Page 12 - Textos de Matemática Vol. 47

2 M. I. GOMES
A few years later my contacts with Coimbra, and also with Nazar´e, whom I already considered a very good friend at the time, were deepened, namely be- cause Nazar´e asked me to supervise Helena Ferreira (my good friend Lena), who was an Assistant at the Department of Mathematics, University of Coimbra, at the time. In Lisboa, DEIOC (Departamento de Estat´ıstica, Investiga¸c˜ao Operacional e Computa¸c˜ao) already had an M.Sc. Degree in Statistics and Operations Research, but in Coimbra, Lena had to be submitted to what we used to call PAPCS (‘Provas de Aptid˜ao Pedag´ogica e Capacidade Cient´ıfica’). Her monograph, entitled ‘Valores Extremos em Esquemas de Dependˆencia Fraca’, was discussed in 1989, and with this supervision my knowledge of ex- tremes of dependent schemes has improved a lot, despite of the fact that I have never worked deeply under such frameworks. Surely, we learn a lot with most of our students, something I miss a bit now that I am retired. And I had bright students working under my supervision not only in the aforementioned topic but also in other topics, among whom I mention Lena, who got a Ph.D. in 1994 (Ferreira, 1994).
Later on, Gra¸ca (Maria da Gra¸ca Temido), another good friend and now Assistant Professor at Universidade de Coimbra, came to Lisboa for M.Sc., at the time on Probability and Statistics, and at DEIO (Departamento de Estat´ıstica e Investiga¸c˜ao Operacional). Gra¸ca worked on Extreme Values in a Normal Model and finished her M.Sc. degree in 1992, under my supervision. She went on for Ph.D., under the supervision of Lu´ısa Canto e Castro and myself, having got her degree in 2000 (Temido, 2000). But despite of having had no more Ph.D. students from Coimbra my contacts and friendship with several colleagues from Coimbra have always been very gratifying.
When trying to find scientific research links with Nazar´e’s work, I could essentially find two topics in which we both have independently done research, Statistical Quality Control and ARCH/GARCH processes. Since I know that Nazar´e has worked with dependent processes for more than 20 years, as can be seen from two recent papers (Gon¸calves and Mendes-Lopes, 2013; Gonc¸alves et al., 2013), and two old ones (Brito et al., 1992; Gon¸calves and Mendes-Lopes, 1993), among more than two handfuls of published articles on this type of dependent processes, I have thus decided to choose the second topic, where I have only developed some work essentially related to the extremal index in ARCH processes. In Section 2 of this article, the notion of extremal index is provided. Section 3 is dedicated to the definition and a few properties of the ARCH processes. In Section 4, a brief review of some EI-estimators is performed. Finally, Section 5 is dedicated to a short, personal tribute and a possible challenge to Nazar´e.