As medical devices become more complex, they routinely collect extensive and complicated data. While classical regressions typically examine the relationship between an outcome and a vector of predictors, it becomes imperative to identify the relationship with predictors possessing functional structures. In this article, we introduce a novel inference procedure for examining the relationship between outcomes and large-scale functional predictors. We target testing the linear hypothesis on the functional parameters under the generalized functional linear regression framework, where the number of the functional parameters grows with the sample size. We develop the estimation procedure for the high dimensional generalized functional linear model incorporating B-spline functional approximation and amenable regularization. Furthermore, we construct a procedure that is able to test the local alternative hypothesis on the linear combinations of the functional parameters. We establish the statistical guarantees in terms of non-asymptotic convergence of the parameter estimation and the oracle property and asymptotic normality of the estimators. Moreover, we derive the asymptotic distribution of the test statistic. We carry out intensive simulations and illustrate with a new dataset from an Alzheimer's disease magnetoencephalography study.

翻译：随着医疗设备日趋复杂，其收集的数据日益广泛且复杂。传统回归分析通常关注结果变量与预测变量向量之间的关系，但识别与具有功能结构的预测变量之间的关系变得至关重要。本文提出了一种新的推断方法，用于检验结果变量与大规模功能型预测变量之间的关系。我们针对广义功能线性回归框架下的功能参数线性假设进行检验，其中功能参数的数量随样本量增长。我们发展了适用于高维广义功能线性模型的估计程序，该程序融合了B样条函数逼近与适应性正则化。此外，我们构建了能够检验功能参数线性组合的局部备择假设的检验程序。我们在参数估计的非渐近收敛性、估计量的Oracle性质及渐近正态性方面建立了统计保证，并推导了检验统计量的渐近分布。我们进行了大量仿真实验，并通过一项阿尔茨海默病脑磁图研究的新数据集进行了实证分析。