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.
Figures & tables
Figure 1: A fully connected neural network with L=3 and p=(5,4,4,3,1) .
Case
Setting
Bias
MAE
EmpSD
SE
CP
AIL
1
n=800,r=⌊n0.85⌋
-0.02
0.36
0.48
0.50
0.87
1.98
n=800,r=⌊n0.90⌋
-0.02
0.21
0.45
0.48
0.92
1.86
n=800,r=⌊n0.95⌋
-0.03
0.11
0.40
0.44
0.95
1.72
2
n=800,r=⌊n0.85⌋
0.11
0.26
0.47
0.49
0.89
1.91
n=800,r=⌊n0.90⌋
0.05
0.14
0.42
0.44
0.93
1.74
n=800,r=⌊n0.95⌋
0.04
0.11
0.36
0.40
0.95
1.59
Table 1: Simulation summary for pointwise inference on g0(x) across test points. EmpSD is the empirical standard deviation of gB(x) over replications. SE is the mean estimated standard error. CP is the empirical coverage probability of nominal 95% intervals, and AIL is the mean interval length.
Figure 2: Estimation and inference summary for Model 3 with n=800 , r=⌊n0.90⌋ , B=1000 , DNN architecture (p0,128,64,1) . Panel (a) plots gB(x) versus g0(x) across test points. Panel (b) compares EmpSD with the estimated SE , highlighting corrected and uncorrected versions when available.
Figure 3: Empirical performance across the subsampling index α . At smaller values ( α≤0.8 ), high MAE degrades the CP, validating the theoretical lower bound αlower . Conversely, as α→1 , the CP drops significantly despite stable MAE, and variance estimates diverge, empirically validating the upper bound αupper .
Case
Setting
Bias
MAE
EmpSD
SE
CP
AIL
1
n=800,r=⌊n0.85⌋
-0.45
0.65
0.69
0.72
0.94
2.83
n=800,r=⌊n0.90⌋
-0.23
0.52
0.62
0.67
0.95
2.61
n=800,r=⌊n0.95⌋
-0.22
0.50
0.55
0.63
0.97
2.46
2
n=800,r=⌊n0.85⌋
-0.17
0.59
0.71
0.69
0.94
2.72
n=800,r=⌊n0.90⌋
-0.02
0.45
0.55
0.64
0.97
2.51
n=800,r=⌊n0.95⌋
0.06
0.43
0.53
0.56
0.96
2.21
Table 2: Simulation summary for inference on the log-hazard ratio contrast ψ(xtest,i,xtest,j) .
Case
n
Method
MAE
EmpSD
SE
CP
AIL
Case 1
800
CoxPH
0.11
0.03
0.14
0.95
0.57
RSF
0.69
0.02
0.20
0.53
1.03
GBR
0.33
0.03
0.35
0.89
1.37
DeepSurv
0.39
0.06
2.52
0.99
2.63
ESM
0.11
0.40
0.44
0.95
1.72
1000
CoxPH
0.10
0.03
0.13
0.95
0.50
Table 3: Estimation Performance and Uncertainty Quantification (Replications R=200 , Bootstrap=200 ) for Model 1 and 3. Pointwise metrics are averaged over the test set xtest .
Method
Mean C
C-index SE
C-index AIL
Mean AUC
AUC SE
AUC AIL
ESM
0.7316
0.0018
0.0074
0.7987
0.0039
0.0171
DeepSurv
0.7284
0.0019
0.0076
0.7928
0.0043
0.0186
CoxPH
0.7223
0.0020
0.0080
0.7944
0.0050
0.0214
RF
0.7221
0.0020
0.0080
0.7928
0.0058
0.0247
GBR
0.7209
0.0019
0.0076
0.7865
0.0043
0.0184
Table 4: Comparative Performance of Survival Models on the BLCS Inference Set
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.
Flexible survival prediction does not by itself ensure valid inference for covariate effects or subject-specific survival probabilities. We develop FLEXI-Haz, a semiparametric deep neural network model for right-censored data that combines an interpretable linear component in the log-hazard with a flexible function of time and adjustment covariates. This formulation permits nonproportional nuisance effects while retaining interpretable regression parameters. Under regularity conditions, we establish nuisance convergence rates that are minimax optimal up to logarithmic factors over composite Holder classes, together with root-n-consistent, asymptotically normal, and semiparametrically efficient estimation of the regression parameter. For adjustment covariates with finite support, we derive the explicit influence function for cumulative hazard and construct a cross-fitted one-step estimator yielding asymptotically valid pointwise confidence intervals for subject-specific survival. Simulations demonstrate reduced bias and improved coverage relative to a proportional hazards comparator under time-varying nuisance effects. An analysis of the Rotterdam breast cancer data illustrates the consequences of flexible adjustment for inference on treatment associations.
Asaf Ben Arie, Malka Gorfine
Department of Statistics and Operations Research, Tel Aviv University, Israel
This paper develops theory for deep neural network (DNN) estimators under dependent data. To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical L2-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary β-mixing data taking values in unbounded sets. I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights. For both DNN classes, I derive general results when the function to be estimated is Hölder smooth and the data are nonstationary, subgaussian, and β-mixing with either exponential or polynomial decay. I then specialize these to nonparametric regression, logistic regression, and quantile regression settings. Under exponential β-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.
Chad Brown
Department of Economics, University of Colorado Boulder, 80309, USA.
Time-to-event data is widespread across the life sciences and engineering, but it is typically encountered together with censoring, which complicates the application of standard machine learning methods. Deep Cox models have emerged as a popular method for analyzing time-to-event data because they gracefully handle censoring and can be used with unstructured data such as clinical text reports, genomic sequences, and pathology images. However, their predicted survival probabilities are often poorly calibrated, thus limiting their practical utility. In this paper, we propose a novel post hoc calibration method for Deep Cox models that uses isotonic regression to refine predicted survival probabilities without affecting discriminative power. We establish favorable theoretical guarantees, including a double-robustness property and asymptotic calibration. Experiments on synthetic and real-world clinical data demonstrate the empirical effectiveness of our method.