Frederico Caeiro

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{Summary: We first consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parametrised in a tuning or control parameter $\alpha$. Such a control parameter enables us to have access for any available sample, to an estimator of the tail index $\gamma$ with a null dominant component of asymptotic bias and consequently with a reasonably flat mean squared error pattern, as a function of $k$, the number of top-order statistics considered.\par Such a control parameter depends on a second-order parameter $\rho$, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro-US\$ exchange rates.}

A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.

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Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.

In this paper we review the properties of the difference-sign randomness test. First we analyse the exact andasymptotic distribution of the test statistic and provide a table with values for the exact distribution function, for samples ofsize n ≤ 32. Then, we also present several moments of the statistic test, under the null hypothesis of randomness and underthe hypothesis of the existence of a linear trend. Finally, we present an illustration of the test difference-sign to a real data set.In this paper we review the properties of the difference-sign randomness test. First we analyse the exact andasymptotic distribution of the test statistic and provide a table with values for the exact distribution function, for samples ofsize n ≤ 32. Then, we also present several moments of the statistic test, under the null hypothesis of randomness and underthe hypothesis of the existence of a linear trend. Finally, we present an illustration of the test difference-sign to a real data set.