#StackBounty: #regression #nonparametric #splines #interpolation Estimating Spline curve by OLS. Is a good idea to fix the knots at Che…

Bounty: 100

I am writing my master’s degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I’ve found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there’s a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.

Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it’s stated that the results regarding the ‘optimality’ of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.

I would like to know if it’s a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $| cdot |_2^2$. In many articles I’ve found phrases like “we will set the knots at the Chebyshev sites, that are well known to be good…” making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.

If your could put some light on the problem I will be very grateful.


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