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

Abstract

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 $t \in [0,T_{\max }],$ where we assume that the distributions F _t are in the domain of attraction of the Fréchet distribution. We observe independent random variables $X_{t_{1}},...,X_{t_{n}}$ associated to a sequence of times $0\leq t_{1}<...<t_{n}\leq T_{\max },$ where $X_{t_{i}}$ has the distribution function $F_{t_{i}}.$ For each $t\in [0,T_{\max }]$, 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 𝜃 _t,τ starting from a threshold τ. We estimate the parameter 𝜃 _t,τ using a nonparametric kernel estimator of bandwidth h based on the observations larger than τ and we propose a pointwise data driven procedure to choose the threshold τ. 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 𝜃 _t,τ are consistent and we determine their rate of convergence. Finally, we study this procedure using simulations and we analyze an environmental data set.