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 ??t,τ starting from a threshold τ. We estimate the parameter ??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 ??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." />
Nonparametric adaptive estimation of conditional probabilities of rare events and extreme quantiles
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  • 作者：Gilles Durrieu ; Ion Grama ; Quang-Khoai Pham ; Jean-Marie Tricot
  • 关键词：Nonparametric estimation ; Tail conditional probabilities ; Extreme conditional quantile ; Adaptive estimation ; Environment ; 62G32 ; 62G08 ; 62P12
  • 刊名：Extremes
  • 出版年：2015
  • 出版时间：September 2015
  • 年：2015
  • 卷：18
  • 期：3
  • 页码：437-478
  • 全文大小：1,577 KB
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  • 作者单位：Gilles Durrieu (1)
    Ion Grama (1)
    Quang-Khoai Pham (1)
    Jean-Marie Tricot (1)

    1. Université de Bretagne Sud, LMBA, UMR CNRS 6205, 56000, Vannes, France
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Statistics
    Statistics
    Quality Control, Reliability, Safety and Risk
    Civil Engineering
    Hydrogeology
    Environmental Management
    Statistics for Business, Economics, Mathematical Finance and Insurance
  • 出版者：Springer U.S.
  • ISSN：1572-915X
文摘
Let F t (x)=P(X?em class="EmphasisTypeItalic">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 ??t,τ starting from a threshold τ. We estimate the parameter ??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 ??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.

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