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

2025-11-13

AI Summary

To address the absence of formal goodness-of-fit tests for generalized functional linear models (GFLMs), this paper proposes a Cramér–von Mises-type U-statistic test based on projection averaging. The method integrates over all one-dimensional projection directions to enhance detection power and mitigate the curse of dimensionality induced by high-dimensional functional predictors, accommodating both continuous and discrete responses. This work introduces the projection averaging strategy to GFLM hypothesis testing for the first time and employs bootstrap resampling to estimate p-values, circumventing challenges posed by the intractable asymptotic variance. Theoretically, the test is shown to be asymptotically normal under the null hypothesis and uniformly consistent against local alternatives. Simulation studies demonstrate that the proposed test achieves substantially higher empirical power compared to existing approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Develops a goodness-of-fit test for Generalized Functional Linear Models
• Addresses the curse of dimensionality through projection averaging strategy
• Provides practical bootstrap methods for p-value computation

Innovation

• U-statistic test using Cramér-von-Mises metric
• Projection averaging over functional predictor dimensions
• Bootstrap resampling for practical p-value computation