Abstract
This survey intends to collect the developments on Goodness-of-Fit for regression models during the last 20 years, from the very first origins with the proposals based on the idea of the tests for density and distribution, until the most recent advances for complex data and models. Far from being exhaustive, the contents in this paper are focused on two main classes of tests statistics: smoothing-based tests (kernel-based) and tests based on empirical regression processes, although other tests based on Maximum Likelihood ideas will be also considered. Starting from the simplest case of testing a parametric family for the regression curves, the contributions in this field provide also testing procedures in semiparametric, nonparametric, and functional models, dealing also with more complex settings, as those ones involving dependent or incomplete data.
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Notes
Note that α is the significance level of the test. Empirical processes are denoted by \(\overline{\alpha_{n}}\) or \(\overline{\alpha _{nh}}\), to make the dependence on h explicit. Along this section, g denotes a target function to test (density, distribution or regression function).
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Acknowledgements
This research has been supported by Project MTM2008-03010 from the Spanish Ministry of Science and Innovation, and by the IAP network StUDyS (Developing crucial Statistical methods for Understanding major complex Dynamic Systems in natural, biomedical, and social sciences), from Belgian Science Policy. The authors also thank the comments and suggestions from the Associate Editor and an anonymous referee.
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González-Manteiga, W., Crujeiras, R.M. An updated review of Goodness-of-Fit tests for regression models. TEST 22, 361–411 (2013). https://doi.org/10.1007/s11749-013-0327-5
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DOI: https://doi.org/10.1007/s11749-013-0327-5
Keywords
- Bootstrap calibration
- Empirical distribution of the residuals
- Empirical regression process
- Likelihood ratio tests
- Smoothing tests