Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are compared in simulations and illustrated in real data examples.
翻译：文献中提议的多数线性减少量方法可以使用适当的散射矩阵来拟订,例如,见Ye和Weiss(2003年)、Tyler等人(2009年)、Bura和Yang(2011年)、Liski等人(2014年)、Luo和Li2016年),然后经常使用一个散射矩阵对另一个散射矩阵的eigen分解法来确定信号子空间的尺寸,并分离数据的信号和噪音部分。三种流行的减少量方法,即主要组成部分分析(PCA)、第四级盲点识别(FOBI)和分片反反回归(SIR),详细考虑了这些方法,头两个点用于测试信号空间的尺寸。讨论限制试验统计数据的空格分布,并建议小样案例采用新的靴式战略。在所有三个案例中,还采用了对信号子空间的信号子层面的一致测试估计值。模拟中比较了无孔和靴式测试,并在真实数据实例中加以说明。
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