功能数据分析中基于锥度的强大测试

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摘要翻译：
提出了一种基于渐缩的检验方法，用于检验泛函线性模型下的全局线性假设。测试统计量被构造为傅立叶系数线性组合平方的加权和，一种锥形的二次型，其中较高的傅立叶频率被向下加权，以强调模型的光滑属性。公式是$q_n^{OPT}=n\sum_{j=1}^{p_n}j^{-1/2}\\boldsymbol{Y}_{n,j}\\^2$。选择用$J^{-1/2}$来实现基于测试速率的自适应最优性，测试速率是一种渐近框架，用于在光滑性约束下测量测试在高维度中的功率保持。已知现有的基于截断或阈值的测试与任何基于渐近的测试相比具有更好的渐近能力；然而，这里显示了高阶效应可能是实质性的，并且基于$q_n^{OPT}$的测试显示出更好的（非渐近）能力来对抗函数数据分析应用中通常关注的选择。提出的测试是为了在实践中使用而开发的，并在一个实例应用中进行了演示。
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英文标题：
《A powerful test based on tapering for use in functional data analysis》
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作者：
Dan J. Spitzner
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
--

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英文摘要：
  A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphasize the smooth attributes of the model. A formula is $Q_n^{OPT}=n\sum_{j=1}^{p_n}j^{-1/2}\|\boldsymbol{Y}_{n,j}\|^2$. Down-weighting by $j^{-1/2}$ is selected to achieve adaptive optimality among tests based on tapering with respect to its ``rates of testing,'' an asymptotic framework for measuring a test's retention of power in high dimensions under smoothness constraints. Existing tests based on truncation or thresholding are known to have superior asymptotic power in comparison with any test based on tapering; however, it is shown here that high-order effects can be substantial, and that a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the sort of alternatives that would typically be of concern in functional data analysis applications. The proposed test is developed for use in practice, and demonstrated in an example application.
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PDF链接：
https://arxiv.org/pdf/801.2224

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关键词：数据分析 Combinations Applications coefficients alternatives 光滑性 测量 weighted 适应 基于