Abstract
We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional 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. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate large conditional quantiles of arbitrary large order. These results are illustrated on finite sample situations.
Highlights
Let (Xi, Yi), i = 1, . . . , n be independent copies of a random pair (X, Y ) in E ×R where E is an infinite dimensional space associated to a semi-metric d
While the nonparametric estimation of ordinary regression quantiles has been extensively studied, less attention has been paid to large conditional quantiles despite their potential interest
Main results The first step towards the estimation of large conditional quantiles is the estimation of small tail probabilities F(yn|x) when yn → ∞ as n → ∞
Summary
We address the problem of estimating q(αn|x) ∈ R verifying P(Y > q(αn|x)|X = x) = αn where αn → 0 as n → ∞ and x ∈ E In such a case, q(αn|x) is referred to as a large conditional quantile in contrast to classical conditional quantiles (or regression quantiles) for which αn = α is fixed in (0, 1). We focus on the setting where the conditional distribution of Y given X = x has an infinite endpoint and is heavy-tailed, an analytical characterization of this property being given . In such a case, the frontier function does not exist and q(αn|x) → ∞ as αn → 0.
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