Nonparametric regression and extreme-value analysis

This note summarizes my contributions to the estimation of extreme level curves. This problem is equivalent to estimating quantiles when covariate information is available and when their order converges to one as the sample size increases. Several estimators of these socalled ”extreme conditional quantiles” are developped and the links with boundary or frontier estimation are emphasized. 1 Extreme-value analysis Extreme value theory is a branch of statistics dealing with the extreme deviations from the bulk of probability distributions. More specifically, it focuses on the limiting distributions for the minimum or the maximum of a large collection of random observations from the same arbitrary (unknown) distribution. Let x1 < · · · < xn denote n ordered observations from a random variable X representing some quantity of interest. A pn-quantile of X is the value qpn such that the probability that X is greater than qpn is pn, i.e. P (X > qpn) = pn. When pn < 1/n, such a quantile is said to be extreme since it is usually greater than the maximum observation xn. To estimate such extreme quantiles requires therefore specific methods to extrapolate information beyond the observed values of X. Those methods are based on Extreme value theory. This kind of issues appeared in hydrology. One objective was to assess risk for highly unusual events, such as 100-year floods, starting from flows measured over 50 years. The decay of the survival function P (X > x) = 1 − F (x), where F denotes the cumulative distribution function associated to X, is driven by a real parameter called the extreme-value index γ. When this parameter is positive, the survival function is said to be heavy-tailed, when this parameter is negative, the survival function vanishes above its right end point. If this parameter is zero, then the survival function decreases to zero at an exponential rate. An important part of our work is dedicated to the study of such distributions. For instance, in reliability, the distributions of interest are included in a semi-parametric family whose tails are decreasing exponentially fast.