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Optimal minimax rates for nonparametric specificaton testing in regression models. (English) Zbl 1033.62042
Summary: In the context of testing the specification of a nonlinear parametric regression function, we adopt a nonparametric minimax approach to determine the maximum rate at which a set of smooth alternatives can approach the null hypothesis while ensuring that a test can uniformly detect any alternative in this set with some predetermined power.
We show that a smooth nonparametric test has optimal asymptotic minimax properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures rate-optimality of the test. We show that, in contrast, a class of nonsmooth tests, which includes the integrated conditional moment test of H. J. Bierens [ J. Econ. 20, 105–134 (1982; Zbl 0549.62076)], has suboptimal asymptotic minimax properties.

MSC:
62G10 Nonparametric hypothesis testing
62J02 General nonlinear regression
62C20 Minimax procedures in statistical decision theory
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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