An asymptotic theory is developed for the estimation of high quantile curves, i.e., sets of points in higher dimensional space for which the exeedance probability is pn, with npn --> 0 (n --> [infinity]). Here n is the number of available observations. This is the situation of interest if one wants to protect against a calamity that has not yet occurred. Asymptotic normality of the estimated quantile curve is proved under appropriate conditions, including the domain of the attraction condition for multivariate extremes.
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