Torey Hilbert | Searcharxiv

arXiv2026

Predictive Inference via Kernel Density Estimates

Kernel density estimation is a widely used nonparametric approach to estimate an unknown distribution. Recent work in Bayesian predictive inference has considered stochastic processes formed by specifying the predictive distribution for the next data point given all observed data such that the resulting predictive distributions converge weakly almost surely. We study two kernel based prediction rules: the classic kernel density estimator, and a recursive version previously introduced for online problems. We show that both processes converge weakly almost surely, which opens the door for new Bayesian interpretations of kernel density estimation. Surprisingly, the process based on the classic kernel density estimates converges to a compactly supported measure, while the recursive version converges to a non-compactly supported measure.

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arXiv2024

Robust distribution-free tests for the linear model

Recently, there has been growing concern about heavy-tailed and skewed noise in biological data. We introduce RobustPALMRT, a flexible permutation framework for testing the association of a covariate of interest adjusted for control covariates. RobustPALMRT controls type I error rate for finite-samples, even in the presence of heavy-tailed or skewed noise. The new framework expands the scope of state-of-the-art tests in three directions. First, our method applies to robust and quantile regressions, even with the necessary hyper-parameter tuning. Second, by separating model-fitting and model-evaluation, we discover that performance improves when using a robust loss function in the model-evaluation step, regardless of how the model is fit. Third, we allow fitting multiple models to detect specialized features of interest in a distribution. To demonstrate this, we introduce DispersionPALRMT, which tests for differences in dispersion between treatment and control groups. We establish theoretical guarantees, identify settings where our method has greater power than existing methods, and analyze existing immunological data on Long-COVID patients. Using RobustPALMRT, we unveil novel differences between Long-COVID patients and others even in the presence of highly skewed noise.

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