Generalized Fiducial Inference for Hybrid Discrete-Continuous Count Data: Theory and Application

2026-09-29

AI Summary

This study addresses the challenge of statistical inference arising from pronounced discrepancies in discrete count distribution shapes within high-frequency athlete tracking data. To this end, it proposes a hybrid generalized confidence inference framework that adaptively selects between exact negative binomial and normal approximation models based on estimated shape parameters. Furthermore, a dynamic partitioning strategy combined with block-diagonal Jacobian computation is designed to enable prior-free uncertainty quantification for hundreds of parameters, supported by a formal proof of the Bernstein–von Mises theorem. Simulations demonstrate that coverage probabilities closely approximate nominal levels, while empirical analyses reveal that positional age effects are substantially confounded by label noise. Collectively, this work provides a reliable statistical inference tool for complex sports tracking data.

Abstract

High-frequency athlete-tracking data can be summarized as quantile cubes (Thomas and Hannig, Journal of Quantitative Analysis in Sports, 2026): counts of time spent in bins of velocity, acceleration and movement angle. In our application each athlete-match session has 100 such bins, with counts from under a hundred to several thousand. Whether a bin's negative binomial distribution is close to normal depends on its shape, the product of mean and dispersion, not on the count alone. We develop a hybrid generalized fiducial framework that uses exact negative binomial structural equations where the estimated shape is small and moment-matched normal approximations where it is large, providing uncertainty quantification for hundreds of regression and dispersion parameters without a prior. Only the normal components contribute to the fiducial Jacobian, which is block diagonal in closed form, and we prove a Bernstein-von Mises theorem for the correctly specified hybrid model. Simulations show near-nominal coverage, agreement with a flat-prior Bayesian analysis, and that the partition should be based on the estimated shape rather than on the mean or the observed responses. In 216 sessions from 17 professional women's soccer athletes, the estimated position and age effects concentrate at the velocity extremes and highest accelerations, but an athlete-level permutation check shows these patterns cannot be distinguished from label noise.

Problem

• hybrid discrete-continuous count data
• generalized fiducial inference
• uncertainty quantification
• negative binomial distribution
• athlete-tracking data

Innovation

• Generalized Fiducial Inference
• Hybrid Discrete-Continuous Model
• Negative Binomial Distribution
• Bernstein-von Mises Theorem
• Uncertainty Quantification