  We introduce a new kind of likelihood function based on the sequence of
moments of the data distribution. Both binned and unbinned data samples are
discussed, and the multivariate case is also derived. Building on this approach
we lay out the formalism of shape analysis for signal searches. In addition to
moment-based likelihoods, standard likelihoods and approximate statistical
tests are provided. Enough material is included to make the paper
self-contained from the perspective of shape analysis. We argue that the
moment-based likelihoods can advantageously replace unbinned standard
likelihoods for the search of non-local signals, by avoiding the step of
fitting Monte-Carlo generated distributions. This benefit increases with the
number of variables simultaneously analyzed. The moment-based signal search is
exemplified and tested in various 1D toy models mimicking typical high-energy
signal--background configurations. Moment-based techniques should be
particularly appropriate for the searches for effective operators at the LHC.
