On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

Abstract

In this paper, we consider the extreme behavior of a Gaussian random field f (t) living on a compact set T . In particular, we are interested in tail events associated with the integral T e f (t) dt. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field f (given that T e f (t) dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of T e f (t) dt is asymptotically equivalent to the tail event of sup T γ(t) where γ(t) is a Gaussian process and it is an affine function of f (t) and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of log b to compute the probability P ( T e f (t) dt > b) with a prescribed relative accuracy.

Efficient simulations for the exponential integrals of Hölder continuous gaussian random fields

“…See Liu and Xu [2012a] for the detailed description of the preceding results. According to the total variation approximation, the high excursion of I(T ) is almost the same as the high excursion of f (t), with a small correction depending on the Hessian matrix.…”

Uniformly efficient simulation for extremes of Gaussian random fields

“…Remark 2. Although the current paper focuses on rare-event simulation for the extremes of Gaussian random fields, the uniform efficiency criterion as well as the proposed method can be easily extended to other Gaussian-related rare-event problems, such as the exponential integrals of Gaussian random fields [e.g., 27,28], where the mean and variance functions are unspecified and we are interested in estimating a family of tail probabilities. Moreover, the proposed method can be extended to the estimation of non-Gaussian tail probabilities.…”

“…suprema [e.g., 3, 7, 9-13, 18, 19, 29-33]. Tail probabilities of other convex functions of Gaussian random fields have also been studied; see [21,23,25,28].…”

“…Importance sampling based efficient simulation procedures have been proposed in the literature to estimate the tail probabilities. Numerical methods for rare-event analysis of the suprema were studied in [1,2]; see also [8,20,24,[26][27][28]34] for related studies.…”

Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix

“…We conducted numerical studies to evaluate the performance of our algorithm. We first took combinations (n, p) ∈ {(100, 10), (100, 20), (500, 20), (1000, 50)}, and β = 1 and 2, respectively. Then we compared our algorithm with other methods and present the results in Table 1 and 2.…”