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**Testing goodness-of-fit in regression via order selection criteria.**
*(English)*
Zbl 0776.62045

A test of the hypothesis that a regression model is a combination of given functions against a general alternative is proposed. The idea is to approximate residuals by means of a Fourier series, coefficients of which are estimated from residuals. The test statistic is based on the number of the estimates of coefficients which are significantly different from zero.

The exact distribution of the test statistic is derived under the hypothesis provided the distribution of residuals is Gaussian. For more general cases (naturally only) large sample distributions are found. The test is consistent against a fixed alternative and a system of local alternatives (converging to the null hypothesis at the rate \(n^{- {1\over 2}})\) may be detected, too. Due to the fact that the test statistic depends on the approximations to the residuals (at given observed points) the test evidently adapts to a given alternative.

The exact distribution of the test statistic is derived under the hypothesis provided the distribution of residuals is Gaussian. For more general cases (naturally only) large sample distributions are found. The test is consistent against a fixed alternative and a system of local alternatives (converging to the null hypothesis at the rate \(n^{- {1\over 2}})\) may be detected, too. Due to the fact that the test statistic depends on the approximations to the residuals (at given observed points) the test evidently adapts to a given alternative.

Reviewer: J.A.Víšek (Praha)

### MSC:

62G10 | Nonparametric hypothesis testing |

62J99 | Linear inference, regression |

62E15 | Exact distribution theory in statistics |

62E20 | Asymptotic distribution theory in statistics |