Model selection criteria with data dependent penalty, with applications to data-driven Neyman smooth tests
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abstract
Tests of no effect in a regression context have been developed using the idea of nonparametric function estimation techniques. L2 error tests based on the difference between the estimated function and a constant perform well as omnibus tests. L2 error tests can be considered as data-driven Neyman smooth tests. In this case, an order (or truncation point) plays an important role in the performance of test statistics and is usually determined by an AIC type criterion (Kuchibhatla and Hart, 1996) or Schwarz's BIC criterion (Ledwina, 1994). When the sample size is small and the underlying function has high frequency behavior, AIC as well as BIC often fail to select a good order for the Fourier series estimator. The failure of correct order selection results in low power for the L2 error tests. To prevent such failure, we propose a Threshold criterion (TIC) and criteria with data-dependent penalty terms (DIC) using the idea of thresholding. These new criteria are more efficient in choosing the correct order in the sense of consistency and convergence rate. We propose new data-driven Neyman smooth tests based on DIC.
