Two methods to distinguish between polynomial and exponential tails are introduced. The methods are based on the properties of the residual coefficient of variation for the exponential and non-exponential distributions. A graphical method, called a CV-plot, shows departures from exponentiality in the tails. The plot is applied to the daily log-returns of exchange rates of US dollar and Japanese yen. New statistics are introduced for testing the exponentiality of tails using multiple thresholds. They give better control of the significance level than previous tests. The powers of the new tests are compared with those of some others for various sample sizes.

Abstract. Research partially supported by the Spanish Ministerio de Educacion y Ciencia, grant MTM2006-01477.: This paper provides precise arguments to explain the anomalous behavior of the likelihood surface when sampling from the generalized Pareto distribution for small or moderate samples. The behavior of the profile-likelihood function is characterized in terms of the empirical coefficient of variation. A sufficient condition is given for global maximum of the likelihood function of the Pareto distribution to be at a finite point.

In this article we show the relationship between the Pareto distribution and the gamma distribution. This shows that the second one, appropriately extended, explains some anomalies that arise in the practical use of extreme value theory. The results are useful to certain phenomena that are fitted by the Pareto distribution but, at the same time, they present a deviation from this law for very large values. Two examples of data analysis with the new model are provided. The first one is on the influence of climate variability on the occurrence of tropical cyclones. The second one on the analysis of aggregate loss distributions associated to operational risk management.

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