Nonparametric testing of lack of dependence in functional linear models

Area of Science:

• Statistics
• Functional Data Analysis

Background:

• Testing dependence between response and functional predictors is crucial in functional linear models.
• Traditional methods using functional principal component analysis require accurate covariance operator estimation, which is challenging with high-dimensional functional data and small sample sizes, often leading to underpowered tests.

Purpose of the Study:

• To propose a novel nonparametric method, Functional Linear models with U-statistics TEsting (FLUTE), to test the dependence assumption in functional linear models.
• To overcome the limitations of existing methods that rely on estimating the covariance operator, particularly in scenarios with small or moderate sample sizes.

Main Methods:

• Developed a nonparametric testing procedure named Functional Linear models with U-statistics TEsting (FLUTE).
• Avoided the computationally intensive estimation of the covariance operator inherent in traditional functional principal component analysis-based methods.
• Established the asymptotic normality of the FLUTE test statistic under both null and local alternative hypotheses.

Main Results:

• The FLUTE test demonstrates superior power compared to benchmark methods in small or moderate sample cases.
• Asymptotic normality of the test statistic is proven, providing theoretical justification for its use.
• Simulation studies and real-world data analyses confirm the practical effectiveness and advantages of the FLUTE method.

Conclusions:

• The FLUTE method offers a powerful and reliable approach for testing dependence in functional linear models, especially when sample sizes are limited.
• This nonparametric approach circumvents the need for estimating the functional predictor's covariance operator, enhancing test power and applicability.
• FLUTE provides a valuable alternative for researchers and practitioners dealing with high-dimensional functional data analysis.