Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models

• URL: http://arxiv.org/abs/2603.23835v1
• Date: Wed, 25 Mar 2026 01:43:12 GMT
• Title: Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models
• Abstract summary: It is unclear how Cox-based optimization error propagates to population risk under partial likelihood.<n>It is also unclear how pointwise bias can be controlled, and how ensemble-based uncertainty behaves under realistic variance regimes.<n>We develop an distribution theory for deep Cox estimators that addresses these issues.
• Score: 7.2275111013288535
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Related papers

• Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression [18.254578337569264]
  This work proposes a two-stage estimator that combines conditional diffusion modeling with a kernel-parametriced conditional moment formulation.<n>We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model.<n> Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches.
  arXiv  Detail & Related papers  (2026-08-08T16:01:24Z)
• An Isotropic Approach to Efficient Uncertainty Quantification with Gradient Norms [7.439615448864818]
  We propose a lightweight method for quantifying predictive uncertainty in neural networks.<n>We use two approximations: a first-order Taylor expansion that expresses uncertainty in terms of the gradient of the prediction, and an isotropy assumption on the parameter covariance.<n>We then use the estimates to investigate when each uncertainty type carries useful signal for predicting answer correctness in question answering with large language models.
  arXiv  Detail & Related papers  (2026-03-31T09:13:09Z)
• Learning bounds for doubly-robust covariate shift adaptation [8.24901041136559]
  Distribution shift between the training domain and the test domain poses a key challenge for machine learning.<n> doubly-robust (DR) estimator combines density ratio estimation with a pilot regression model.<n>This paper establishes the first non-asymptotic learning bounds for the DR estimator.
  arXiv  Detail & Related papers  (2025-11-14T06:46:23Z)
• Regularizing Extrapolation in Causal Inference [12.057981453189505]
  We propose a unified framework that directly penalizes the level of extrapolation.<n>We derive a worst-case extrapolation error bound and introduce a novel "bias-bias-variance" tradeoff.
  arXiv  Detail & Related papers  (2025-09-21T18:05:15Z)
• In-Context Parametric Inference: Point or Distribution Estimators? [66.22308335324239]
  We show that amortized point estimators generally outperform posterior inference, though the latter remain competitive in some low-dimensional problems.<n>Our experiments indicate that amortized point estimators generally outperform posterior inference, though the latter remain competitive in some low-dimensional problems.
  arXiv  Detail & Related papers  (2025-02-17T10:00:24Z)
• Deep Partially Linear Transformation Model for Right-Censored Survival Data [6.315323176162257]
  This paper introduces a deep partially linear transformation model (DPLTM) as a general and flexible regression framework.<n>The proposed method is capable of avoiding the curse of dimensionality while still retaining the interpretability of some cocensors of interest.<n> Comprehensive simulation studies demonstrate the impressive performance of the proposed procedure in terms of both accuracy and the predictive power.
  arXiv  Detail & Related papers  (2024-12-10T15:50:43Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate.<n>We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Quantifying predictive uncertainty of aphasia severity in stroke patients with sparse heteroscedastic Bayesian high-dimensional regression [47.1405366895538]
  Sparse linear regression methods for high-dimensional data commonly assume that residuals have constant variance, which can be violated in practice. This paper proposes estimating high-dimensional heteroscedastic linear regression models using a heteroscedastic partitioned empirical Bayes Expectation Conditional Maximization algorithm.
  arXiv  Detail & Related papers  (2023-09-15T22:06:29Z)
• Optimal variance-reduced stochastic approximation in Banach spaces [114.8734960258221]
  We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. We establish non-asymptotic bounds for both the operator defect and the estimation error.
  arXiv  Detail & Related papers  (2022-01-21T02:46:57Z)
• Learning to Estimate Without Bias [57.82628598276623]
  Gauss theorem states that the weighted least squares estimator is a linear minimum variance unbiased estimation (MVUE) in linear models. In this paper, we take a first step towards extending this result to non linear settings via deep learning with bias constraints. A second motivation to BCE is in applications where multiple estimates of the same unknown are averaged for improved performance.
  arXiv  Detail & Related papers  (2021-10-24T10:23:51Z)
• Near-optimal inference in adaptive linear regression [60.08422051718195]
  Even simple methods like least squares can exhibit non-normal behavior when data is collected in an adaptive manner. We propose a family of online debiasing estimators to correct these distributional anomalies in at least squares estimation. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.
  arXiv  Detail & Related papers  (2021-07-05T21:05:11Z)
• Amortized Conditional Normalized Maximum Likelihood: Reliable Out of Distribution Uncertainty Estimation [99.92568326314667]
  We propose the amortized conditional normalized maximum likelihood (ACNML) method as a scalable general-purpose approach for uncertainty estimation. Our algorithm builds on the conditional normalized maximum likelihood (CNML) coding scheme, which has minimax optimal properties according to the minimum description length principle. We demonstrate that ACNML compares favorably to a number of prior techniques for uncertainty estimation in terms of calibration on out-of-distribution inputs.
  arXiv  Detail & Related papers  (2020-11-05T08:04:34Z)
• Distributionally Robust Parametric Maximum Likelihood Estimation [13.09499764232737]
  We propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric nominal distribution. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.
  arXiv  Detail & Related papers  (2020-10-11T19:05:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.