Motivated by the need to statistically quantify differences between modern (complex) data-sets which commonly result as high-resolution measurements of stochastic processes varying over a continuum, we propose novel testing procedures to detect relevant differences between the second order dynamics of two functional time series. In order to take the between-function dynamics into account that characterize this type of functional data, a frequency domain approach is taken. Test statistics are developed to compare differences in the spectral density operators and in the primary modes of variation as encoded in the associated eigenelements. Under mild moment conditions, we show convergence of the underlying statistics to Brownian motions and construct pivotal test statistics. The latter is essential because the nuisance parameters can be unwieldy and their robust estimation infeasible, especially if the two functional time series are dependent. In addition to these novel features, the properties of the tests are robust to any choice of frequency band enabling also to compare energy contents at a single frequency. The finite sample performance of the tests are verified through a simulation study and are illustrated with an application to fMRI data.
翻译:现代(复合)数据集通常是通过高清晰度测量不同连续过程而得出的,因此,对现代(复合)数据集之间的差异进行统计性量化,这种差异通常导致对连续过程进行高清晰度测量,为此,我们提出新的测试程序,以检测两个功能时间序列的第二顺序动态之间的相关差异。为了将功能性数据特点的功能性动态考虑在内,采用了一种频率域法。开发测试统计数据是为了比较光谱密度操作器的差异,以及作为相关电子元素编码的主要变异模式的差异。在温和的瞬间条件下,我们展示了潜在统计数据与布朗尼运动的趋同,并构建了关键测试统计数据。后者至关重要,因为细微的参数可能不易操作,而且其稳健的估计不可行,特别是如果两个功能性时间序列取决于这些新特点,则测试的特性对于任何选择的频率频带的特性都具有很强性能,从而能够以单一频率比较能源内容。测试的有限样品性能通过模拟研究得到验证,并与FMRI数据的应用加以说明。