Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions

Jonathan El Methni (1) , Laurent Gardes (2) , Stéphane Girard (1)

1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
2 IRMA - Institut de Recherche Mathématique Avancée

In this paper, we introduce a new risk measure, the so-called Conditional Tail Moment. It is the moment of order a>0 of the loss distribution above the upper alpha-quantile. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments such as Value-at-Risk, Conditional Tail Expectation, Conditional Value-at-Risk or Conditional Tail Variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where alpha converges to 0). It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France).

Domains: Statistics [math.ST]; Statistics Theory [stat.TH]

Jonathan El Methni, Laurent Gardes, Stéphane Girard. Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions. 2013. ⟨hal-00830647v3⟩