# Is a spline interpolation considered to be a nonparametric model?

I am aware of the basic differences between nonparametric and parametric statistics. In parametric models, we assume the data follows a distribution and fit it onto it using a fixed number of parameters. With KDE for instance, this is not the case because we don’t assume that the modeled distribution has a particular shape.

I am wondering how this relates to interpolation in general, and to spline interpolation in specific. Are all interpolation approaches considered to be nonparametric, are there “mixed” approaches, what is the case with spline interpolation?

These models are nonparametric in the sense that using them does not involve reported quantities like $$\widehat{\beta}, \widehat{\theta}\widehat{\beta}, \widehat{\theta}$$, etc. (in contrast to linear regression, GLM, etc.). Smoothing models are extremely flexible ways to represent properties of $$yy$$ conditional on one or more $$xx$$ variables, and do not make a priori commitments to, for example, linearity, simple integer polynomial, or similar functional forms relating $$yy$$ to $$xx$$.
If I understand you, your “mixed” approaches are what are called “semi-parametric models”. Cox regression is one highly-specialized example of such: the baseline hazard function relies on a nonparametric estimator, while the explanatory variables are estimated in a parametric fashion. GAMs—generalized additive models—permit us to decide which $$xx$$ variables’ effects on $$yy$$ we will model using smoothers, which we will model using parametric specifications, and which we will model using both all in a single regression.