Generalised Pareto distribution: impact of rounding on parameter estimation

Abstract

Problems that occur when common methods (e.g. maximum likelihood and L-moments) for fitting a generalised Pareto (GP) distribution are applied to discrete (rounded) data sets are revealed by analysing the real, dry spell duration series. The analysis is subsequently performed on generalised Pareto time series obtained by systematic Monte Carlo (MC) simulations. The solution depends on the following: (1) the actual amount of rounding, as determined by the actual data range (measured by the scale parameter, σ) vs. the rounding increment (Δx), combined with; (2) applying a certain (sufficiently high) threshold and considering the series of excesses instead of the original series. For a moderate amount of rounding (e.g. σ/Δx ≥ 4), which is commonly met in practice (at least regarding the dry spell data), and where no threshold is applied, the classical methods work reasonably well. If cutting at the threshold is applied to rounded data—which is actually essential when dealing with a GP distribution—then classical methods applied in a standard way can lead to erroneous estimates, even if the rounding itself is moderate. In this case, it is necessary to adjust the theoretical location parameter for the series of excesses. The other solution is to add an appropriate uniform noise to the rounded data (“so-called” jittering). This, in a sense, reverses the process of rounding; and thereafter, it is straightforward to apply the common methods. Finally, if the rounding is too coarse (e.g. σ/Δx~1), then none of the above recipes would work; and thus, specific methods for rounded data should be applied.