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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?

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    $\begingroup$ Why not use one of the many well-understood robust regression procedures on the data directly? $\endgroup$ – whuber Feb 11 at 20:34
  • $\begingroup$ 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... $\endgroup$ – Scott Feb 11 at 21:03

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