##
###
Volume of tubes and distribution of the maxima of Gaussian random fields
[unknown]

Satoshi Kuriki, Akimichi Takemura

2009
*
American Mathematical Society Translations: Series 2
*
unpublished

when c is large. Our approach is to approximate (1.3) using integral-geometric quantities of the submanifold M . We refer to (1.2) as a Karhunen-Loève (KL) expansion, although it is a generalization of the classical Karhunen-Loève expansion ([2]). 1.2. Tube method and Euler characteristic heuristic. Two methods are known to be applicable for approximating (1.3). One is the tube method and the other is the Euler characteristic heuristic. We give expositions of these methods based on works by the
## more »

... sed on works by the present authors ([26], [29], [42], [43]), J. Taylor and R. Adler ([44], [5]), and Taylor, et al. [45]. Section 2 presents an exposition of the tube method. A tube means a tubal neighborhood on the unit sphere in a Euclidean space. H. Hotelling [14] pointed out the relation between the distribution of the likelihood ratio test for a nonlinear regression model and the volume of a tube about a curve on the unit sphere and computed significance levels of the likelihood ratio test by explicitly computing the volume of the tube. H. Weyl [46] generalized the result to a general dimension. Their result is now known as the Hotelling-Weyl theorem. Although the Hotelling-Weyl theorem played an important role in the development of differential geometry ([23], [12]), it has been forgotten in statistics for a long time. More recently, Knowles and Siegmund [22] and Sun [38], [39] revived Hotelling's method and applied the result to some problem by pointing out that the probability (1.3) can be reduced to the volume of a tube when the KL-expansion (1.2) is a finite sum. This is the tube method. We provide a detailed exposition of the Euler characteristic heuristic in Section 3. If the sample paths of a one-dimensional stochastic process X(t), t ∈ T ⊂ R are smooth, then we can evaluate the expected number of upcrossings, that is, the number of times the graph (t, X(t)) crosses the horizontal line X(t) = c from below ([21], [7] ). This expected value for a large c has been traditionally used as an approximation to the tail probability P max t∈T X(t) ≥ c of the maximum of the stochastic process in the field of signal processing and other fields. As a generalization of this method to the general dimension, the Euler characteristic heuristic was proposed by Adler and A. M. Hasofer (e.g., [4] , [13], [1]) and further developed as a practical method by Worsley (e.g., [48], [49] ). The relation between the tube method and the Euler characteristic heuristic was initially not understood ([3]). However, the authors ([26], [42]) proved that these two methods are equivalent for Gaussian random fields with a finite KLexpansion. The Euler characteristic heuristic is more general than the tube method. However, the tube method has many applications to multivariate analysis ([29], [30], [24]) and it presents more concrete geometric pictures in terms of the tubes. Furthermore, the evaluation of the error term by the tube method can be extended to the Euler characteristic heuristic. Therefore, we present separate expositions of the two methods. The results on the Euler characteristic heuristic including the most recent ones are extensively reviewed in the new book by Adler and Taylor [5] . In statistics, there is a strong need to evaluate the distribution of the maximum of a random field. Suppose that X(p) is a test statistic for each p. Then, (1.3) corresponds to the adjustment of the p-value due to multiple testing (multiple comparisons). In hypothesis-testing problems, where the parameter under an alternative hypothesis is restricted to a cone, the asymptotic null distribution of the log

doi:10.1090/trans2/227/02
fatcat:roufiweuojfgbohl3r5nnvdjf4