The first author and H. Rootzén [J. Stat. Plann. Inference 35, No. 1, 1-13 (1993; Zbl 0770.62026)]constructed confidence intervals for extreme quantiles, i.e., quantiles outside the scope of the sample under extreme-value conditions. Estimation of extreme quantiles is necessary, e.g., when determining the height of a projected sea-dike: on the basis of high tide water levels observed during 100 years one has to design the height of the dike in such a way that the return period of a flood is 10,000 years.
In this paper we consider the multidimensional problem. The requirement is now, e.g., that the return period of a flood at either one of two places along the coast is 10,000 years. This leads to the problem of estimation of extreme quantile curves in two-dimensional distributions. We formulate the problem and its solution in two dimensions in order to keep the notation relatively simple. Generalization to higher dimensions is straightforward.
Our aim is to estimate the curve of all values \((x(p)\), \(y(p))\) for which \(p = 1 - F(x(p)\), \(y(p))\), where \(F\) is some unknown distribution function from which a sample has been taken.