M-Estimators for Regression with Changing Scale

Christopher S. Withers () and Saralees Nadarajah ()
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Christopher S. Withers: Industrial Research Limited
Saralees Nadarajah: University of Manchester

Sankhya B: The Indian Journal of Statistics, 2016, vol. 78, issue 2, No 3, 238-286

Abstract: Abstract 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.

Keywords: M-estimator; Regression; Robust; Trend in scale; Primary 62G20; Secondary 62F12 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s13571-016-0122-x

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