I have a real data set ($n=50$). I would like to fit some parametric models to this data set and then compare the maximum log-likelihood values with my spline model which is a nonparametric model. Could I use AIC criteria as a model selection? If not, which model selection criteria can I use?


You shouldn't generally use AIC to choose between parametric and nonparametric models. Parametric and nonparametric models have different modeling assumptions. The traditional AIC is based on a function of the likelihood. Likelihoods of parametric and nonparametric models are not always comparable.

An alternative that in theory can naturally compare parametric and nonparametric is a generalized degrees of freedom based criteria. See Ye's paper or Huang and Chen's paper for geostatistical data

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    $\begingroup$ You can't generally use AIC to compare parametric and non-parametric models. But for many spline models, it is a fairly reasonable thing to do (assuming it's not a penalized spline). $\endgroup$ – Cliff AB Nov 13 '15 at 4:41
  • $\begingroup$ Yes, but in practice most spline based models are penalized to control overfitting. Also, AIC may be used to choose the appropriate spline based model to use, but that's a different problem of the one stated by shany $\endgroup$ – Roberto Rivera Nov 14 '15 at 14:12
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    $\begingroup$ If AIC can discriminate between different smoothing parameters, why can't it discriminate between a smoothing parameter of infinity and a smoothing parameter of 0? I guess this assumes an unpenalized intercept. $\endgroup$ – generic_user Feb 16 '17 at 20:59

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