Tail Of Empirical Distribution Research Articles

Fitting heavy-tailed mixture models with CVaR constraints

Abstract Standard methods of fitting finite mixture models take into account the majority of observations in the center of the distribution. This paper considers the case where the decision maker wants to make sure that the tail of the fitted distribution is at least as heavy as the tail of the empirical distribution. For instance, in nuclear engineering, where probability of exceedance (POE) needs to be estimated, it is important to fit correctly tails of the distributions. The goal of this paper is to supplement the standard methodology and to assure an appropriate heaviness of the fitted tails. We consider a new Conditional Value-at-Risk (CVaR) distance between distributions, that is a convex function with respect to weights of the mixture. We have conducted a case study demonstrating e˚ciency of the approach. Weights of mixture are found by minimizing CVaR distance between the mixture and the empirical distribution. We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution.

Efficient bootstrap estimates for tail statistics

Abstract. Bootstrap resamples can be used to investigate the tail of empirical distributions as well as return value estimates from the extremal behaviour of the sample. Specifically, the confidence intervals on return value estimates or bounds on in-sample tail statistics can be obtained using bootstrap techniques. However, non-parametric bootstrapping from the entire sample is expensive. It is shown here that it suffices to bootstrap from a small subset consisting of the highest entries in the sequence to make estimates that are essentially identical to bootstraps from the entire sample. Similarly, bootstrap estimates of confidence intervals of threshold return estimates are found to be well approximated by using a subset consisting of the highest entries. This has practical consequences in fields such as meteorology, oceanography and hydrology where return values are calculated from very large gridded model integrations spanning decades at high temporal resolution or from large ensembles of independent and identically distributed model fields. In such cases the computational savings are substantial.

Estimation of the probability of strongest seismic disasters based on the extreme value theory

This paper presents the review of the experience in applying the approach based on the limiting distributions of the extreme value theory (the generalized Pareto distribution, GPS, and generalized extreme value distribution, GEV) for deriving the distributions of maximal magnitudes and related ground accelerations from the earthquakes on the future time intervals of a given duration. The results of analyzing the global and regional earthquake catalogs and the ground peak accelerations during the earthquakes are described. It is shown that the magnitude of the strongest possible earthquake M max (and analogous characteristics for other types of data), which is often used in seismic risk assessment, is potentially unstable. We suggest a stable alternative for M max in the form of quantiles Q q (τ) of the maximal possible earthquake, which could occur during the future time interval of length τ. The quantity of the characteristic maximal event M c, which has been introduced in our previous publications, is another helpful robust scalar parameter. All the cases of approximation of the tails of empirical distributions, which were studied in our works, turned out to be finite (bounded); however, the rightmost point of these distributions, M max, is often poorly detectable and unstable. Therefore, the M max parameter has a low practical value.

Gamma-Laguerre Formalism: Rigorous Approach and Application to Hydrologic Time Series

Previous work by various authors has shown that gamma-Laguerre distributions are appropriate for hydrologic applications because they offer the flexibility necessary to fit both the bulk and tails of empirical distributions, as well as a handy estimation formalism. The present work solves problems that still remain in gamma-Laguerre parameter estimation formalism: evaluating the highest degree of Laguerre polynomial and avoiding decreasing intervals in fitted probability distribution functions. From a pool of over 200 maximal rainfall and maximal discharge time series from all over the world that have been analyzed, a couple of examples, located in Mexico City and Niamey, respectively, are presented to illustrate the functionality of the formalism.