Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:00 pm
Date: 18th May
Place: 3001

Yannick BARAUD (Université du Luxembourg) – A Robust Alternative to Least Squares in Regression

 Abstract: 

In collaboration with Guillaume Maillard, we study the estimation of a regression function under weak assumptions on the error distribution. In particular, we do not assume that the errors are i.i.d., nor that they have finite variance or exponential moments; we only require them to be independent and centered (and hence integrable). In particular, when the errors are square-integrable, they may, for instance, be heteroscedastic.

Within this statistical framework, we introduce a generic estimation method that yields estimators whose performance automatically adapts to the integrability properties of the errors. For these estimators, we establish non-asymptotic risk bounds for the L_1-loss. When the regression function belongs to a linear space and the errors are Gaussian but not necessarily i.i.d., these estimators exhibit remarkable robustness properties: they may converge at parametric rate (up to a logarithmic factor) in situations where classical least squares is not even consistent.

Nevertheless, we mainly illustrate their properties in the context of estimating a regression function under a shape constraint, such as monotonicity, unimodality, or convexity. We show that the proposed estimator is not only robust with respect to this a priori shape assumption, but also exhibits adaptation properties which are similar to those established for the least squares under the additional assumptions that the errors are i.i.d. and square-integrable.

Organizers:

Anna KORBA (CREST), Vincent DIVOL (CREST) , Jaouad MOURTADA (CREST)

Sponsors:
CREST-CMAP