Page 11 - Textos de Matemática Vol. 47

NAZARE´ AND ARCH PROCESSES: EXTREMAL INDEX ESTIMATION
M. IVETTE GOMES
To my good friend Nazar´e Mendes-Lopes, a token of friendship
Abstract. AfterafewmemoriesrelatedtomyfriendNazar´e,definitions of extremal index and ARCH processes are put forward. A brief revision of classical and corrected bias generalized jackknife estimation of the extremal index is further considered. Finally, we dedicate a short and personal tribute to Nazar´e, speaking about a possible co-operation, ‘the heart of Science’.
1. Memories and scientific scope of the article
As far as I remember, I met Nazar´e for the first time thirty years ago, in Lagos, Algarve, 1984, during the “III Col´oquio de Estat´ıstica e Investiga¸c˜ao Operacional”. Nazar´e spoke on recent results she had already published at Publications de l’Institut de Statistique de l’Universit´e de Paris (Mendes-Lopes, 1983, 1984), and it is still quite vivid in my memory her talk on what I used to call ‘marked point processes’. I have indeed found quite interesting the French nomenclature, ‘chromatiques’, for this type of processes, a terminology that I have never forgotten. She had been working for M.Sc. in France, at the Uni- versit´e de Paris VI (Pierre et Marie Curie), having begun there her Ph.D. project under the supervision of Jean Ge↵roy, a pioneer in the field of extreme value theory (EVT), my elected field of research. Despite of having got a de- gree of ‘Docteur de Troisi`eme Cycle’, in Paris, 1982, she had to prepare a Ph.D. thesis in Portugal. And I had the honor to be in ‘Sala dos Capelos’, Universi- dade de Coimbra, for the first time in 1985, to discuss her Ph.D. thesis. I also vividly recall the table where we had to place our documents, small enough for Professor Jean Ge↵roy only . . .
Accepted: 17 February 2015.
2010 Mathematics Subject Classification. Primary 60G70; Secondary 60G10, 60G17. Key words and phrases. ARCH processes, extremal index, friendship, generalized jackknife,
statistics of extremes.
The work was supported by National Funds through FCT (Funda¸c˜ao para a Ciˆencia e a
Tecnologia), PEst-OE/MAT/UI0006/2014 project.
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