Graphical characterisation of probability distribution tails

Abstract

The purpose of this paper is to present a graphical method to characterise the nature of a distribution (exponential or algebraic). In the algebraic case, this statistical tool provides an estimation procedure of the parameter characterising the decrease of the survival function. The realizations of the random variable under study being available in the form of time series, this method is based on the relationship between the duration of exceeding an intensity threshold and the accumulation of the realizations of the random variable during this length of time. The behaviour of the duration-accumulation graphs (when the threshold of reference increases indefinitely) results in a function, the limit of which only depends on the parameter characterising the algebraic decrease of the probability distribution. The estimate of this parameter is biased but can be corrected effectively by numerical methods. We applied this method to two rainfall series differing by their geographical origin (Dédougou in Burkina Faso and a station on the Island of La Réunion) and their time step (respectively 1 day and 76 seconds). For both of them, the behaviour of tail distributions is shown to be algebraic and the values of the parameter characterizing the algebraic decrease of the probability distribution of the two series are very close. This would tend to justify the assumption of a multifractal nature for these series.

Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: II. Recherche empirique sur de longues séries de précipitations

“…The results of this study also converge with other recent studies such as those by Chaouche (2001), Chaouche et al (2002), Coles et al, (2003) and Sisson et al (2003), which all verify the EV2/Pareto behaviour in the tail of the distribution of rainfall extremes and exclude the possibility of EV1/exponential behaviour. In particular, Chaouche (2001) exploited a data base of 200 rainfall series of various time steps (month, day, hour, minute) from the five continents, each including more than 100 years of data.…”

Bayesian comparison of different rainfall depth–duration–frequency relationships

“…Recently, Koutsoyiannis and Baloutsos (2000), Chaouche et al (2002), , , Sisson et al (2006), Koutsoyiannis (2004a, b) and Bacro and Chaouche (2006) have shown that extreme rainfall quantiles can be seriously underestimated by the Gumbel distribution. This discussion has significant practical consequences, particularly for high return periods used for the design of major hydraulic constructions or the estimation of risk of extreme floods.…”

Incertitude d'estimation des pluies extrêmes du pourtour méditerranéen: illustration par les données de Marseille

“…La procédure de calcul des bornes de cet intervalle procède de la délimitation du domaine permis pour le paramètre ξ, conditionnellement à l'échantillon observé (Chaouche & Bacro, 2004a); la statistique qui dérive de cette délimitation, à savoir Ce premier résultat, que nous allons remettre en discussion dans ce qui suit, signifie entre autres que le maximum des cumuls journaliers appartient au domaine d'attraction de Fréchet, c'est-à-dire admet une queue de distribution qui est à décroissance algébrique; ceci rejoint des résultats de travaux menés sous l'égide de P. Hubert (Chaouche et al, 2002) avec une approche non-paramétrique, concernant la caractérisation des queues de distribution de séries pluviométriques ainsi que ceux de Koutsoyiannis (2004a,b).…”