  We consider the problem of finding anomalies in a $d$-dimensional field of
independent random variables $\{Y_i\}_{i \in \left\{1,...,n\right\}^d}$, each
distributed according to a one-dimensional natural exponential family $\mathcal
F = \left\{F_\theta\right\}_{\theta \in\Theta}$. Given some baseline parameter
$\theta_0 \in\Theta$, the field is scanned using local likelihood ratio tests
to detect from a (large) given system of regions $\mathcal{R}$ those regions $R
\subset \left\{1,...,n\right\}^d$ with $\theta_i \neq \theta_0$ for some $i \in
R$. We provide a unified methodology which controls the overall family wise
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 non identically 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.
