MULTIDIMENSIONAL MULTISCALE SCANNING IN EXPONENTIAL FAMILIES: LIMIT THEORY AND STATISTICAL CONSEQUENCES

We consider the problem of finding anomalies in a d-dimensional field of independent random variables {Y-i}(i is an element of{1,...,n}d), each distributed according to a one-dimensional natural exponential family F = {F-theta}(theta is an element of Theta). Given some baseline parameter theta(0) is an element of Theta, the field is scanned using local likelihood ratio tests to detect from a (large) given system of regions R those regions R subset of {1, ..., n}(d) with theta(i) not equal theta(0) for some i is an element of R. We provide a unified methodology which controls the overall familywise error (FWER) to make a wrong detection at a given error rate. Fundamental to our method is a Gaussian approximation of the distribution of the underlying multiscale test statistic with explicit rate of convergence. From this, we obtain a weak limit theorem which can be seen as a generalized weak invariance principle to nonidentically distributed data and is of independent interest. Furthermore, we give an asymptotic expansion of the procedures power, which yields minimax optimality in case of Gaussian observations.