Kong Dehan, Staicu Ana-Maria, Maity Arnab
Department of Biostatistics, University of North Carolina, Chapel Hill, NC, 27599, U.S.A.
Department of Statistics, North Carolina State University, Raleigh, NC 27695, U.S.A.
J Nonparametr Stat. 2016;28(4):813-838. doi: 10.1080/10485252.2016.1231806. Epub 2016 Aug 20.
We extend four tests common in classical regression - Wald, score, likelihood ratio and F tests - to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.
我们将经典回归中常用的四个检验—— Wald 检验、得分检验、似然比检验和 F 检验——扩展到函数线性回归,用于检验原假设,即标量响应与函数协变量之间不存在关联。使用函数主成分分析,我们将函数线性模型重新表示为标准线性模型,其中函数协变量的效应可以通过函数主成分得分的有限线性组合来近似。在此背景下,我们考虑应用这四个传统检验。当主成分数量发散时,针对密集观测的函数协变量,从理论上研究了所提出的检验程序。利用备择假设下检验的理论分布,我们开发了一种在函数线性回归背景下计算样本量的程序。在模拟实验中以及使用两个实际数据应用,对密集和稀疏观测的有噪声函数数据,进一步对这四个检验进行了数值比较。