Non-parametric 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 de-fined as the moment of order a ≥ 0 of the loss distribution above the upper α-quantile whereα ∈ (0, 1). Estimating the Conditional Tail Moment permits to estimate all risk measuresbased on conditional moments such as Conditional Tail Expectation, Conditional Value-at-Risk or Conditional Tail Variance. Here, we focus on the estimation of these risk measures incase of extreme losses (where α → 0 is no longer fixed). It is moreover assumed that the lossdistribution is heavy-tailed and depends on a covariate. The estimation method thus combinesnonparametric kernel methods with extreme-value statistics. The asymptotic distribution ofthe estimators is established and their finite sample behavior is illustrated both on simulateddata and on a real data set of daily rainfalls.