Quantile predictions for elliptical random fields

In this article, we consider elliptical random fields. We propose some quantile predictions at one site, given observations at some other locations. To this end, we first give exact expressions for conditional quantiles, and discuss problems that occur when computing these values. A first affine regression quantile predictor is presented in detail, an explicit formula is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. As the performance of this regression quantile turns out to be very poor for extremal quantile levels, a second predictor is proposed. We prove that this new extremal predictor is asymptotically equivalent to the true conditional quantile. Through numerical illustrations, we show that quantile regression may perform poorly outside the usual Gaussian random field setting. This justifies the use of the proposed extremal quantile predictions.