Kernel estimators of extreme level curves. (English) Zbl 1367.62159
Summary: We address the estimation of extreme level curves of heavy-tailed distributions. This problem is equivalent to estimating quantiles when covariate information is available and when their order converges to one as the sample size increases. We show that, under some conditions, these so-called “extreme conditional quantiles” can still be estimated through a kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. Making use of this result, some kernel estimators of the conditional tail-index are introduced and a Weissman type estimator is derived, permitting to estimate extreme conditional quantiles of arbitrary large order. These results are illustrated through simulated and real datasets.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62G07 | Density estimation |
| 62G30 | Order statistics; empirical distribution functions |
| 62E20 | Asymptotic distribution theory in statistics |
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