Locally monotonic regression is a recently proposed technique for the deterministic smoothing of finite-length discrete signals under the smoothing criterion of local monotonicity. Locally monotonic regression falls within a general framework for the processing of signals that may be characterized in three ways: regressions are given by projections that are determined by semimetrics, the processed signals meet shape constraints that are defined at the local level, and the projections are optimal statistical estimates in the maximum likelihood sense. the authors explore the relationship between the geometric and deterministic concept of projection onto (generally nonconvex) sets and the statistical concept of likelihood, with the object of characterizing projections under the family of the p-semi-metrics as maximum likelihood estimates of signals contaminated with noise from a well-known family of exponential densities. >