Our purpose is to present a new method for adaptively estimating a regression function when little is known about the shape and scale of the errors and which can cope with error distributions as different as Gaussian, Uniform, Cauchy or even with unimodal unbounded densities.
In favorable cases and when the true distribution belongs to the model, the estimator is asymptotically equivalent to the M.L.E. and, nevertheless, still behaves reasonably well when the model is wrong, even in cases for which the least-squares do not work.
The assumptions that are needed to get our results are rather weak, in particular no moment condition is required on the errors, and this is why the method can adapt to both the regression function, the shape of the errors and their scale.
Moreover, it appears that the practical results obtained by simulation are surprisingly good as compared to more specific estimators.
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