函数数据的单向ANOVA检验及其图形解释
A one-way ANOVA test for functional data with graphical interpretation
摘要：
提出了一种新的函数方差分析检验方法，并对结果进行了图形化解释。该测试是Myllymaki等人引入的全局包络测试的扩展（2017，空间过程的全局包络测试，J. R。中央集权主义者Soc.B79，381- 404，doi：10.1111/rssb.12172）。图形化的解释是实现了一个全球性的信封，这是联合绘制的所有样本的功能。如果根据经验数据计算的均值函数在给定包络之外，则以预定显著性水平$α$拒绝原假设。建议的单向功能方差分析的优点是，它确定的域的功能是负责潜在的拒绝。我们介绍两个版本的测试：第一个给出了一个图形解释的测试结果在原始空间的功能和第二个立即提供了一个事后检验，通过识别组之间的显着的成对差异。所提出的检验依赖于函数的离散化，因此该检验也适用于多维ANOVA问题。在文章的实证部分，我们通过分析欧洲国家的财政分权来展示该方法的使用。实证分析的目的是捕捉不同的欧洲国家集团之间的政府支出分散化比率的水平之间的差异。根据现有文献，其背后的想法很简单：欧洲一体化历史较长的国家应该分散更多的政府支出。我们使用1995年至2016年期间29个欧盟和EFTA国家的政府支出集中率，根据欧洲经济和政治一体化的假设水平分为三组。
英文摘要：
A new functional ANOVA test, with a graphical interpretation of the result, is presented. The test is an extension of the global envelope test introduced by Myllymaki et al. (2017, Global envelope tests for spatial processes, J. R. Statist. Soc. B 79, 381--404, doi: 10.1111/rssb.12172). The graphical interpretation is realized by a global envelope which is drawn jointly for all samples of functions. If a mean function computed from the empirical data is out of the given envelope, the null hypothesis is rejected with the predetermined significance level $α$. The advantages of the proposed one-way functional ANOVA are that it identifies the domains of the functions which are responsible for the potential rejection. We introduce two versions of this test: the first gives a graphical interpretation of the test results in the original space of the functions and the second immediately offers a post-hoc test by identifying the significant pair-wise differences between groups. The proposed tests rely on discretization of the functions, therefore the tests are also applicable in the multidimensional ANOVA problem. In the empirical part of the article, we demonstrate the use of the method by analyzing fiscal decentralization in European countries. The aim of the empirical analysis is to capture differences between the levels of government expenditure decentralization ratio among different groups of European countries. The idea behind, based on the existing literature, is straightforward: countries with a longer European integration history are supposed to decentralize more of their government expenditure. We use the government expenditure centralization ratios of 29 European Union and EFTA countries in period from 1995 to 2016 sorted into three groups according to the presumed level of European economic and political integration.
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