Confidence regions for level curves and a limit theorem for the maxima of Gaussian random fields

Consider a Gaussian random field over a continuous two-dimensional region of interest. The level curve for this process at level u is defined to be all locations where the process takes the value u. From the observed data one can create a prediction surface over the region of interest and estimate the level curve. We present two methods for constructing confidence regions for the level curve of a random field.;In the first method a series of rectangular confidence regions are constructed along the estimated level curve which individually intersect the true level curve with high confidence. The boxes extend in directions perpendicular to the estimated level curve and the widths of the boxes are chosen so that the edges of each box touch the neighboring boxes. The heights of the boxes are chosen by simulating realizations of the random field conditional on the observed data, and taking the appropriate quantiles of the set of nearest level crossings for the realizations.;The second method constructs a confidence region for the true level curve through hypothesis testing, adjusting the critical value to control the simultaneous Type I error rate. A confidence region S is constructed by testing H0 : Z( s) = u versus Ha : Z(s) u, and taking S to be the union of all s for which we fail to reject the null hypothesis that Z(s) = u. Using kriging, we construct a test statistic which has a standard normal distribution. The critical value is adjusted to control the simultaneous Type I error rate through empirical simulation of the test statistic.;We conclude by introducing a limit theorem for the distribution of the maxima of a triangular sequence of stationary Gaussian random fields on an n x n lattice. The result is an extension of the work presented by Hsing et al. (1996) to two dimensions. Under certain dependence and limiting conditions we show that the maximum of the random fields exhibits extremal clustering in the limit.