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arXiv · 2609.38611

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

Kendall Thomas·Jan Hannig
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Abstract

High-frequency athlete-tracking data can be summarized as quantile cubes (Thomas and Hannig, 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 ranging from under a hundred to several thousand. Whether a bin's negative binomial distribution is well approximated by a normal distribution depends on its shape parameter, the product of its mean and dispersion, rather than 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. This approach provides uncertainty quantification for hundreds of regression and dispersion parameters without specifying a prior distribution. Only the normal components contribute to the fiducial Jacobian, which has a closed-form block-diagonal structure, 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 the importance of partitioning by estimated shape rather than by the mean or observed responses. In 216 sessions from 17 professional women's soccer athletes, the estimated position effects concentrate at the velocity extremes, while the estimated age effects follow an acceleration gradient. However, an athlete-level permutation check indicates that these patterns are not distinguishable from those obtained under athlete-label permutations.

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Kendall Thomas, Jan Hannig. 2026-10-19. Generalized Fiducial Inference for Hybrid Discrete-Continuous Count Data: Theory and Application. https://arxiv.org/abs/2609.38611

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