A Lack-Of-Fit Test with Screening in Sufficient Dimension Reduction
A Lack-Of-Fit Test with Screening in Sufficient Dimension Reduction
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DOI：
10.5705/ss.202018.0176
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发表时间：
2020
影响因子：
1.4
通讯作者：
Yaowu Zhang;Wei Zhong;Liping Zhu
中科院分区：
文献类型：
--
作者：
Yaowu Zhang;Wei Zhong;Liping Zhu
研究人员经常需要推断反应的条件均值如何随预测因子而变化。足够的降维技术通过识别原始预测因子的线性组合的最小集合来降低维度，而不损失信息。本研究检验了原始超高维协变量的给定少量线性组合是否足以表征响应的条件均值。首先，我们引入了一种新的一致的失拟检验统计量的情况下，协变量的维数是中等的。所提出的测试被证明是n一致的零假设下，根n一致的备择假设下。利用Bootstrap方法对检验的p值进行了近似，并从理论上研究了检验的一致性。为了处理非线性维数，我们引入了一个两阶段的失拟检验与筛选（LOFTS）程序，数据分裂策略的基础上。数据被随机分成相等的两半。在第一阶段，我们应用鞅差相关筛选的一半的数据，并选择一个温和的协变量集。中国统计：预印本doi：10.5705/ss.202018.0176
Researchers often need to infer how the conditional mean of a response varies with the predictors. Sufficient dimension-reduction techniques reduce the dimension by identifying a minimal set of linear combinations of the original predictors, without loss of information. This study tests whether a given small number of linear combinations of the original ultrahigh-dimensional covariates is sufficient to characterize the conditional mean of the response. We first introduce a novel consistent lack-of-fit test statistic for the case when the dimensionality of the covariates is moderate. The proposed test is shown to be n-consistent under the null hypothesis, and root-n-consistent under the alternative hypothesis. A bootstrap procedure is developed to approximate the p-values, and the consistency of the test is studied theoretically. To deal with the ultrahigh dimensionality, we introduce a two-stage lack-of-fit test with screening (LOFTS) procedure, based on a data-splitting strategy. The data are randomly partitioned into two equal halves. In the first stage, we apply the martingale difference correlationbased screening to one half of the data, and select a moderate set of covariates. Statistica Sinica: Preprint doi:10.5705/ss.202018.0176