# Flexible Deep Neural Networks for Partially Linear Survival Data: Estimation and Survival Inference

arXiv: https://arxiv.org/abs/2512.10570 · HTML: https://mlsanity.com/paper/2512.10570/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 9 of 9)

- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network](https://mlsanity.com/paper/1606.00931/) (2016)
- [Time-to-Event Prediction with Neural Networks and Cox Regression](https://mlsanity.com/paper/1907.00825/) (2019)
- [A Scalable Discrete-Time Survival Model for Neural Networks](https://mlsanity.com/paper/1805.00917/) (2018)
- [Image-based Survival Analysis for Lung Cancer Patients using CNNs](https://mlsanity.com/paper/1808.09679/) (2018)
- [Deep Convolutional Neural Networks for Imaging Data Based Survival Analysis of Rectal Cancer](https://mlsanity.com/paper/1901.01449/) (2019)
- [Conditional Distribution Function Estimation Using Neural Networks for Censored and Uncensored Data](https://mlsanity.com/paper/2207.02384/) (2022)
- [Optimal Cox Regression Subsampling Procedure with Rare Events](https://mlsanity.com/paper/2012.02122/) (2020)
- [Confidence Intervals and Simultaneous Confidence Bands Based on Deep Learning](https://mlsanity.com/paper/2406.14009/) (2024)

## Abstract (arXiv)

We propose a flexible deep neural network (DNN) framework for modeling survival data within a partially linear regression structure. The approach preserves interpretability through a parametric linear component for covariates of primary interest, while a nonparametric DNN component captures complex time-covariate interactions among nuisance variables. We refer to the method as FLEXI-Haz, a FLEXIble Hazard model with a partially linear structure. In contrast to existing DNN approaches for partially linear Cox models, FLEXI-Haz does not rely on the proportional hazards assumption. We establish theoretical guarantees: the neural network component attains minimax-optimal convergence rates over composite H"older classes, the linear estimator is sqrt-n-consistent, asymptotically normal, and semiparametrically efficient, and we develop a cross-fitted one-step estimator of the cumulative hazard and survival function for a new subject, together with pointwise asymptotic confidence intervals. To the best of our knowledge, this is the first frequentist asymptotic pointwise inference result for a survival function in a DNN survival model, with or without a linear component. Simulations and real-data analyses demonstrate the utility of FLEXI-Haz as a principled and interpretable alternative to methods based on proportional hazards.