Goodness-of-fit Test for Generalized Functional Linear Models via Projection Averaging

TL;DR

This paper develops a new goodness-of-fit test for Generalized Functional Linear Models using projection averaging, which effectively handles high-dimensional functional data and shows strong performance in simulations.

Contribution

It introduces a novel projection-averaged U-statistic based test for GFLMs, with asymptotic properties and bootstrap methods for practical p-value computation.

Findings

Test demonstrates good power in simulations

Effective in high-dimensional settings

Outperforms existing methods

Abstract

Assessing model adequacy is a crucial step in regression analysis, ensuring the validity of statistical inferences. For Generalized Functional Linear Models (GFLMs), which are widely used for modeling relationships between scalar responses and functional predictors, there is a recognized need for formal goodness-of-fit testing procedures. Current literature on this specific topic remains limited. This paper introduces a novel goodness-of-fit test for GFLMs. The test statistic is formulated as a U-statistic derived from a Cram\'er-von-Mises metric integrated over all one-dimensional projections of the functional predictor. This projection averaging strategy is designed to effectively mitigate the curse of dimensionality. We establish the asymptotic normality of the test statistic under the null hypothesis and prove the consistency under the alternatives. As the asymptotic variance of the limiting null distribution can be complex for practical use, we also propose practical bootstrap resampling methods for both continuous and discrete responses to compute p-values. Simulation studies confirm that the proposed test demonstrates good power performance across various settings, showing advantages over existing methods.