M-Estimators for Regression with Changing Scale

M-estimation provides a class of estimators for the ‘signal plus noise’ problem, where the signal has a parametric form and the distribution of the noise is unspecified. Here, we extend this to modeling observations subject to trends in both location and scale, that is, to the model observation = ( location signal ) + ( scale signal ) × ( noise ) , $$ \text{observation} = (\text{location signal}) + (\text{scale signal}) \times (\text{noise}), $$ where the location signal and scale signal are smooth functions of an unknown q-vector 𝜃 say, and the components of the noise have some unknown cumulative distribution function (cdf) F say. We define the scaled M-estimator of 𝜃 with respect to a given smooth function ρ : ℝ → ℝ $\rho : \mathbb {R} \to \mathbb {R}$ . When the scale is not changing this reduces to the usual unscaled M-estimator requiring that F be suitably centered with respect to ρ.