Abstract

In model checking for regressions, nonparametric estimation-based tests usually have tractable limiting null distributions and are sensitive to oscillating alternative models, but suffer from the curse of dimensionality. In contrast, empirical process-based tests can, at the fastest possible rate, detect local alternatives distinct from the null model, yet are less sensitive to oscillating alternatives and rely on Monte Carlo approximation for critical value determination, which is costly in computation. We propose an adaptive-to-model hybrid of moment and conditional moment-based tests to fully inherit the merits of these two types of tests and avoid the shortcomings. Further, such a hybrid makes nonparametric estimation-based tests, under the alternatives, also share the merits of existing empirical process-based tests. The methodology can be readily applied to other kinds of data and construction of other hybrids. As a by-product in sufficient dimension reduction field, a study on residual-related central mean subspace and central subspace for model adaptation is devoted to showing when alternative models can be indicated and when cannot. Numerical studies are conducted to verify the powerfulness of the proposed test.

摘要

在回归模型检查中，基于非参数估计的测试通常具有易于处理的极限零分布，并且对振荡的替代模型敏感，但会受到维数灾难的影响。相比之下，基于经验过程的测试可以以尽可能快的速度检测与空模型不同的局部备选方案，但对振荡备选方案不那么敏感，并且依赖蒙特卡洛近似来确定临界值，这在计算上是昂贵的。我们提出了一种基于矩和条件矩的测试的自适应模型混合体，以充分继承这两种测试的优点并避免缺点。此外，这种混合使得基于非参数估计的测试，在替代方案下，也共享现有的基于经验过程的测试的优点。该方法可以很容易地应用于其他类型的数据和其他混合体的构建。作为充分降维领域的副产品，与残差相关的中心均值子空间和用于模型自适应的中心子空间的研究致力于显示何时可以指示替代模型，何时不能指示替代模型。进行数值研究以验证所提出测试的有效性。