Abstract

The goodness-of-fit problem is explored, when the test statistic is a linear combination of squared Fourier coefficients’ estimates coming from the Fourier decomposition of a probability density. Common examples of such statistics include Neyman’s test statistics and test statistics, generated by L2-norms of kernel estimators. We prove the asymptotic normality of the test statistic for both the null and alternative hypothesis. Using these results we deduce conditions of uniform consistency for nonparametric sets of alternatives, which are defined in terms of distribution or density functions. Results on uniform consistency, related to the distribution functions, can be seen as a statement showing to what extent the distance method, based on a given test statistic, makes the hypothesis and alternatives distinguishable. In this case, the deduced conditions of uniform consistency are close to necessary. For sequences of alternatives—defined in terms of density functions—approaching the hypothesis in L2-metric, we find necessary and sufficient conditions for their consistency. This result is obtained in terms of the concept of maxisets, the description of which for given test statistics is found in this publication.

中文翻译：

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关于 Neyman 型非参数检验的一致一致性

摘要

当检验统计量是来自概率密度傅立叶分解的平方傅立叶系数估计值的线性组合时，将探讨拟合优度问题。此类统计的常见示例包括 Neyman 的测试统计和由核估计器的 L2 范数生成的测试统计。我们证明了原假设和备择假设的检验统计量的渐近正态性。使用这些结果，我们推导出非参数替代方案集的一致一致性条件，这些条件是根据分布或密度函数定义的。与分布函数相关的均匀一致性结果可以看作是一种陈述，表明基于给定检验统计量的距离方法在多大程度上使假设和备选方案可区分。在这种情况下，一致一致性的推导条件几乎是必要的。对于替代序列（根据密度函数定义）接近 L2 度量中的假设，我们找到了它们一致性的充分必要条件。这个结果是根据最大集的概念获得的，在本出版物中可以找到对给定测试统计数据的描述。