Nonparametric adaptive estimation of conditional probabilities of rare events and extreme quantiles

Let F (t) (x)=P(X <= x|T = t) be the conditional distribution of a random variable X given that a covariate T takes the value t is an element of[0. T-max]where we assume that the distributions F-t are in the domain of attraction of the Frechet distribution. We observe independent random variables X-t1 ,...X-tn associated to a sequence of times 0 <= t <= ... < tn <= T-max,T- where X-ti has the distribution function For each , we propose a nonparametric adaptive estimator for extreme tail probabilities and quantiles of F-t . It follows from the Fisher-Tippett-Gnedenko theorem that the tail of the distribution function F-t can be adjusted with a Pareto distribution of parameter oee integral(t,tau) starting from a threshold tau. We estimate the parameter oee integral (t,tau) using a nonparametric kernel estimator of bandwidth h based on the observations larger than tau and we propose a pointwise data driven procedure to choose the threshold tau. A global selection of the bandwidth h based on a cross-validation approach is given. Under some regularity assumptions, we prove that the non adaptive and adaptive estimators of oee integral (t,tau) are consistent and we determine their rate of convergence. Finally, we study this procedure using simulations and we analyze an environmental data set.