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

Let Ft(x)=P(X≤x|T=t) be the conditional distribution of a random variable X given that a covariate T takes the value t∈[0,Tmax],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t \in [0,T_{\max }],$\end{document} where we assume that the distributions Ft are in the domain of attraction of the Fréchet distribution. We observe independent random variables Xt1,...,Xtn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X_{t_{1}},...,X_{t_{n}}$\end{document} associated to a sequence of times 0≤t1<...<tn≤Tmax,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leq t_{1}<...<t_{n}\leq T_{\max },$\end{document} where Xti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X_{t_{i}}$\end{document} has the distribution function Fti.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F_{t_{i}}.$\end{document} For each t∈[0,Tmax]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in [0,T_{\max }]$\end{document}, we propose a nonparametric adaptive estimator for extreme tail probabilities and quantiles of Ft. It follows from the Fisher-Tippett-Gnedenko theorem that the tail of the distribution function Ft can be adjusted with a Pareto distribution of parameter 𝜃t,τ starting from a threshold τ. We estimate the parameter 𝜃t,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 𝜃t,τ are consistent and we determine their rate of convergence. Finally, we study this procedure using simulations and we analyze an environmental data set.