# Loess preprocessing for robust quadratic regression?

It seems to me that we could fit a quadratic curve to the predictions of a LOESS curve in order to obtain a parametric model approximating the non-parametric LOESS model. Is this something that is done to perform robust quadratic regression? Are there any texts that describe such a preprocessing step?

Specifically, the steps would be, fit a LOESS curve to $$X$$, $$\mathbb{y}$$. Obtain predictions $$\mathbb{\hat{y}}$$ from the LOESS smoother evaluated at the training set $$X$$. Fit a quadratic regression on $$X$$, $$\mathbb{\hat{y}}$$. For bandwidths approaching 0, the resulting quadratic would approach the un-preprocessed quadratic regression. For bandwidths approaching $$\infty$$, the resulting quadratic would approach a line (at least within the range of the training data). Thus, this seems like a valid way to make a "robust" quadratic regression.

Is this process or something like it examined in any literature/textbooks?

• Why not use one of the many well-understood robust regression procedures on the data directly? – whuber Feb 11 at 20:34
• My interest is mostly in the theoretical properties of the LOESS estimator. It occurred to me that something like the above would be straightforward to construct, and that the "polynomial-approximatability" (whatever that means) of LOESS curves might be studied somewhere. I am surprised that I haven't found anything after searching all day, since such a thing doesn't seem utterly unreasonable... – Scott Feb 11 at 21:03