Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models.

*(English)*Zbl 0673.62017Summary: D. R. Cox and E. Koh [A smoothing spline based test of model adequacy in polynomial regression. Tech. Rep. 787, Dept. Statistics, Univ. Wisconsin-Madison (1986)] considered the model \(y_ i=f(x(i))+\epsilon_ i\), \(\epsilon_ i\) i.i.d. \(N(0,\sigma^ 2)\), with the (parametric) null hypothesis f(x), \(x\in [0,1]\), a polynomial of degree m-1 or less, versus the alternative f is “smooth”, based on the Bayesian model for f which leads to polynomial smoothing spline estimates for f. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We extend their result to generalized smoothing spline models and to partial spline models.

We also show that the test statistic has an intimate relationship with the behavior of the generalized cross validation (GCV) function at \(\lambda =\infty\). If the GCV function has a minimum at \(\lambda =\infty\), then GCV has chosen the (parametric) model corresponding to the null hypothesis; we show that if the LMP test statistic is no larger than a certain multiple of the residual sum of squares after (parametric) regression, then the GCV function will have a (possibly local) minimum at \(\lambda =\infty\).

We also show that the test statistic has an intimate relationship with the behavior of the generalized cross validation (GCV) function at \(\lambda =\infty\). If the GCV function has a minimum at \(\lambda =\infty\), then GCV has chosen the (parametric) model corresponding to the null hypothesis; we show that if the LMP test statistic is no larger than a certain multiple of the residual sum of squares after (parametric) regression, then the GCV function will have a (possibly local) minimum at \(\lambda =\infty\).

##### MSC:

62F03 | Parametric hypothesis testing |

62G10 | Nonparametric hypothesis testing |

62H15 | Hypothesis testing in multivariate analysis |