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I am a little bit familiar with quantile regression but I was wondering what is the difference between parametric quantile vs non-parametric quantile. Also, the "rq" function is R is based on parametric version or nonparametric version?

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  • $\begingroup$ I've never heard of 'parametric' QR. QR is always discussed as a robust, distribution-free, nonparametric modeling method. Would you supply a reference? Hyndman has a good suggestion for implementing QR in this paper, Probabilistic time series forecasting with boosted additive models: an application to smart meter data ... robjhyndman.com/papers/sig-alternate.pdf $\endgroup$ – Mike Hunter Nov 22 '17 at 14:51
  • $\begingroup$ @ DJohnson, Hi, thanks for the reply. If you check "package ‘quantreg in R" they prepared the package for Description Estimation and inference methods for models of conditional quantiles: "Linear and nonlinear parametric and non-parametric". I know that the nonparametric quantile regressions are local constant or smoothing spline. $\endgroup$ – shad Nov 22 '17 at 19:16
  • $\begingroup$ I'd want to hear your definition of nonparametric before commenting. The word has been debased and stretched over decades to the point of becoming fairly useless at best. Often "parametric" just seems to mean "assuming a conditional normal distribution" where "assuming" means "working best under". But quantile regression is focused on parameter estimation. It's likely to be robust in any default flavour of estimating conditional medians; when estimating extreme quantiles, however, it's as likely as anything else to be sensitive to wild observations. $\endgroup$ – Nick Cox Nov 23 '17 at 15:59
  • $\begingroup$ I can't comment on R functions I haven't used or read about: what they say about themselves should be consistent with their own authors' definitions of nonparametric. If any method is centred on estimating parameters, it's hard to see that "nonparametric" is a helpful label, but nothing much hinges on what it's called. It's what it does that is crucial. $\endgroup$ – Nick Cox Nov 23 '17 at 16:02
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I think in the context of Koenker's quantreg package, he meant non parametric for non parametric regression, which means the form of covariates is not fixed. Rq assumes a linear form so it's not non parametric. On the other hand, rqss uses a penalized approach so is non parametric in that sense.

To note, you can also say non parametric when referring to a distribution of random variable as was pointed out in comments but here, due to regression framework, there are parameters to estimate making this a semi parametric, not non parametric.

To add, I've also came across some literature assuming some distribution e.g. Laplace, normal mixtures to force a likelihood inference. In that sense, these are fully parametric models like classical mean regression.

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