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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.

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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