002
0.0
0.1
0.2
0.3
θtµt
Fig. 2. Daily maxima of ozone concentration at Merced from 1990 to 2002, posterior means for ( ��� , � � ) and 90%posterior probability interval for � �
$ � n(�|EFLM4@8:' . 0 R�,/*PEF*50OWgE6R�25'V1�0 =61�0�*50 )?2>LM4FQ�)FD�0�*�,9)@R<0�^�0 R�'�W�2�460zR9'|L�Q�2�,<'#)T=+Q�1MQ�8:0O2�0 1P� � nX¨B0JL '�8:=6R90O2�0s25460z8:' . 0 Rt|SaLO'�)+*-,/. 0 1�,9)6D:25460h=61�,<'#1�*�� Z $ � H b � \ � )�Z � ®�� � \r® ��m�VZ� � ®X� � \r®X| ��m�VZ� � ®!� � \rn`¬`.6. ,<25,9'�)FQ�R<R9S���I�0;Q#*5*5E68:0254+Q�2 Z � ®-��\XQ�150>Qs=F15,9'�1�,6,9)F. 0 =!0 )+. 0 )?2�IK460�150KR9'�D Z � \XQ�)F.:�~Wg'�R9R9'�I£Q��3'�1�8:Q�RF. ,/*f2�15,9t6E 2�,<'#)���,$n 0�n�� Z � \ ��� )[Z ��+®����6\Q�)+.�� Z ��\ ��)[Z ®�� \On��)�A},<D#E61�0vmTI(0]=F150�*-0�)#2z2�460]1�0�*5E6R<2�*z'�W�xF2-25,9)6Da2�46,/*;�s� � l3xF1M*-2J'�1M. 0 1J=!'�R9S|)6'�8:,/Q�R�Ä �_�nCoP460Vx+D�E61�0]*-4F'�I>*[email protected]�2�Q Z =+'#,<)?2�*M\r��=!'#*-250 1�,9'�1h8h0�Q�)F*hWg'#1:� � Q�)F. $ � Q�)+.�2�460aj�k�� =+'?*f2�0 1�,<'#1;=61�'�t+Q�t6,9R<,<2fSb,9)#2�0 1�^�Q�R�Wg'�1 $ � n£oP4F0. 0�LO1�0�Q�*503,9)TR<0�^�0 R/*PQ�L 15'?*5*�25,98:0~,/*Kw?E6,<250s)6'�2�'�1�,<'#EF*�n}oP460�*-0~0�*-25,98VQ�2�0�*P,9R<R9EF*-251MQ�250`254+Q�2K'�E61P8:' . 0 R/*KQ�1�03^#0 1�S:pF0OU ,9t6R90,9)T1�0 =F150�*-0�)#2�,<)FDh. S|)FQ�8:,9Ls*-251�EFLr2�E6150~Wg'#1K0OU|251�0 8:0z^�Q�R9E60�*�n¬3*�I(034FQ�^�0K=61�0 ^|,9'�EF*5R9S;8:0 )?2�,<'#)60�.��#254F0>EF*503'�W�2�460`. ,/*-251�,<t6E625,9'�)V,9) Z G�\�Wg'�1(0OU|251�0 8:0�*{,9*(tFQ#*-0�.:'�)]Q�*5S|8:= 25'�25,/L *�noP4FQ�2�,9*�� � ,9*�Q�*�*5E68:0�.;25'Jt!0>2�460>8VQ�U ,987E68�'�W��@,9)F. 0�=+0�)F. 0 )?2�1�Q�)F. '�8 ^�Q�1�,9Q�t6R<0�*XQ�)+.��?Q�W¾250�1�)6'#158VQ�R9,9Y�Q�2�,<'#)���IK460�)�a2�0 )F.6*�2�':,<) x+)6,u2fS#�|25460J�s� � ,/*�'�t 2MQ�,9)60�.�n{��)H=61MQ�Lr2�,9L 0`I�0~4FQ�^�0~)60 ,<254F0 1P,<)+. 0 =!0 )F.60 )FL 0~)F'�1KQ7^#0 1�S:R9Q�15D#03)|E687t!0 1'�WC'�tF*50 1�^�Q�25,9'�)+* n}o�':LM460�L y 2�460`D#'|'|.6)60�*�*X'�W�x62P'�W�2�460s�s� � 8:' . 0�R��?254F0~E+*-EFQ�R!Q�=6=61�'#Q#LM4V,9*(25';'�tF*50 1�^�0>25460sQ�t6,9R9,u2fS'�WC25460z8:' . 0�R�25':=61�0�. ,/Lr2P2�460z0 8:=6,915,/L Q�R�w?EFQ�)?25,9R90�*P'�WC25460J.FQ�2�QFn �`E61K8:'|.60 R�=61�' . EFL 0�*(25,98:0Ol%^�Q�15S|,9)6DVw?EFQ�)?25,9R90�*��6*-'+�254F0 1�0~,/*P'�)FR<SV'#)60s'�tF*50 1�^�Q�25,9'�)TQ�^�Q�,9R9Q�t6R<0325':'�t62�Q�,9)H25460z0�8h=F,<1�,9L�Q�R�w?EFQ�)?2�,<R90zQ�2>Q�)|SVD�,9^�0�)]2�,<8:0�nXoC':'�^�0�1�L '�8:0>2�46,9*. ,��]LOE6R<2fS@I(0hLO'�)+*-,/. 0 132fI('H. ,9Q�D�)6'?*f2�,9L�* n`oP460JxF1M*-23'�)60;,9*`=61�0�*50 )?250�.@,<)BAC,9D�EF150 �6n>¨�0;'#tF*50 1�^�0s254+Q�2`8:'?*f2~'�W{254F0.6Q�2�Q]Q�150JL '�)?2�Q�,<)60�.@IK,u2�46,9)a25460 Z 2�,<8:0Ol%^�Q�1�S|,<)6D|\P,9)#2�0 1�^�Q�R/*>IK46'?*-0JR9'�I�0 1>Q�)+.aE6=6=!0 1`t+'#E6)F.6*~Q�1�0s25460:Ã��qQ�)F.Nj#Ã��w?EFQ�)?2�,<R90�*P1�0�*5=+0�Lr2�,<^#0 R9S�n¬>) Q�Ru2�0 1�)FQ�2�,<^#0T. ,/Q�D#)6'#*-25,/LT,9*V25460a'#)60@tFQ�*50�.£'�)c254F0a8h0 2546' .�=61�'�=!'#*50�.[,<)�_s,98 0 2vQ�R$n Z G�j�j � \rn �0 2! � �� �rZ���� \r®{z � 2�460 )��>Q�L L '�1M. ,9)6D�25' �K'?*-0�)|t6R9Q�2-2 Z G�j#Ã#m�\O�(,<Ws25460d8:' . 0�R3,/*V=61�'�=!0 1�R<S£*-=!0�L ,uxF0�.£25460�)�25460B*-0�w#EF0 )FL 0" �*546'�E6R/.Vt!0`,9)F. 0�=+0�)F. 0�)#2KQ�)+.v. ,/*f2�15,9t6E 2�0�.]Q#*(Q7E6)6,<Wg'�1�8¤,<) Z k6�9G�\On�&(R<0�Q�1�R<S#��Wg'#1('#E61(8:' . 0�R+I�034FQ�^#03QJIK4F'�R90`*�Q�8:=6R90'�W# � Wg'#1~0�Q�LM4 � n _s,<8 0 2sQ�R�n Z G�j�j � \K=F15'#=+'?*-0725'@. 'HQTw|w�l%=6R9'�2s'�W%$ ��� Z m H � �[kF¯ à H \O�!IK4F0 1�0&$ ,/*`25460VL E687E6R/Q�2�,<^#0. ,/*f2�15,9t6E 2�,<'#):WgE6)FLO25,9'�)V'�W�QJ*-2�Q�)F.6Q�1�.h)6'#158VQ�R$nCoP4F0>150�*-EFRu2M*{Wg1�'�8 2�4603Q�)FQ�R9S *-,/*{'�W�2�460' � Q�150K=61�0�*50 )?2�0�.h,<)]A},9D�E61�0K³Fn¨�03'�t+*-0�15^#0(2�4FQ�2�� Q�L L '�1M. ,9)6D~25'z2�46,9*�. ,/Q�D#)6'#*-25,/L���25460`8h' . 0�RF,/*X=61�'�^|,9.6,<)6DJQzW�Q�,915R9S;D#'|'|.hxF2�'�W�2�4603.6Q�2�Q6n���)V1�0 D?Q�1M.6*25'T2�460h2�Q�,9R{t!0 4FQ�^|,9'�1s'�W(_d0OU ,9L 'N&(,u2fSa'#Y '�)F0;0 U?2�150�8:0�*���QHj?Ã���=!'#*-250�15,9'�1s,9)#2�0 1�^�Q�R}Wg'�1s2�460V*-4FQ�=+0:=FQ�1�Q�8:0O250�1`�H,9*Z �>kF¯<G � jF® �>k6¯9G ³?²#\O��IK46,/LM4TL '�1�150�*-=!'�)+.6*(25'VQh¨B0�,<t6EFR<R�W�Q�8:,<R9Sv'�W}. ,/*-251�,<t6E625,9'�)F*�n
�`)60z'�W}2�46078:'?*f23,98:=+'#1-2MQ�)?2`w?E60�*f2�,<'#)F*>Q�t!'�E623'�Y�'�)607.FQ�,9R<SH8:Q�U ,<8VQV,/*K2�'v. 0O2�0 1�8:,<)607,uW{2�460 1�0J,/*3Q]. 0�LO1�0�Q#*-,9)6D251�0 )+.�n7A6'#132�46,/*s=6E61�=+'?*-0#�!I�0:LO'�)+*-,/. 0 1�0�.NQdÀ ¼Mº8M^Q P�M�·XP#¼O·)(�M�»+*^Q M^Zh¹�¶�» Z ¨�0�*-2JQ�)F.��3Q�1�1�,9*5'�)��}G�j#j?Â�\KÄ C_ Wg'#1s� � n
Time-Varying Models for Extreme Values 5
Time
ozon
e
1990 1992 1994 1996 1998 2000 2002
0.0
0.1
0.2
0.3
95%5%
Fig. 3. Daily maxima of ozone concentrations. 95% and 5% quantiles for time-varying model.
Lag
AC
F
(a)
0 20 40 60 80 100
0.0
0.4
0.8
(b)
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−2 0 2
−2
02
4
Fig. 4. Diagnostic for the time-varying model fitted to the data observed at Merced between 1990 and 2002. (a)Autocorrelation function of
��� � ������ � (b) qq-plot of � ����� ��� � ������ � � .
6 G. Huerta and B. SansooP46,/*>*50�LO'#)F.H'�1M. 0 1P=!'�R9S?)F'�8:,9Q�R�,9*K.60OxF)60�.Tt?S]25460z0�w#E+Q�25,9'�)+* �� � � � � �&% �� � � � � ��� ��� � ��� ��� �� � � � � ��� ��� �IK,<254:254F0>2fS|=6,/L Q�RF)6'�1�8VQ�RF. ,/*-251�,<t6E625,9'�)]Q�*�*5E68:= 25,9'�)F*{Wg'�1{2�460>0 1�1�'�1�250�158V*�nC��)]Ä �_ )6'�2MQ�25,9'�)h2�46,9*�8:'|.60 RF,/*XIK1�,<2-250�)
" #� � Z G�®�k#\r� $ #� � Z � � ®!� � \r�� � � � G Gk G � ®
% �s� )[Z k6® \~Q�)F.D+ #� � Z � � ®�� � \ � )�Z k6®.- � \rn � � ,/*~25460hR90 ^�0�R{Q�)F.�� � ��� 1�0 =F150�*-0�)#2M*3,9)FL 150�8h0�)?2�Q�R}D�1�'�IP254NWg'�1~254F0*50 1�,<0�*K� � n(¨�07LO'#)F*5,9. 0�150�.HQV.6,9*�LO'#E6)?2>W�Q�LO25'�1>Q�=6=61�'#Q#LM4v2�'V8h' . 0�R�2�460J0 ^#'�R9E 25,9'�)@^�Q�1�,/Q�)FL 0�- � nP¬ . ,/*5L '�E6)?2KW�Q�LO25'#18:0�Q#*-E61�0�*�254F0sQ�8:'�E6)?2P'�W�,9) Wg'�1�8VQ�2�,<'#)vR9'#*�* �|t!0O2fI�0 0 )v25,98:0 � �bG~Q�)F.V2�,<8:0 � �|Wg'�1�254F0s*-2�Q�250~=FQ�1MQ�8:0 250 1M*X'�W}QhÄ C_Z ¨�0�*-27Q�)+.��`Q�1�15,/*5'�)���G�j#j? �C&(46=�n]³|\rn]A6'#1s25460H.6Q�,9R<SN8:Q�U ,<8VQT'�WP'�Y�'�)60VQ�)+.dWg'#1JQa. ,9*�LO'#E6)?2sW�Q�Lr2�'�1z'�WPkF¯ j?à ��I�0'�t62�Q�,9)60�.hQs=!'#*-250�15,9'�1{0�*f2�,<8VQ�250PWg'�1 � � 254FQ�2X150�*-0�87t6R90�*}2�460>R90 ^�0�R60�*-25,98VQ�2�0�.hIK,u2�4:25460>xF1�*-2X'�1M. 0�1{=+'#R<S|)6'#8h,/Q�RF8h' . 0�R�n�>'�I�0 ^#0 1���2�460T0�*-25,98:Q�250�.�,9)FLO1�0 8:0 )?2MQ�R�D#15'�IP2�4b,9*h=61MQ�LO25,/L Q�R<R9S�L '�)F*-2�Q�)?2;,9)b25,98:0HIK,<254£Qd^�Q�R9E60HL R<'?*-0v25'�k6neoP46,9*,98h=FR<,90�*P2�4FQ�2K2�460 1�0s,9*>R<,<2-2�R<0z0�^?,/. 0�)FLO0s'�W{*5,<D#)6,ux!L Q�)?2>*546'�152>LM4FQ�)6D#0�*�,9)TR90 ^#0 R$noC'dQ�*�*-0�*5*~2�460v*5,<D#)6,ux!L Q�)+LO0V'�W>Q@R9'�)6Da1MQ�)FD�0:251�0 )+.��CI�0vL '�)F*5,9.60 1�0�.�Qa150�D�1�0�*�*-,9'�)B8:'|.60 RXWg'#1J� � ��IK,u2�4�Q@25,98:0^�Q�1�S|,<)6Dh,9)#2�0 1MLO0�= 2>t6E 2>IK,<254aQ:LO'#)F*f2MQ�)?2>*-R9'�=!0� � � $ � ��� Z � ���� \��&% �
$ � � $ � ��� �,+ �IK460�150% � � )�Z k6® \r��+ � � )�Z k6®!| \r®X|N¦§k6� % � Q�)F.;+ � Q�1�0JQ�*�*-EF8h0�.@,<)+. 0 =!0 )F.60 )?2`Q�)+.��� � Z G��`\ Z� ����� � � \K*5'9�1�0 =61�0�*50 )?2�*3254F0hLM4FQ�)6D�0hR90 ^�0�RC=!0 1sE6)F,u2z'�W(25,98:0�n7oP46,/*~)60 I�8h' . 0�R{'�)B� � ,9*s)6'�2sQTÄ �_ *5'H25460VA�A�±P�NQ�R9D�'#15,<25468. '|0�*3)6'�2sQ�=6=FR<Sa. ,9150�Lr2�R<SH25'H'�t 2MQ�,9)d=+'?*f2�0 1�,<'#1>,9) Wg0 1�0 )FL 0�*3'�)D�[Q�)F. $ � n �3'�I(0�^�0 1��6)6'�25,/LO07254FQ�2~LO'#)F. ,<25,9'�)FQ�R�'#) $ � �� � � $ � . 0 xF)60�*sQH*f2MQ�)F.6Q�1�.d150�D�1�0�*�*-,9'�)a8:' . 0 R�IK,<254eLO'#)F*f2MQ�)?2~*5R9'�=!0hQ�)F.dY�0 1�'],<)?2�0 1MLO0 =62�n7AF'�1sQv)6'#) l%,<) Wg'#158VQ�25,9^�0=61�,<'#1�� F Z �C\�� G#��25460HWgE6R9RPL '�)F.6,u2�,<'#)FQ�R(Wg'�1\�§Wg'#R<R9'�I>*7QN)6'�1�8VQ�RP. ,9*-251�,9t6E 25,9'�)�n�AFE61-2�460 1�8:'�1�0��XL '�)F. ,<25,9'�)+Q�R�'#)&�X�� � �6� Z � ���� \(Wg'�R9R<'�I>*PQ;xF1M*f2K'#1�. 0�1P=+'#R<S|)6'#8h,/Q�R�Ä �_ Q�)F.HQ�2K2546,/*KLO'#)F. ,<25,9'�)FQ�R�*f2�0 =��+A}A�±K�HQ�=6=6R9,90�*�25']*�Q�8:=6R90 $ � n��)@254F07L Q#*-0z'�W}2�4607.6Q�,<R9Sv8VQ�U|,98VQ:'�W{'�Y�'�)60#�FA},<D#E61�0;Ã:*-4F'�I>*P254607=+'?*f2�0 1�,<'#1>. ,9*-251�,9t6E 25,9'�)a'�W�25460J=+Q�1MQ�8:0O2�0 10�Xn�%23,/*3L R<0�Q�1K2�4FQ�2`25460�150J,/*>=FR<0�)#2fSH'�W{=!'#*-250 1�,9'�1>8VQ�*�*Kt!0 R9'�I Y 0 1�'Fn���)aW�Q#Lr2��F25460;=+'?*f2�0 1�,<'#1K=61�'�tFQ�t6,<R9,<2fS]254+Q�2a�[©ik:,9*k6¯¿Â�j#k#m7Q�)F.@*5';2�460 1�0s,9*3*-'#8h0sR9'�)6D�l�2�0 1�8¤,<)+. ,9L�Q�2�,<'#)v2�4FQ�23.6Q�,<R9Sv8VQ�U|,98VQh'�Y '#)60~R<0�^�0 R/*P4FQ�^#0~. 0�LO1�0�Q#*-0�.�n¬3*KQ�)aQ�.6. ,<25,9'�)+Q�R�0OU6Q�8h=FR<0#� I(0zLO'#)F*5,9. 0�1K8VQ�U ,<8VQh8:'�)?254FR<SH1�Q�,<) W�Q�R<R�^�Q�R<E60�*�Wg15'#8��?Q�)|EFQ�1�SaG�j#²6G`25':�>'�^#0 8;t+0�1G�j#j�j:2MQ y 0 )NQ�2`2�460h_NQ�,/w#EF0O2��� QH*f2MQ�25,9'�)NR<' L Q�250�.NQ�2`25460]�|,98��'�)�±('#R��� ^�Q�1`¬>,915=!'�152~)F0�Q�1J&�Q�1MQ�L�Q�*�� � 0�)60 Y�E60 R/Q6n;&('#R<0�*0O2aQ�R$n Z m�k#k��?\V=61�0�*50 )?2TQ�)iQ�)+Q�R9S|*5,/*]'�W725460�*-0�.6Q�2MQ6�3IK4F,9LM4iQ�150B^�0�15S£150�R<0�^�Q�)?2T. EF0N2�'[2�460�L�Q�2MQ�*-251�'�=646,/La0�^�0 )?2M*254+Q�2H' L�LOE61�150�.£,<) 2�460N)F'�1525460�15)§L '#Q�*-2]'�W � 0�)60 Y�E60 R/Qb,9)iÄ`0�LO0�87t!0 1�G�j#j�j6n oP460B1�Q�,<) W�Q�R<R`^�Q�R9E60�*HQ�1�0N*-4F'�IK)£'#)A},<D#E61�0H² Z Q#\On��%2:4FQ#*7t!0 0�)c*5E6D�D#0�*-250�.b254FQ�2;2�460T_d'�)?2546R9S,�>'#1-2�4c¬�25R/Q�)?2�,9L �~*5L ,<R9R9Q�25,9'�) Z �3¬ �s\7��)F. 0 U[L Q�)[460 R9=0OU =6R/Q�,9)d1MQ�,9) W�Q�R9R}=6150�LO,9=6,<2�Q�2�,<'#)F*~Q�LO1�'#*�*K2�460:¬�25R/Q�)?25,/L �~L 0�Q�)BQ�)F.dL '�E6R/.NQ�R9*5']0 U|=FR9Q�,<)d1MQ�,9) W�Q�R<R}=6460�)6'�8:0 )+Q])60�Q�1�t?S254F0 � 0 )60�Y E60�R9Q�)T¬�2�R9Q�)#2�,9LzL '#Q#*f2 Z =!0 1M*-'#)FQ�R�L '�8:87EF)6,9L�Q�2�,<'#)vWg1�'�8 �0 R9S *>�~E60 )6)F,/\OnX±�Q#*-0�.v'#)v2�46,/*KI(0zL '�)F*5,9.60 1P254F0�3¬ �¤,<)+. 0OUNQ#*zQHL '�^�Q�15,/Q�2�0�nJoP460:�`¬ � ,9)F. 0OUB,9*`254F0]. , ��0 1�0 )+LO0h'�W()F'�1�8:Q�R<,9Y 0�.N*50�QHR<0�^�0�R�Q�,<1s=61�0�*�*5E6150�*3t!0O2fI�0 0�)Á�'�)?2MQ]Ä30�R<D?Q�.6QF��¬>Y '#150�*3Q�)F. �K0�S y�� Q�^|, y ���fL 0 R/Q�)F.�ns_a'#150;,<) Wg'#158VQ�25,9'�)d'#)a25460 �3¬ � ,<)F.60OUd,/*`Q�^�Q�,9R/Q�t6R90zWg1�'�8 254F0 � =jlk�� g!kmj^r ��i(gpe4k�i x"! q w j#�Wqng!tfuWo�v3uTk3j4k iWtmj imvWq4k��lvFn�¬c25,98h0>*50 1�,90�*}=6R9'�2X'�W+254600�`¬ ��,9)F. 0 U7,9*�*-46'�IK);,<)VA},9D�E61�0² Z t!\rnXA61�'�8�Qz^|,9*5EFQ�RFLO'#8h=+Q�1�,9*5'�)h'�W�A},<D#E61�03² Z Q?\%l Z t+\O��2�460 1�0>,/*X)6'z'#t?^|,9'�EF*X150�R9Q�25,9'�)F*546,9=ht!0O2fI�0 0�):25460a�3¬ �i,<)+. 0OU^�Q�R9E60�*KQ�)F.T8VQ�U ,<8VQh1MQ�,9) W�Q�R9R�^�Q�R<EF0�*>Q�2K_NQ�,/w#EF0O2��� QhQ�,915=!'�152�n¨�0JLO'�)+*-,/. 0 1�0�.TQ:8:' . 0 R�IK4F0 1�0%$ � ����� ]Z � � ® � ®-��\r� $ � ,/*P25460z1MQ�,9) W�Q�R<R�^�Q�R<E60JQ�23_dQ�,/w?E60O2��� Q;*-2�Q�25,9'�)aQ�2K2�,<8:0 �Q�)+.v� � Wg'�R9R<'�I>*KQ:. S|)FQ�8h,/Ls1�0 D�1�0�*�*5,<'#)]IK,u2�4T25460z8:'�)?2�46R<S2�3¬ � ,<)F.60OU � � �FQ�*�2�460z'�)FR<SvL '�^�Q�1�,9Q�250 �� � � $ � ��� � � � ��% �
$ � � $ � ��� �,+ � �� � � � � ��� �,+ � � ¯oP460~EF*5EFQ�R�Q�*�*5E68:= 25,9'�)F*(Wg'#1h% � Q�)F.:+ � � �X }��G#®�m7Q�1�0`8VQ�.60~Q�)F.V2�460s0 ^�'#R<E625,9'�)]^�Q�1�,/Q�)FL 0�*�Q�1�0`. 0 xF)60�.H^|,9QhQ;E6)6,/w#EF0. ,/*5L '�E6)?2KW�Q#Lr25'#1�nPA6'#13Q]. ,/*�LO'�EF)#2KW�Q#Lr2�'�1>'�W{kFn j � �FA},9D�E61�0J² Z Q#\K*546'�I>*�254F07=!'#*-250 1�,9'�1K8:0�Q�)a'�W�� � '�^#0 15l$,98:=+'?*-0�.V2�'
Time-Varying Models for Extreme Values 7
β
Den
sity
−4e−05 −2e−05 0e+00 2e−05
010
000
2000
030
000
Fig. 5. Posterior distribution of�
for ozone data.
(a)
Time
1960 1970 1980 1990 2000
040
8012
0
rainfallµt
(b)
Time
NAO index
1960 1970 1980 1990 2000
−4
−2
02
4
Fig. 6. (a) Maiquetıa monthly observations from 1961 and posterior mean of ��� . (b) Monthly values of NAO index from1961.
8 G. Huerta and B. Sanso
time
β t
1960 1970 1980 1990
0.0
0.5
1.0
Posterior median90% probability
Fig. 7. Posterior median of dynamic regression coefficient� � and 90% probability bands
254F0;1MQ�,9) W�Q�R9R�.6Q�2MQ6ns¨B0h'�t+*-0�15^#07Q�)d,9)FL 150�Q�*50J254+Q�2s,9*~LO'#8h=+Q�25,9t6R90;IK,<254N25460;W�Q�LO2`2�4FQ�2zQ])|E68;t+0�1s'�WXR/Q�1�D�078VQ�U|,98VQQ�1�0HL R<E+*f2�0 1�0�.�,9)c25460TR9Q#*f2V. 0�L Q�.60�n �>Q�,<) W�Q�R<RP,9) �3'�1525460�15) � 0 )60�Y E60�R9QB,9*:QB^�0�15Sb*50�Q�*5'�)+Q�R�=6460 )F'�8:0 )6'#)��XIK,<254£Q. 1�SNQ�)+.�Q@I�0O27*50�Q#*-'#)d2�4FQ�27Q�150:EF*5EFQ�R9R9SN^�0�15SB. ,9*-25,9)6D#E6,9*54FQ�t6R<0#nV¨B0HLO'�)+*-,/. 0 1�0�.N254F0V)60 0�.B'�WPQ#.6. ,9)6DaQa*-0�Q�*5'�)FQ�RLO'#8:=+'#)60 )?2~25'T25460:8:' . 0 R{tFE 2JI(0;Wg'�EF)F.N)F'H=!0 1�,<' . ,/L Q�R�pFEFLO25EFQ�25,9'�)F*~,9)N2�460V*-0�15,90�*~'�W�� � 25'a*5E6=6=!'�152`2�46,/*zLOR/Q�,98Tn¬>R/*5'F� I�0sL�Q�R/LOE6R/Q�2�0�.v25460z. ,���0 1�0 )FL 0�*Pt!0O2fI�0 0 )H254F0z8h0�. ,/Q�)T'�WC� � Q�)F.v254F0JLO'�1�1�0�*5=+'#)F. ,9)6D7'#tF*-0�15^�Q�25,9'�)HQ�)F.HI(0z. ,/.)6'�2`'�t+*-0�15^#07Q�)|S@=+0�15,9' . ,/LO,<25,90�*�n~A},9D�E61�0:Â]*-46'�I>*3254F0;=!'#*-250�15,9'�1`8h0�. ,/Q�)d'�W � � Q�)+.Nj�k �¤=F15'#tFQ�t6,9R9,u2fS@,9)#2�0 1�^�Q�R/*>Wg'#10�Q#LM4h25,98h0`=+'#,<)?2�n}¨�03'#tF*50 1�^�0�254FQ�2�Q�W¾250 1`G�j|Â�k`25460~LO'|0 �]L ,<0�)#2(,9*(*-,9D�)6,<x+L�Q�)?25R9S7=!'#*5,u2�,<^#0�n{oP46,/*X,9)F. ,/L Q�250�*X254FQ�2X254F0 1�0,/*X0 ^|,/. 0 )+LO0>'�W�Qz*5,9D�)6,<x+L Q�)?2XR<,9)60�Q�1(Q#*5*5' LO,/Q�25,9'�)ht!0O2fI�0 0 ):2�4603. S|)FQ�8:,9L�*X'�W�2�460>8:'�)?254FR<Sh8VQ�U ,<8VQzQ�)F.h254603t+0�4FQ�^|,<'#1'�W(25460:�`¬ � ,9)F. 0OU�n:oP460]*-251�0 )6D�254N'�W�*5EFLM4eQ�*�*-' LO,/Q�2�,<'#)NLM4FQ�)6D�0�*`IK,<254B25,98:0�nVoP460:=!'#*-250 1�,9'�1z. ,/*f2�15,9t6E 2�,<'#)B'�W�254F0*54FQ�=!0s=FQ�1MQ�8:0O2�0 1P� Z )6'�23*-46'�IK)!\�,/*K8:'#*-25R9SvL '�)FL 0 )?251MQ�2�0�.]t!0O2fI�0 0�)en �:Q�)+.Bn¿Â|n � 0�15SVWg0�I *�Q�8:=6R90�*�Wg0�R<R�'�E 2M*-,/. 0s254F0,9)#2�0 1�^�Q�R Z kF�<G�\rn{oP46,/*(,/*PQ�)H,<)F.6,9L�Q�25,9'�)H254+Q�2�25460z. ,/*f2�15,9t6E 2�,<'#)v'�W�254F0`8VQ�U ,98VQ7,/*PA61�¥0�LM460 2�Q�)F.V2�4FQ�2K,<2P,/*�E6)|t+'#E6)F. 0�.Q�t!'�^�0#n3. GEV distribution with parameters varying in space and time¨�0PL�Q�)h^�0�15Ss)FQ�25E61MQ�R9R<Sz0 U|250 )+.J2�460P8h' . 0�R =61�0�*50 )?250�.J,9)72�460P=6150�^|,<'#EF*�*-0�Lr25,9'�)725'z.6Q�2�Q`,<):*-=+Q�LO0PQ�)+.J2�,<8:0�n�¨�0PtFQ�*50'�EF1�Q�=6=F15'?Q�LM4h,<)V=61�' LO0�*5*XL '�)|^�'#R<E625,9'�)F*{Q#*{,<) �3,<D?. '�) Z m�k#k#m�\On���):W�Q#Lr2��?=61�' LO0�*�*XL '�)|^�'#R<E 2�,<'#)F*�=61�'�^|,9. 0>QzL '�)|^�0�)6,90 )?2R9,<)60�Q�1`150�=6150�*-0�)?2�Q�2�,<'#)@'�W(�sQ�E+*5*5,9Q�)a=61�'|L 0�*�*-0�* n`&('#)F*5,9. 0�1>2�4FQ�2sQ�2>2�,<8:0 � I(0J4+Q�^�07QV^#0�LO25'�1 � � � Z � � � � ®�¯ ¯�¯O® ����� � � \��Wg'�1KIK4F,9LM4@'�t+*-0�15^�Q�2�,<'#)F*�I(0�150~1�0�LO'#1�.60�.vQ�2K25460J*5,<250�*'� � ® ¯ ¯�¯ ®�� ��� n{¬�*-=+Q�LO0 l�2�,<8:0~Wg'�1�87EFR9Q�25,9'�)HWg'�1 � � ,9*K. 0 xF)60�.TQ#* �
� � � � �� � �&% �� � � � � ��� ��� �IK460�150� � ,/*KQ � � 1��T8VQ�251�,uUvD#,<^#0 )Ht?S � ���� �'/ Z � � �D+ � \r® � � G�®�¯ ¯�¯ ®! �� IK460�150�/],/*KQhD�,9^�0�) y 0�15)F0 R$n�oP4F,9*P8:' . 0 R,/*>,9)FLOR9EF. 0�.@IK,u2�46,<)@2�4607L R9Q#*5*K'�W�Ä �_N*>EF*50�.@,9)a=61�0 ^|,<'#EF*K*50�Lr2�,<'#)F*PWg'#1>250�8:=+'#1�Q�R�8:'|.60 R9,<)6D+n(¨�07Q#*5*5E68:0s254FQ�2>254F0'�t+*-0�15^�Q�2�,<'#)@0 1�15'#1s% �n� )�Z k6® � �� � � � \>Q�)+.@25460;0 ^#'�R9E 25,9'�)a0�151�'�1 � � � )�Z k6® � �� ��� \r�!Wg'�1 � ��G#® ¯�¯ ¯O®X Bn3oP4F07=61�,<'#1>Wg'#1254F0J*f2MQ�250s^#0�Lr2�'�1 � � ,9* � � ��)[Z k6® � �� ��� \On¬3*>8h0�)?25,9'�)60�.TQ�t+'�^#0��|2�460J0 R90 8:0�)#2M*s/ Z�� �6+ � \KQ�150z. 0 xF)60�.@t|STQV*-8:'|'�2�46,<)FD y 0 1�)60�R�IK460 1�0*+ � ® ¯ ¯�¯O® + � Q�1�0s254F0*5=FQ�2�,9Q�R�*-,<250�*K'�1 y )F'�2�*>IK460 1�0s25460 y 0 1�)60�R9*KQ�150JL 0 )?250�150�.�n��|,9)FL 0s25460J0�151�'�1P2�0 1�8V*>Q�1�0s)6'�2`*5=FQ�25,/Q�R9R<STLO'�1�1�0 R/Q�250�.��|254F0*5=FQ�2�,9Q�RP.60 =!0 )F. 0�)FLOS['�W>2�46,9*:8:' . 0�R�,/*VLO'#8:=6R<0 250�R<S�. 0 xF)60�.[t|S���n£�|'�8:0T150�LO'�8:8:0 )+. 0�. y 0 1�)60�R9*hQ�1�0��>¶^[|ÀMÀr¹�¶^Q/ Z ��\ � 0 U|=��+� H H � H H � �m� ��� ��� � (4MWQ!¼ QF¸$¹�¶�» / Z ��\ � 0OU =!�F� H H � H H �� �"� �<�`Wg'�1a*5'�8:0 �¤¦ kcQ�)F. Y ($#6¼ ·r¹�ºM¶�»0/ Z ��\ �
Time-Varying Models for Extreme Values 9� GK� H H � H H � �� ��� � ��� �����$�FWg'�13Q:D#,<^#0 )��?n�Ä3'#,<)FD]QhL '�)|^�'#R<E625,9'�)@Q�2 + � ® ¯�¯ ¯O®!+ � =61�'�^|,9.60�*KQ;xF)6,<2507Q�=6=61�'�U ,<8VQ�2�,<'#)]2�'254F0]LO'�)?2�,<)|E6'#EF*zLO'#)?^#'�R9E 25,9'�)�n:AF'�1s2�460],9)#2�0 D#1�Q�R�L '�)|^�'#R<E625,9'�)N,<27,/*s=+'?*5*5,9t6R<0:2�'@xF)F.eQ�)�0�w?E6,9^�Q�R90 )FL 0Vt+0 2fI(0�0 )�254F0y 0�15)F0 R/*�Q�)F.H25460JL '�^�Q�1�,9Q�)FLO03WgE6)FLr2�,<'#)a'�WC25460z=61�' LO0�*5*�nX�|0 0�_s0 1�) Z m�k#k�k?\{Wg'#1>8:'�1�0~. 0 2�Q�,<R/* n��)s'�1M. 0 1�2�'K8:0 1�D�0�254F0�*-=+Q�25,/Q�R�Ä �_ IK,<254s2�460�Q�)FQ�R9S *-,/*�'�W?25460�0OU|251�0 8:0{^�Q�R9E60�*���Q�*�*-E68:0}254+Q�2 �� � � �����vZ � � � ® � ®-��\Wg'�1 �~��G#® ¯�¯ ¯O®��[Q�)F. � ��G�® ¯�¯ ¯O®X BnXoP4F0J. ,9*-251�,9t6E 25,9'�)HWgE6)FLO25,9'�)@Wg'#1 �� � � ,9*� � � Z��� � � �-� � � ®5� ® � \�� 0OU =V�3���%G(�b�V� � � � �B� � �� �>� ���-�5 ¡ ¢ ¯A6'#1X0�Q#LM4 � R90O2�� � � Z � � � � ®-� � � � ® ¯�¯ ¯ ®5��� � � \��$nC¨�0>)6'�I£. 0Ox+)603QzÄ �_ '�):� � EF*-,9)6Ds2�460K=61�'|L 0�*�*{LO'#)?^#'�R9E 25,9'�):Q�=F=615'?Q�LM4. 0�*5L 15,9t+0�.v=F150�^?,9'�E+*-R9S�n � � � � � $ � �&% � �
$ � � $ � ��� ��� �IK460�150v25460T*f2MQ�250H^�0�Lr25'#1 $ � � Z $ � � � ® ¯ ¯�¯ ® $ � � � \ � ��% � � Z % � � � ®�¯ ¯�¯ ®X% � � � \ � � � � � Z � � � � ® ¯�¯ ¯�® � � � � \ � n�¬`.6. ,<25,9'�)FQ�R<R9S���% ��)�Z k6® � �� � ��\zQ�)+. � �*� )[Z k6® � �� ��� \OnT`*-,9)6DdQd�sQ�EF*5*5,/Q�) y 0�15)60�RXI�0V4FQ�^�0h254+Q�2J2�460���1 �e8VQ�2�15,<U � �{,/*JD�,9^�0�)Bt|S� � � �§0OU = Z ��� H H � � � + �mH H � �m�\CIK460 1�0 � � ,9*{25460K=!'#*5,u2�,<'#);'�W�*-2�Q�25,9'�)\ 8�W+ � ,/*}254603L 0 )?250�1{'�W+25460���2�4 y 0 1�)60 R��h� G#® ¯ ¯�¯O® �Q�)+.��V,/*>Qh1MQ�)6D#0`=FQ�1�Q�8:0O250�1�nA6'#1K254F07=FQ�1�Q�8h0 250�1�*P,9)@254607xF1M*f2`*-2�Q�D�0z'�W}2�46078:' . 0�R�I�07*-=!0�L ,uWgSH254F07=61�,<'#13. ,/*f2�15,9t6E 2�,<'#)F*'� Z � \ � � )�Z � � ®�� � \Q�)+. � Z ��\ � )�Z � ®�� \OnvA6'#1s25460V=FQ�1�Q�8:0O250�1�*~,9)�25460]*50�L '�)F.�*f2MQ�D#0�� $ � � )�Z k6® � �� � � \���G� � �� � ��� ; �!Z� � ®!� � \��G� � �� ����� ; ��Z� � ®!� � \�Q�)F.]G � �� ����� ; ��Z� � ®!� � \rn�oP460�R<'#D�l%R<, y 0�R<,946'|' .~Wg'�1}254F0�xF1M*-2-l�*f2MQ�D�0�8h' . 0�R|=FQ�1MQ�8:0O2�0 1M*,/*
� Z $ \ � � ��;R9'�D � � °���� �
�� � �
� G(��� � � � � ��� � �� �>� ���5�5 ¡� � G(� G� � °���� �
�� � �
R<'#D � G(�b� � � � � ��� � �� �K� ¡Á�'#*-250�15,9'�1K,9) Wg0�150�)FLO07Q�)F.d*5,<8;E6R/Q�25,9'�)@Wg'#132�46,/*`*5=FQ�L 0Ol$25,98:0J8:'|.60 RCWg'�R9R<'�I>*`Q]*-,98:,<R/Q�1~*f2�15E+Lr25EF1507Q#*>Wg'#1>2�460h�s� �25,98:0Ol%^�Q�1�S?,9)6Da8:' . 0 R$n@oP460VWgE6R9R�L '�)F.6,u2�,<'#)FQ�R{Wg'#1J� � � � � Q�)F.e�aQ�1�0]*5Q�8:=6R<0�.�0�Q�LM4eIK,<254[Qa_d0O251�'�=!'�R9,/*fl �3Q#*f2�,<)6D?**-250 =�n{¨B0s. 1MQ�I�25460z*-2�Q�2�0`^#0�LO25'�1 $ � IK,u2�4TAF'�1�I�Q�1�.VA},9Ru2�0 1�,<)FDh±PQ�L y IPQ�1M.]�|,987E6R/Q�2�,<'#)�n{oP4603WgEFR<R�L '�)F. ,<25,9'�)+Q�R/*�'�W � �� �� �� � � �� Q�1�0h*�Q�8:=6R90�.NIK,u2�4N��)|^�0�1�*50h�sQ�8:8:QT. ,/*f2�15,9t6E 2�,<'#)F*�n7o�1MQ�.6,u2�,<'#)FQ�R9R<S#�!Wg'�1s=61�'|L 0�*�*`L '�)|^�'#R<E625,9'�)F*�� y 0 1�)60�R9*~Q�1�0LO0�)?250 1�0�.aQ�15'#E6)F.T=!'�,9)#2M*3'#)dQV1�0 D�EFR9Q�1>D�1�,9.@Q�)F.@254F0J1�Q�)6D�0z=+Q�1MQ�8:0O2�0 1>,/*>xFU|0�.aQ�*��V�! #"C�!IK4F0 1�0zIK460 1�0$"B,9*3254F0. ,/*f2MQ�)FL 0st+0 2fI(0�0 ) y 0�15)60�R�Q�)F.� ~,9*KQ:L '�)F*-2�Q�)#2�� EF*5EFQ�R9R<Svt+0 2fI(0�0 )eG�´�m7Q�)F.am n¨�0~L '�)F*5,9.60 1X2�460s=61�'�t6R90 8 '�WC8h' . 0�R<,9)6Dh.6Q�,9R<SV8VQ�U|,98VQJ'�W�'�Y '#)603,9)H_d0OU ,9L ':&(,<2fS]. E61�,<)FDJ254F0`S#0�Q�1~G�j#j�jFnCÄ~Q�,9R<S8VQ�U ,98:Q~Q�1�0�'#t 2�Q�,<)F0�.zWg1�'�8 2�460 �K0�.h¬>E625'�8��Q�25,/L Q~. 0K_a'�)6,<25'#150�'`¬387t6,90 )?2�Q�R Z �>¬`_d¬`\}46'#E61�R<Ss8:0�Q#*-E61�0 8:0�)#2M*C254FQ�2LO'#8:0sWg15'#8 G�jh8:'�)F,u2�'�1�,<)6DV*-2�Q�25,9'�)F*�n�_a'#150z,9) Wg'�1�8VQ�2�,<'#)TQ�t+'#E 2 �>¬3_N¬ Q�)F.T2�460J.6Q�2�QV*50O2�*3*f2�EF. ,90�.@,9)H2�46,/*>=FQ�=+0�1Q�1�0>Q�^�Q�,9R9Q�t6R90(Wg1�'�8�25460` � Ki ! r k g �my w g]r&%�n�¬§*5=FQ�2�,<'�l�2�0 8:=!'�1MQ�R 8:' . 0 RFWg'�1{2�4603Q�LO25EFQ�R646'#E615R9S7'#Y '#)60K'�tF*50 1�^�Q�2�,<'#)F*,/*3. 0�^�0 R9'�=!0�.T,9) �>E60�1-2MQ:0O2`Q�R�n Z m�k#k�³|\rn(oP4+Q�2>=FQ�=+0�1>Wg' LOEF*50�*>'�)@,<)?2�0 1�=+'#R9Q�25,9'�)aQ�)F.@*546'�152K250�158�Wg'�1�0�L�Q�*-25,9)6D:'�W�254F046'#E615R9SJ'#Y '�)F0(R90 ^#0 R/*{Q�)F.;)6'�2X,9):0OU|251�0 8:0�* n}oP460K8:'|.60 R ,/*{tFQ�*50�.;'�);254F0>Q�8:=6R9,u2�EF. 0�*X'�W!.FQ�,9R<Szp+EFLr2�EFQ�2�,<'#)F*{'�W!'�Y�'�)60#n�|EFLM4eQ�8:=6R9,<25EF. 0�*zQ�1�0hQ#*5*5E68:0�.d25'@^�Q�1�Sa,<)e*5=FQ�L 0hQ�)F.d2�,<8:0VQ�)+.B. 0�=+0�)F.B'�)eLO'�^�Q�1�,9Q�250�*>2�4FQ�2JQ�R/*5'H^�Q�1�Sa,9)�*5=FQ�L 0Q�)+.v2�,<8:0#n(��)aA},<D#E61�0 � I(0z=61�0�*50 )?2K2�460 �>¬`_d¬ *-2�Q�2�,<'#)F*KQ�R9'�)FDVIK,u2�4H2�460 y 0 1�)60 R�=!'#*5,<25,9'�)F*>*50 R90�LO250�.v2�']. 0Ox+)60z254F0=61�' LO0�*�*sLO'#)?^#'�R9E 25,9'�)N'�)B25460VR9' L Q�2�,<'#)N=FQ�1�Q�8:0O250�1�*�n;oP4F0h1�0�*5E6R<2�*`254+Q�2JI�0;=61�0�*50 )?2z,9)N2�46,9*s=+Q�=!0 1JQ�1�0htFQ�*50�.B'�)�Q*5,uU|250�0 )e=!'�,9)#2M*zD�1�,/.�nV¨B0]x62-2�0�.�254F0V8h' . 0�RXEF*5,<)FDdQ@)6,<)F0]Q�)F.eQTWg'�EF1z=+'#,<)?2�*JD�1�,9.eQ�)+.N'#tF*-0�15^#0�.B)6'd*5E6tF*-2�Q�)?2�,9Q�R. ,��!0�150�)FLO0�*3,9)a2�460h.6,9Q�D�)6'?*f2�,9L�*3Q�)+.dw?EFQ�)?25,9R9070�*-25,98VQ�2�0�*�n � Q�1�S|,<)FD]254F0;LM46'#,9L 07'�W{2�4607=+Q�1MQ�8:0O2�0 1M*K254+Q�2~.60OxF)60h'�EF1=61�,<'#1�*PQ�R9*5':=615' . E+LO0�.vW�Q�,91�R<S]1�'�tFEF*f2K1�0�*5E6R<2�*�n��)BA},9D�E61�0;jHI�0;*546'�I 25460V.6Q�2�Q]Wg'�1`Wg'�E61z*-2�Q�2�,<'#)F*~Q�)F.d25460h0�*-25,98VQ�250�*~'�WX2�460;=!'#*-250�15,9'�1~8:0�. ,/Q�)+*`'�WX254F072�,<8:0Ol^�Q�1�S|,<)6Da�s� � . ,/*f2�15,9t6E 2�,<'#)F*stFQ#*-0�.d,9)�QTR90�Q�^�0 l$'#)60Ol%'�E62`Q�)FQ�R<S *5,9*�n;�|,9)FL 0;2�460 1�0VQ�1�0;)F'H1�0 =6R9,/L Q�2�0;'#tF*50 1�^�Q�25,9'�)F*`Wg'#1Q�)|S:D�,9^�0�)v.6Q�S#�#,<2�,/*�4FQ�1M.:2�'7,9)#2�0 1�=61�0O2(=F150�LO,/*-0�R<Sh254F0`8:0�Q�)6,9)6D7'�W�25460~8:0�. ,/Q�)TQ�2(25,98h0 � n��>0�^�0 15254F0 R90�*�* �?I(0~0OU =!0�LO2254+Q�232�460:LOE61�^�0JIK,9R9R�*5=6R9,u2`25460:LOR9'�E+.a'�WX=!'�,9)?2�*��!1�'�EFD�46R9S��!,<)d2fI('H4FQ�R9^�0�* n3oC'HQ#*5*50�*>25460h^�Q�R9,9.6,u2fS@'�W{2�46078:' . 0�R���I�0'�t62�Q�,9)60�.h25460~=61�0�. ,/Lr2�,<'#)F*{Wg'#1�0�Q#LM4V'�)603'�W�254F03R9'|L�Q�2�,<'#)F*�Q�W¾250 1(R<0�Q�^|,<)6DJ25460�8�'#E 2��|'#)603Q�2PQs25,98:0�n{A6'#1(0�Q�LM4V*-2�Q�2�,<'#)I�0>=+0�1-Wg'#158:0�.]QJ. ,/Q�D#)6'#*-25,/LKtFQ#*-0�.:'�):254F0 � 2�1�Q�)F*-Wg'�1�8:Q�25,9'�)��#,<)HQ�)FQ�R<'#D�Sz2�'J254F0`Q�)+Q�R9S|*5,/*X=61�0�*50 )?250�.:,<)T�|0�LO25,9'�)vm n
10 G. Huerta and B. Sanso
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.30
19.3
519
.40
19.4
519
.50
19.5
5
longitude
latit
ude
ω1 ω2 ω3 ω4
ω5 ω6 ω7 ω8
ω9 ω10 ω11 ω12
ω13 ω14 ω15 ω16
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAH
CUA
TPN
CHA
TAX
Fig. 8. RAMA stations, kernel and interpolation grid positions. The stations are denoted with a three letters code. Thekernel centers are denoted with ��� . The grid points correspond to the locations used to obtain the values shown inFigure 11. The continuous line corresponds to the boundaries of Mexico City.
¬3*>Wg'�1K2�460JE6)F,<^�Q�1�,/Q�250;L Q�*50sI(0;. ,9.aQ:D#1�Q�=646,/L Q�R�Q�)FQ�R<S *5,9*P'�W�254607Q�E 25' L '�1�150�R9Q�25,9'�)F*PQ�)+.T254607w|w#l%=6R<'�2�*�n�oP4F0J150�*-EFRu2M*Wg'�1`254F07Wg'#E61z*f2MQ�25,9'�)+*`L '�)F*5,9.60 1�0�.a,9)BA},9D�E61�0;jHQ�1�07=61�0�*50 )?250�.a,9)BA},<D#E61�0vG�kFn �`^#0 1MQ�R9R�2�460h.6,9Q�D�)6'?*f2�,9L�*3I�0 1�0JW�Q�,<1�R<SQ�L�LO0�= 2�Q�t6R<0#n���)HQ�.F. ,u2�,<'#)�� I�03)6'�25,/LO0�.v254+Q�2�25460~x62�254FQ�2KEF*-0�.vQ�R<R!254F0s.FQ�2�Q7=61�'|.6EFLO0�.]1�0�*5E6Ru2M*(254FQ�2KQ�150`^�0�15SV*5,<8:,9R9Q�125'h2�46'#*50z'�t 2MQ�,9)60�.TQ�W¾2�0 1>. 0�R<0 25,9)6D]Q�)|S]'#)60z*f2MQ�2�,<'#)�noP460V*5=FQ�2�,9Q�R�Q�)F.N250 8:=!'�1MQ�R�^�Q�1�,9Q�t6,9R<,<2fSd'�W(25460:1�0�*5E6R<2�*s,/*~*546'�IK)�,<)eA},<D#E61�0HG#G�nh¨B0:'�t+*-0�15^#0hQTLOR90�Q�1~D#1�Q#. ,90 )?2,9)FLO1�0�Q#*-,9)6DJ,9)V254F0@�`�{lf�|¨ . ,9150�Lr2�,<'#)�n}oP4F,9*�,/*(L '�)F*5,9*-250�)?2(IK,<254V2�460`1�0�*5E6R<2�*(,<)��>E60�1-2MQJ0 2�Q�R$n Z m�k#k�³|\rn�oP460~.6,<1�0�LO25,9'�)'�W}254F0JD�1MQ�. ,90 )?2>1�0 8VQ�,9)F*P0�*5*50 )?25,/Q�R9R9SvE6)FLM4+Q�)6D#0�.T,<)T2�,<8:0��!t6E 2325460J8:0�. ,9Q�)F*K^�Q�1�SH*5E6tF*-2�Q�)?25,/Q�R9R<SVWg1�'�8 .6Q�SV2�'v.6Q�S#n¬>)i0OU =6R9'�1MQ�25,9'�)�'�Wz25460B=+'?*f2�0 1�,<'#1v. ,/*f2�15,9t6E 2�,<'#)§'�WJ��*546'�I>*:2�4FQ�2H25460�*-4FQ�=+0B,9*HLOR90�Q�15R9S£Q�)60 D?Q�25,9^�0a=+Q�1MQ�8:0O2�0 1��,9)F. ,/L Q�2�,<)FD:254FQ�2>25460;. ,9*-251�,9t6E 25,9'�)at!0 R9'�)FD#*P2�':25460J¨�0 ,9t6E6R9R�W�Q�8:,<R9S�nP¬ j#Ã�� =!'#*-250�15,9'�1K=61�'�t+Q�t6,9R<,<2fS],9)?250 1�^�Q�R�Wg'#1>�:,9*Z �>kF¯ m6G�jF® �>k6¯9G � ²#\On�oP46,9*P,98:=6R9,<0�*�2�4FQ�2P2�460~.60 )F*5,u2fS]x62-2�0�.v25';25460z.6Q�2�QJ4+Q�*PQ;*5E6=6=!'�152�254FQ�2P,/*�t+'#E6)F. 0�.HQ�t+'�^#0�n�A6'#1Q�)|Sv*-2�Q�25,9'�)��|2�460zt!'�E6)F.T,/*P25,98:0Ol%^�Q�1�S?,9)6D:Q�)F.H,/*KD�,9^�0�)Ht|S]� � � � � �� n&('�R90�*30O2sQ�R$n Z G�j�j#j#\P=F15'#=+'?*-07Qv*-E68:8VQ�1�ST'�W{254F0;0 U?2�150�8VQ�RC.60 =!0 )F. 0�)FLO0h'�W{2fI�']1MQ�)F.6'�8 ^�Q�15,/Q�tFR<0�* � Z � ® $;\Kt|S251MQ�)+*fWg'#158VQ�2�,<'#)]2�'h^�Q�15,/Q�t6R90�*��6*�Q�S�� Z�� ® \�IK,<254@E6)F,uWg'#158�8:Q�15D#,<)+Q�R/* n�oP46,/*P,9*KD#,<^#0 )Tt|S� Z �\X�im~� R<'#D�� � Z�� © C® © �\R9'�D�� � Z�� © �\ k � � G�¯oP46,/*K,/*>Q�)aQ�=F=615'�U ,98VQ�25,9'�)H25'�� � Z ¦ H � ¦ �\On�¨�07. 0Ox+)60JQh25,98:0Ol%^�Q�15S|,9)6D:^�0�1�*5,<'#)H'�WC2�46,/*>8:0�Q#*-E61�0~Wg'�13Q:=+Q�,91'�W�R9'|L�Q�2�,<'#)F* �JQ�)+. � � EF*5,9)6Dh25460z=61�0�.6,9LO25,9^�0s=+'?*f2�0 1�,<'#1>. ,/*f2�15,9t6E 2�,<'#)F*P'�W�'�E61K*5=FQ�L 0Ol$25,98:0~8:' . 0�R�nXoP4|EF*
� �rZ C®���®�� � \X� m~� R9'�D� � Z�(� � © }® ��� � # © H b \R<'#D�� � Z�� � � © H b \ k � �iG�¯ Z m#\IK460�150 �(� � � � � � � � Z���� � � \r® �b� �#®�� �KQ�)F. b .60 )6'�250�*:Q�R<RKQ�^�Q�,9R9Q�t6R90H.6Q�2�Q6n[�%W~Q�*5Q�8h=FR<0HWg15'#8 25460 � '#,<)?2V=+'?*f2�0 1�,<'#1. ,/*f2�15,9t6E 2�,<'#)e'�W�25460]=+Q�1MQ�8:0O2�0 1M*`,/*JQ�^�Q�,9R/Q�t6R90��}*5Q�S#��� � ���� � ®-� � ��� ® � � ��� ®! ~� G�® ¯�¯ ¯O®�����2�460 ) Z m�\sL Q�)et+0vQ�=6=61�'�U ,<8VQ�2�0�.t|S�L '�8:=6E 2�,<)FD@25460H=615'#=+'#1-2�,<'#)B'�WK=FQ�,91�* Z�� � ���� � ® � � ���� � # \zR<0�*5*s2�4FQ�)�Qax6U 0�. [Q�)F.b. ,9^?,/. ,9)6D@254F,9*;=61�'�=!'�1525,9'�)�t?SB254F0
Time-Varying Models for Extreme Values 11
AZC
time
ozon
e
0 50 100 150 200 250 300
0.05
0.10
0.15
0.20
0.25 Data
Median
XAL
time
ozon
e
0 50 100 150 200 250 300
0.05
0.10
0.15
TPN
time
ozon
e
0 50 100 150 200 250 300
0.05
0.10
0.15
0.20
TAX
time
ozon
e
0 50 100 150 200 250 3000.
050.
100.
150.
20
Fig. 9. Daily maxima for 1999 (dots) and posterior estimate of the �� � quantile of the GEV distribution at each timepoint. (continuous line). The quantile is based in a leave-one-out analysis. The chosen stations correspond to locationsin the NE, NW, SE and SW respectively, as shown in Figure 8.
AZC
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−3 −2 −1 0 1 2 3
−3
−2
−1
01
23
XAL
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−3 −2 −1 0 1 2 3
−2
−1
01
2
TPN
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−3 −2 −1 0 1 2 3
−3
−2
−1
01
2
TAH
Theoretical Quantiles
Sam
ple
Qua
ntile
s
−3 −2 −1 0 1 2 3
−4
−3
−2
−1
01
2
Fig. 10.� � diagnostics for four of the stations using the posterior predictions based on leaving one station out at a time.
12 G. Huerta and B. Sanso
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 47
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 61
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 63
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 69
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 127
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 191
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 258
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 263
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
−99.3 −99.2 −99.1 −99.0 −98.9
19.2
519
.35
19.4
519
.55
day 270
LAGTAC
EAC
SAG
AZC
TLA XAL
MER
PEDCES
PLA
HAN
UIZBJU
TAX
CUA
TPN
CHA
TAH
Fig. 11. Posterior estimate of the �� quantile of the space-time GEV distribution for a � � � resolution grid. Dotscorrespond to the locations of the stations. The boundary of Mexico City is shown as a reference. The contour linescorrespond to fixed ozone levels in ppm.
Time-Varying Models for Extreme Values 13
u
χ(u)
TACXALTPNTAH
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 12. Dependency between the maxima at AZC and TAC, XAL, TPN and TAH for��� � � � and different values of
�.
=61�'�=!'�1525,9'�)d'�W(L Q#*-0�*~IK460 1�0 � � ���� � © }n7A},<D#E6150TG�mH*546'�I>*>2�460:D�1MQ�=64N'�W � � �.� Z C®��#®�� �9\3Wg'#1 �VL '�1�150�*-=!'�)+. ,<)FDV25'T254F0R9'|L�Q�2�,<'#)d'�W(*-2�Q�25,9'�)�¬���& Q�)F. � �C25'HWg'�EF1`'�25460�1s.6, ��0 1�0 )?2z*f2MQ�25,9'�)+* nzoP460;W�Q�LO2`2�4FQ�2zQ�R9R{LOE61�^�0�*>2�0 )F.d25'TY 0 1�']Wg'#1) LOR9'#*50P25'vG��|*-E6D#D�0�*f2M*�254FQ�2X254F0>8:'|.60 R!,9*�=615' . E+LO,9)6DJQ�*5S|8h=625'�2�,9L>,<)F.60 =!0 )F. 0�)FLO0#n��>0 ^#0 1525460�R<0�*5*���I�0>'#tF*-0�15^#0�254FQ�2�254F0LOEF15^#0�*~4FQ�^�0ht!0 4FQ�^|,9'�1M*32�4FQ�27.60 =!0 )F.e*-251�'�)6D#R<Sa'#)N2�460:R<' L Q�25,9'�)F*s'�W�2�460]*f2MQ�2�,<'#)F* ��Wg'�1JQTIK,9.60h1MQ�)FD�0h'�W# b^�Q�R<E60�* noP46,/*K,9)F. ,/L Q�2�0�*�2�4FQ�2��6,<)@=61MQ�LO25,/LO0#�|25460z8:' . 0 R�,/*>Q�t6R<0~2�']L Q�=625E61�0z. , ��0 1�0 )?2P2fS|=+0�*K'�W}=FQ�,91�IK,9*50z. 0 =!0 )F.60 )FL ,<0�* n4. Discussion¨�074FQ�^#0J=61�0�*50 )?250�.dQVD�0�)60 1MQ�R�Wg1�Q�8:0 I�'�1 y Wg'�1~8:' . 0 R9,<)FDv0OU|2�150�8h07^�Q�R<EF0�*3254FQ�2`0OU 46,9t6,<2sQV25,98h0 l$^�Q�15S|,9)6D]t!0 4FQ�^|,9'�1�noP460V8:' . 0 R/*zQ�1�0:tFQ�*50�.B'�)�2�460VEF*50h'�W(2�460v�s� � .6,9*-251�,<tFE 25,9'�)eQ�)F.eQ#*5*5E68:0h254FQ�2z25460VR9' L Q�2�,<'#)B=+Q�1MQ�8:0O2�0 1s^�Q�1�,<0�*IK,<254H25,98h0zQ#L LO'#1�.6,<)6D725'VQhÄ C_bn6A6'#1(2�460zE6)6,9^�Q�1�,9Q�250sL Q#*-0`I(0s=61�0�*50 )?250�.VWg'#E61K. , ��0 1�0 )?2K8:' . 0 R/*KQ�)F.]2fI(':. ,��!0�150�)?2.6Q�2�Qa*50O2�*�n �`E617Q�)+Q�R9S|*5,/*s150�^�0�Q�R9*~2�4FQ�2z2�460]=61�'�=!'#*50�.e. S|)FQ�8:,9LVQ�=6=61�'#Q�LM4�=61�'�^|,9. 0�*zQTpF0OU ,9t6R90]I�Q�SN'�W>L Q�= 25E61�,9)6D*546'�1523Q�)F.@R9'�)FD]1�Q�)6D�0~2�150�)F.6*3Q#*KI�0 R9RCQ�*>25,98:0Ol%^�Q�15S|,9)6D]. 0�=+0�)F. 0 )+LO,90�*�nPoP4F07Q�=6=F15'?Q�LM4aL Q�)Tt!070�Q�*5,<R9Sv0 U|250 )+. 0�.T2�'Q�L�LO'#E6)?2(Wg'#1>*-=FQ�25,/Q�R�^�Q�1�,9Q�t6,9R<,<2fSVt|S]EF*5,<)6Dh=61�' LO0�*�*PLO'#)?^#'�R9E 25,9'�)+* n�oP460zQ�*�*-EF8h=625,9'�)T'�W}Q:�s� � . ,/*-251�,<t6E625,9'�)HWg'�1P254F0.6Q�2�Q],/*~Q�)NQ�=6=61�'�U|,98VQ�2�,<'#)��Ft6E62~8:0O2�46' .6*~tFQ�*50�.d'�)Nw?EFQ�)#2�,<R90�*`L Q�)dt+0hEF*50�.@2�'TQ#*5*50�*�*P2�460;D#'|'|.6)60�*�*>'�WXx62`'�WX254F08:' . 0 R$n3oP460JE+*-0;'�W�25460:�s� � . ,9*-251�,9t6E 25,9'�)a8VQ y 0�*3,<)6Wg0 1�0 )FL 0zWg'�1~w#E+Q�)?25,9R<0�*3*-251MQ�,9D�4?2-Wg'#15IPQ�1M.��6*5,<)+LO0z2�4607,9)|^�0 1M*50z'�W254F0:L�.|W(,/*sQ�^�Q�,9R9Q�t6R<0;,<)eLOR9'#*50�.dWg'�1�8@nJ¨ ,u2�46,9)�QT_B&�_B& Q�=6=61�'#Q#LM4dQ�R9R}254FQ�2s,/*s)60�0�. 0�.d,/*32�460:0 ^�Q�R9EFQ�25,9'�)N'�W(*5EFLM4,9)?^#0 1M*-0:Wg'#170�Q�LM4 � '�,9)?2;*�Q�8:=6R90]'�WK2�460]=FQ�1�Q�8h0 250�1�*�nd�|'F��8:' . 0 RPLM460�L y ,9)6Dd*-2�Q�2�,9*-25,/L *;Q�1�0]0�Q#*-SN25'NL '�8:=6E 2�0�nNoP4F0Q�*5S|8:= 25'�25,/L~.60 =!0 )F. 0�)FLO07,<)F.6EFLO0�.Ht|SV2�460J8:' . 0 R�)60 0�.6*�25'Vt!0s25460�'�1�0O2�,9L�Q�R9R<SvQ#*5*50�*�*50�.�n{�X8:=6,91�,9L�Q�R�8:0�Q#*-E61�0�*Pt+Q�*50�.'�)@_a'#)#2�0;&�Q�1�R<':Q�=F=615'�U ,98VQ�25,9'�)+*�,9)F. ,/L Q�250s254+Q�2K254F0z8h' . 0�R�,/*�W�Q�,915R9SVpF0 U|,9t6R90�n_dQ�)?Sz=!'#*�*5,<t6R90P0OU|250�)F*5,<'#)F*�Q�1�0(=!'#*�*-,9t6R90�25's8:'|.60 R/*{LO'�)+*-,/. 0 1�0�.7,<)h2�46,9*�=+Q�=!0 1�n}¬`*{Q�):Q�R<250 1�)FQ�25,9^�0�'�1�,<):Q�.F. ,u2�,<'#)25'HQ:25,98:0Ol%^�Q�15S|,9)6D]R9' L Q�2�,<'#)aI(0;L Q�)dL '�)F*5,/. 0 1K2�,<8:0 l$^�Q�1�S|,<)FDv*-4+Q�=!07Q�)F.a*�L Q�R<0#n>oP46,/*>IK,9R<R}=61�'�^?,/. 0z8:'�1�0zpF0 U ,<t6,9R9,u2fSWg'�1}2�460K. 0�*�LO1�,<=625,9'�)7'�W6254F0�0 U|251�0 8VQ�R|t!0 4FQ�^|,9'�1M* n�AF'�1}254F0P*�L Q�R<0�I�0�L�Q�);L '�)F*5,/. 0 1�Q�);0 ^�'#R<E625,9'�)7R9, y 0(25460P'�)F0(=61�'�^|,9.60�.t|ShQ7Ä C_ Q�W¾250�1X2�1�Q�)F*fWg'#158VQ�25,9'�)h25'z2�460>R9'�D7*�L Q�R<0#n���)V2�4603*5=FQ�2�,<'�l�2�0 8:=!'�1MQ�RFL�Q�*50KI(03L�Q�)]L '�)F*5,9.60 1�*�L Q�R<0>Q�)F.V*-4+Q�=!0=FQ�1�Q�8h0 250�1�*}2�4FQ�2�^�Q�15S;IK,u2�4VR<' L�Q�25,9'�)�nC¨�03I�'�E6R/.hR<, y 0>25'z8:0 )?25,9'�):2�4FQ�2PQz*f2MQ�2�,<'#)Vt?S:*-2�Q�2�,<'#)VQ�)FQ�R<S *5,9*{'�W�25460>'#Y '#)608VQ�U ,98:QV.6,9.a)F'�2>=61�'�^|,9. 0zD#15'#E6)F.6*KWg'�13*5EFLM4dQ�)@0 U?2�0 )F*5,9'�)a,9)@25460J=+Q�1525,/LOE6R/Q�13L�Q�*50z'�W�254F07.6Q�2MQ]LO'#)F*5,9. 0�150�.T,<)@2�46,9*=FQ�=+0�1�n
14 G. Huerta and B. SansoA},<)+Q�R9R<Sb,<2V,9*h=!'#*�*-,9t6R90H2�'�L '�)F*5,/. 0 1:8:0O2�46' .6*htFQ�*50�.c'�)[25461�0�*546'�R/.�8:' . 0�R9*�ncoP46,/*h=61�0�*50 )?2�*;254F0@LM4+Q�R9R<0�)6D�0H'�W. 0�Q�R9,<)6D:IK,<254@=!'#*�*-,9t6R90`2�,<8:0 l$^�Q�1�S|,<)FD;'#1KR<' L Q�25,9'�)T. 0 =!0 )F.60 )?2>254F150�*-46'#R9.F* n�¨�0~I�'�EFR9.HR<, y 0~2�'V8h0�)?25,9'�)H254FQ�2KWg'�1P254F0'�Y�'�)60~.6Q�2MQJI�03xF2-250�.]2�1�Q#. ,<25,9'�)FQ�RF2�46150�*-4F'�R/.V8:'|.60 R/*�*-2�Q�25,9'�)]t|S]*-2�Q�25,9'�)�n�¨�0`'#tF*-0�15^#0�.��#Wg'�1P*5'�8:0~*-2�Q�25,9'�)F*��|*f2�15'#)6D*50 )F*5,u2�,<^|,<2fS�25'a2�460TLM4F'�,/LO0]'�WK25460v25461�0�*546'�R/.�nN¬3R9*5'NI�0]'#tF*-0�15^#0�.B254+Q�2:'�= 2�,<8VQ�R�LM46'�,/LO0�*J'�WK25461�0�*546'�R/.6*;*-E6D#D�0�*f2�0�.^�Q�R9E60�*�2�4FQ�2>^�Q�15,90�.T*-EFtF*f2MQ�)?25,/Q�R9R9SVWg'�1K)60�Q�1�t|S]*f2MQ�25,9'�)+* nAcknowledgmentsoP460hQ�E 2�46'�1M*>IK,9R<R}R9, y 0J25'TQ�L y )6'�IKR90�. D#0s25460;460 R9=N'�W �0�R<S *`�~EF0 )6)6,�'#)a25460h. ,/*5L EF*5*5,9'�)a'�W{2�460 � 0 )60�Y E60�R9Q�)d1�Q�,<)6W�Q�R9R.6Q�2�Q6nB¨B0TQ�R/*-'d254FQ�) y Q�)[Q�)6'�)|S|8:'�EF*7150 Wg0 1�0 0]IK46'?*-0HLO'�8:8:0 )?2M*7R90�Q#.e25'�Q�)�,98:=615'�^#0�.e^#0 1M*-,9'�)e'�WK2�460T'�1�,<D#,<)FQ�R=FQ�=+0�1�n{oP460J*50�L '�)F.TQ�E 2�46'�1KIPQ�*�=FQ�1-2�,9Q�R<R9Sv*5E6=6=!'�15250�.vt|S2�~� A�D#1�Q�)?2PÄ~_N�?l�k#Ã�k�³ � à G#nReferences¬3*�*5E6)��L �Q�'F�`&`n�±`nh�Jn9�`�sQ�8h0�158VQ�)��KÄ7nPQ�)F. �'�=!0�*�� �JnPA�n Z m�k�k�³?\V±PQ�S#0�*5,9Q�)£Q�)FQ�R9S *-,/*V'�WJ0 U|251�0 8:0d0 ^#0 )?2�*vIK,u2�425461�0�*546'�R/.H0�*-25,98:Q�25,9'�)�nhY!¸%¶�¸�¹/Àr¸$¹�ºM¶�» � M�P�¼ »¾»¿¹RQfS����C�Fm�m?Â��6m�³�³+n&�Q�15250�1��F&`nFQ�)F.=_s'�46)�� �zn Z G�j�j�³|\ �`)@�~,9t6tF*>*�Q�8:=6R9,9)6DhWg'�1>*-2�Q�2�0z*-=FQ#LO0s8:' . 0 R/* n U>¹]M^ZV¼ ¸�·r¹ V#¶������!�FÃ�³+G�6Ã�à �Fn&�Q�*�*5'�)�� �>n+Q�)F.a&('�R90�*��F��n Z G�j�j#j#\��|=FQ�25,/Q�R�1�0 D�1�0�*�*5,<'#)v8:' . 0 R/*�Wg'�1K0 U?2�150�8:0�*�n � ��¸$·5¼ Z:¼OÀr���+�F³�³#j�?³?² � n&('�R90�*�����n Z m�k�kFG�\��aQ� �QF¸$·XM�P^[Fº ¸�¹]MWQV¸pM*Y+¸�¶�¸$¹/ÀM¸$¹�ºM¶�» � M�P�¼ »¿¹RQfS\M�� � ��¸$·5¼ Z:¼��+¶�» [F¼rÀrnl�>0�I��X'�1 y ��`� ¬ �#�|=61�,<)FD�0 15l � 0 1�R9Q�DFn&('�R90�*��C��n9���>0 ��0 1�)FQ�)�� �+n�Q�)+.Bo}Q�IK)�� �+n Z G�j�j#j#\`Ä30�=+0�)F. 0�)FLO0]8:0�Q�*5E6150�*`Wg'�1J0 U?2�150�8:0h^�Q�R<E60]Q�)FQ�R9S *-0�* n � �#¸$·5¼ Z:¼OÀr���� ��� � ��j�� ��²?à n&('�R90�*��6��nFQ�)F.Ho}Q�IK)��#�+n Z G�j#j�²?\�¬ ±PQ�S�0�*-,/Q�)vQ�)FQ�R9S *-,/*�'�W}0OU|251�0 8:0s1�Q�,<)6W�Q�R9R�.FQ�2�QFn��#( (�»¿¹�¼8P Y!¸%¶�¸�¹/Àr¸$¹�ºrÀr�������|³?²����|³| � n&('�R90�*�����n��hn9��Á�0�15,/L LM4F,�� Xn��zn�Q�)F.J� ,9*�*-'#)�����n�¬Jn Z m�k�k��?\!¬eWgE6R9R<S~=61�'�tFQ�t6,<R9,/*f2�,9L{Q�=6=61�'#Q�LM4325'>0 U|251�0 8:0{^�Q�R9E60{8:'|.60 R9R<,9)6DFn� MW[ ·�Q!¶�»NM���� *3P�·XM�» MXS *�� ���� �|³|Ã!� Ã�k6nÄ~Q�^?,/*5'�)��P¬Jn�&`n(Q�)F. �|8:,<254�� �zn Xn Z G�j�j#k#\h_a' . 0�R9*:Wg'#1V0OU6LO0�0�.6Q�)FLO0�*:'�^�0�1h4F,<D#4c2�461�0�*546'�R/.6* Z IK,<254§L '�8:8:0 )?2�*M\rn� MW[ ·�Q!¶�»NM��s¸ #6¼#"sM *�¶�»cY!¸%¶�¸�¹/Àr¸$¹�ºM¶�»<Y<M�º ¹�¼O¸ *�$�Y�¼ ·r¹�¼OÀ@U%$ � ¼ ¸ #�M�PmM�» M.S�¹�ºM¶�»¿�&� � � ��j ���?³#³?m6nÄ`E6=6E6,/* �XÄ;n Z m�k#k#Ã#\ �`Y '#)60vLO'#)FLO0�)?251MQ�25,9'�)+* �hQa15'#t6EF*-2;Q�)+Q�R9S|*5,/*J'�WK8;E6Ru2�,<^�Q�15,/Q�2�0]0OU|251�0 8:0�* n('+¼Mº #lQ M^ZV¼ ¸�·r¹�ºOÀrnBoC'Q�=6=!0�Q�1�nA61W`E64|IK,91-2�4 l��6LM46)FQ�25250�1��P��n Z G�j�j�³?\;Ä`Q�2�QbQ�E6D�8:0 )?2MQ�25,9'�)�Q�)+.£.6S?)+Q�8:,9LTR9,<)F0�Q�1v8h' . 0�R9*�n � M^[ ·�Q�¶�»sM��)'�¹RZV¼�Y�¼ ·r¹�¼rÀ�aQ!¶�» *�ÀM¹/Àr���*����G � ��� m�k#m6n�sQ�0 2�Q�)��3&`n(Q�)+.§�~15,9D�'#R<0 2-25'+��_�n Z m�k�k�³?\V�|8:'|'�2�46,<)FD�*5Q�8h=FR<0d0 U|251�0 8:0�*VIK,<254§.6S?)+Q�8:,9Ld8h' . 0�R9*�n � �#¸�·5¼ ZV¼rÀr� � �m�m G+� m���²6n�sQ�8:0�158VQ�)��6Ä7n Z G�j#j?Â�\ � ¶�·XVmM,.- #6¶�¹RQ � MWQF¸�¼�-X¶�·r» M�n �'�)F. '#)��6s_ ��&(4FQ�=68VQ�)TQ�)F. �`Q�R9R�n�~,9R<R90 R/Q�)F.����>n9�c�>S LM4 y Q6�CÄ7n�Q�)F.��6LM46)60 ,/. 0�1���7n Z m�k#k�³|\`�|=+Q�25,/Q�R�8:' . 0 R/*~Wg'�1~2�460V. ,9*-251�,9t6E 25,9'�)�'�W�0OU|251�0 8:0�* n/'+¼Mº #�0·5¼ (�09���~0 '#=64|S|*5,/L Q�R��?2�Q�25,/*f2�,9L�*;Á�15' � 0�Lr2����s&�¬ �zn��|E68:8:0 17��)+*f2�,u2�E 250TQ�272�460@&(0�)?250 1;'�)[�~R9'�tFQ�RP&(4FQ�)6D�0#�}Ä`E y 0>)6,9^�0�1�*5,u2fS#n
�>0��!0�15)+Q�)��#�+n?Q�)F.vo}Q�IK)����Fn Z m�k�k�³|\X¬ LO'�)+. ,u2�,<'#)FQ�R�Q�=F=615'?Q�LM4:Wg'#1(8;E6Ru2�,<^�Q�15,/Q�2�030OU|2�150�8h0`^�Q�R9E60�* Z IK,<254@. ,/*5L EF*5*5,9'�)+\On� 0�"�0cY!¸%¶�¸�¹/Àr¸10NY M�º�0cU>��2�2��F³#j?Â!� Ã�³#²6n
�>,9D#.6'�)��!Ä7n Z m�k�k#m#\P�|=FQ�L 07Q�)+.a*-=FQ#LO0 l�2�,<8:0z8:' . 0 R9,<)FD]EF*5,<)6Dv=615' L 0�*�*>LO'#)?^#'�R9E 25,9'�)+* n(��)43n[F¶WQF¸$¹¾¸%¶�¸�¹5,�¼ � ¼O¸ #4M�P�À6�AM�·-�[ ·r·�¼AQ+¸��nQ�,�¹¾·XM^Q�Z:¼ QF¸�¶�»7 MÀMÀ�[F¼OÀ Z 0�.6*�nX&`n{¬3)F. 0 1M*5'�)�� � n{±�Q�15)F0O2-2��XÁ�n�&`nX&(4+Q�2fIK,9)cQ�)+.[¬Jn��7n���Rulf�|4FQ#Q�1MQ�IK,/\O��?Â��6Ã�²Fn �'�)F. '#)����|=61�,9)6D�0�1 � 0�15R/Q�D+n
�>EF0 152�Q6�?�hn<�#�6Q�)F* �'F��±~n�Q�)F.h�|251�'�EF.�� �Fn �zn Z m�k#k�³?\�¬ *-=+Q�25,9'�l$250�8:=+'#1�Q�R#8:' . 0 R|Wg'#1X_a0OU ,/LO'sLO,<2fSJ'�Y�'�)60�R90 ^�0�R9*�n��#(�(!»¿¹�¼8PY+¸�¶�¸$¹/ÀM¸$¹�ºOÀM�8� �Fm �FG� m�³ � n
Time-Varying Models for Extreme Values 15_s0 1�)�� �+n Z m�k�k�k?\�U`¶ *�¼OÀr¹�¶^Q]µ�·XM�ºM¼rÀMÀ
�-�M^Q�,^M�» [ ¸�¹]MWQ �#(�(!·XM�¶�º #6¼OÀ3¸{M@Y (F¼MºO¹ ��*�¹RQTSY (F¶�¸$¹�¶�»��h¼ (F¼ Q P�¼ Q!ºM¼0Y+¸$·�[FºO¸ [ ·5¼ n ÁX4�n Ä7n25460�*-,/* �F��)F*f2�,u2�E 250z'�W{�|2�Q�2�,9*-25,/L *KQ�)F.@Ä30�LO,/*-,9'�)a� L ,<0�)FLO0�* �FÄ`E y 0s3)6,9^�0 1M*5,u2fS#�6Ä3E61�4FQ�8T���>'�15254d&�Q�1�'�R9,9)FQ6� `�6¬Jn
_s,98T����n<���|4F0 =64FQ�1�.��c�Jn�Q�)F.e&(46,9t��C��n Z G�j�j � \3�|25' LM4FQ�*-25,/L;^�'�R/Q�2�,<R9,<2fS �3R<, y 0�R<,946'|' .a,<)6Wg0 1�0 )FL 0hQ�)F.NL '�8:=FQ�1�,/*-'#)dIK,u2�4Q�1MLM4v8:' . 0 R/* n�"`¼+,0�½�¹RQ�0NY!¸ [�P 09��2���� ��²FG� ��j��Fn�0�.|Wg'�1M.���¬7n�Q�)+.;o}Q�IK)�� �+n Z G�j�j�²?\C�|2�Q�2�,9*-25,/L *}Wg'#1{)60�Q�1�,<)F.60 =!0 )F. 0�)FLO0K,9):87E6R<25,9^�Q�1�,9Q�250�0OU|251�0 8:0P^�Q�R9E60�*�n U>¹]M^ZV¼O¸$·r¹ V#¶��� ��G�²#j��!G �  n
�Z G�j�j|Â�\(_a' . 0 R9,9)6DV. 0 =!0 )F.60 )FL 0zIK,u2�46,<) � '#,<)?2P2�Q�,<R�150�D�,9'�)F*�n � 0�" 0NY!¸%¶�¸�¹/Àr¸10 Y M�º�0cU>�8����� ³ Â�Ã��?³#j#j6nÁX,/L y Q�)F.F* � �+n Z G�j? G�\�oP460s2fI�'�l�. ,<8:0�)F*-,9'�)+Q�R�Á�'�,/*�*-'#)v=61�' LO0�*�*KQ�)+.v0 U?2�150�8VQ�R�=61�' LO0�*5*50�*�n � M^[ ·�Q!¶�»NM��#�#( (�»¿¹�¼8PVµ�·XM��
�¶�� ¹¾»¿¹¾¸�*��8����Â�³?Ã��6Â�Ã�²6n�
Z G�j?Â�Ã�\��?2�Q�25,/*f2�,9L�Q�R�,9) Wg0 1�0 )+LO0zEF*5,<)FDh0 U|251�0 8:0s'�1M. 0 1>*-2�Q�25,/*f2�,9L�* n�' #6¼ �aQ�Q�¶�» À M���Y+¸�¶�¸$¹/ÀM¸$¹�ºOÀM� ��G#G�j���G �FG�n�K'?*-0�)?tFR9Q�2-2���_bn Z G�j?Ã�m#\ �>0 8VQ�1 y *z'�)bQ@87EFRu2�,<^�Q�1�,/Q�250:2�1�Q�)F*-Wg'�1�8:Q�25,9'�)�n.�aQ�Q�¶�» ÀLM�� � ¶�¸ #F¼AZV¶�¸$¹�ºM¶�» Y+¸�¶�¸$¹/Àr¸�¹�ºOÀr� �� �³|Â�k�?³ Â�m n�|8:,<254�� �Jn Xn Z G�j � j#\(�{U|251�0 8:0s^�Q�R9E60zQ�)FQ�R<S *5,9*�'�WC0 )|^|,915'#)68:0 )?2�Q�R+2�,<8:0J*50 1�,<0�* �{¬3)@Q�=6=6R9,9L�Q�2�,<'#)]25':2�150�)F.T. 0O2�0�Lr2�,<'#),<)@D�1�'�EF)F.|l%R<0�^�0 R�'#Y '�)F0 Z &K´ � �FÁ �? � l ��j��?\rn�Y!¸%¶�¸�¹/Àr¸$¹�ºM¶�»<Y�ºO¹�¼ Q!ºM¼ ����� ��²?Â!� �|Â�Â|n�|8:,<254�� �zn Xn9��o}Q�IK)����Fn�¬Jn�Q�)F.7&('#R<0�* ����n��hn Z G�j�j?Â�\�_dQ�1 y '�^`LM4FQ�,9)s8h' . 0�R9*�Wg'�1�2�461�0�*546'�R/.~0OU6LO0�0�.6Q�)FLO0�* nmU>¹]M^ZV¼O¸$·r¹ V#¶�����C�Fm�³?j�� m�² � n¨�0�*-2���_bn�Q�)F. �3Q�1�1�,9*5'�)�� �Fn Z G�j�j|Â�\sU`¶ *�¼OÀr¹�¶^Q�½NM�·�¼�ºM¶�ÀM¸$¹RQfSa¶WQ P�� *^Q!¶WZ;¹�º � M�P�¼ » Àrn��>0 I �X'#1 y �3�|=F15,9)6D�0�1 � 0�15R/Q�D+�*-0�LO'�)+.v0�. )�n