Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sattwik Ghosal, Xuran Meng, Yi Li

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.

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

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.
Paper Structure (12 sections, 7 theorems, 50 equations, 5 figures, 4 tables)

This paper contains 12 sections, 7 theorems, 50 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Multilayer Neural Networks
  5. Nonparametric Cox Models and Hölder Functional Classes
  6. Inference on DNN Estimates via an Ensemble Subsampling Learner
  7. Estimation and Inference via Ensemble Subsampling
  8. Theoretical Results
  9. Simulation Studies
  10. Comparison with Competing Methods
  11. Real Data Analysis
  12. Conclusion

Key Result

Theorem 1

Let the true function $g_0$ be specified in eq:true_functional_class with $g_0(\mathbf{0}) = 0$. Under Condition as:DNN_structures, it holds that where $\eta = \frac{2\delta\gamma_{i_{\text{eff}}}^*}{t_{i_{\text{eff}}}c_{\text{eff}}}$, with $c_{\text{eff}}$ and $i_{\text{eff}}$ defined in junk.

Figures (5)

  • Figure 1: A fully connected neural network with $L = 3$ and $\mathbf{p} = (5,4,4,3,1)$.
  • Figure 2: Estimation and inference summary for Model 3 with $n=800$, $r=\lfloor n^{0.90}\rfloor$, $B=1000$, DNN architecture $(p_0,128,64,1)$. Panel (a) plots $\widehat{g}_{B}(\mathbf{x})$ versus $g_0(\mathbf{x})$ across test points. Panel (b) compares $\mathrm{EmpSD}$ with the estimated $\mathrm{SE}$, highlighting corrected and uncorrected versions when available.
  • Figure 3: Empirical performance across the subsampling index $\alpha$. At smaller values ($\alpha \le 0.8$), high MAE degrades the CP, validating the theoretical lower bound $\alpha_{\text{lower}}$. Conversely, as $\alpha \to 1$, the CP drops significantly despite stable MAE, and variance estimates diverge, empirically validating the upper bound $\alpha_{\text{upper}}$.
  • Figure 4: Comparative evaluation of model performance. The ensemble shows strong ranking stability and stable estimation accuracy over time.
  • Figure 5: Stage-specific contrast HR studies. In each panel, curves correspond to Stages 2, 3, and 4, with shaded regions indicating IJ-based 95% confidence bands.

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Theorem 1: Bound of Optimal Approximation Error
  • Theorem 2
  • Corollary 1: Pointwise Mean Squared Error Bound
  • Theorem 3
  • Theorem 4
  • Corollary 2
  • Theorem 5