Fitting heavy-tailed mixture models with CVaR constraints

Giorgi Pertaia,Stan Uryasev

doi:10.1515/demo-2019-0019

Giorgi Pertaia, Stan Uryasev

Open Access

https://doi.org/10.1515/demo-2019-0019

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Abstract

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.

Highlights

IntroductionFinite mixtures (or mixture distributions) allow to model complex characteristics of a random variable
Finite mixtures allow to model complex characteristics of a random variable
We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution

Summary

Introduction

Finite mixtures (or mixture distributions) allow to model complex characteristics of a random variable. They are frequently used in the cases where data are not normally distributed. Nite mixtures are well suited for modeling heavy tails. Another application of nite mixtures is to model multi-modal random variables. Finite mixtures are frequently used in these elds to model a wide variety of random variables. Paper [14]estimates Value-at-Risk (VaR) for a heavy-tailed return distribution using a nite mixture. Paper [1] models the error distribution of the GARCH(1,1) with a nite mixture, the resulting model is called NM-GARCH. Finite mixtures are frequently used in machine learning for clustering and classi cation of the data

Objectives

Conclusion