Distribution-free Detection of a Submatrix

We consider the problem of detecting the presence of a submatrix with larger-than-usual values in a large data matrix. This problem was considered in (Butucea and Ingster, 2013) under a one-parameter exponential family, and one of the test they analyzed is the scan test. Taking a nonparametric stance, we show that a calibration by permutation leads to the same (first-order) asymptotic performance. This is true for the two types of permutations we consider. We also study the corresponding rank-based variants and precisely quantify the loss in asymptotic power.