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On the underfitting and overfitting sets of models chosen by order selection criteria. (English) Zbl 1070.62516

Summary: For a general class of order selection criteria, we establish analytic and non-asymptotic evaluations of both the underfitting and overfitting sets of selected models. These evaluations are further specified in various situations including regressions and autoregressions with finite or infinite variances. We also show how upper bounds for the misfitting probabilities and hence conditions ensuring the weak consistency can be derived from the given evaluations. Moreover, it is demonstrated how these evaluations, combined with a law of the iterated logarithm for some relevant statistic, can provide conditions ensuring the strong consistency of the model selection criterion used.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
62M40 Random fields; image analysis

Software:

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References:

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