Graphical characterisation of probability distribution tails

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 (Dedougou in Burkina Faso and a station on the Island of La Reunion) 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.