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

In this paper, we introduce a new risk measure, the so-called Conditional Tail Moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α-quantile where α ∈ (0, 1). 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 α→ 0 is no longer fixed). 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 CévennesVivarais region (France).