A geometric interpretation of the multiresolution criterion in nonparametric regression

A recent approach to choosing the amount of smoothing in nonparametric regression is to select the simplest estimate for which the residuals 'look like white noise'. This can be checked with the so-called multiresolution criterion, which Davies and Kovac [P.L. Davies and A. Kovac, Local extremes, runs, strings and multiresolutions (with discussion and rejoinder), Ann. Stat. 29 (2001), pp. 1-65.] introduced in connection with their taut-string procedure. It has also been used in several other nonparametric procedures such as spline smoothing or piecewise constant regression. We show that this criterion is related to a norm, the multiresolution norm (MR-norm). We point out some important differences between this norm and p-norms. The MR-norm is not invariant w.r.t. sign changes and permutations, and this makes it useful for detecting runs of residuals of the same sign. We also give sharp upper and lower bounds for the MR-norm in terms of p-norms.