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On model-free conditional coordinate tests for regressions. (English) Zbl 1352.62088
Summary: Existing model-free tests of the conditional coordinate hypothesis in sufficient dimension reduction [R. D. Cook, Regression graphics. Ideas for studying regressions through graphics. New York, NY: Wiley (1998; Zbl 0903.62001)] focused mainly on the first-order estimation methods such as the sliced inverse regression estimation [K.-C. Li, J. Am. Stat. Assoc. 86, No. 414, 316–342 (1991; Zbl 0742.62044)]. Such testing procedures based on quadratic inference functions are difficult to be extended to second-order sufficient dimension reduction methods such as the sliced average variance estimation [R. D. Cook and S. Weisberg, J. Am. Stat. Assoc. 86, No. 414, 328–332 (1991; Zbl 1353.62037)]. In this article, we develop two new model-free tests of the conditional predictor hypothesis. Moreover, our proposed test statistics can be adapted to commonly used sufficient dimension reduction methods of eigendecomposition type. We derive the asymptotic null distributions of the two test statistics and conduct simulation studies to examine the performances of the tests.

MSC:
62H15 Hypothesis testing in multivariate analysis
62H25 Factor analysis and principal components; correspondence analysis
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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