Theoretical extensions of ultra-high-dimensional oracle variable selection for the Cox model with interval-censored data: left truncation, transformation models, grouped penalties, diverging sparsity, and zero-consistency of the initial estimator

Extend the oracle-variable-selection theory of Pak, Zhang, Wu, Weng, and Li (arXiv:2405.01275; published in the Journal of Statistical Planning and Inference, 2026), who developed the first variable selection methods for the Cox model with interval-censored data that enjoy the oracle property in the ultra-high-dimensional regime where the covariate dimension can grow exponentially with the sample size ($\log p = O(n^{\delta})$). Their theory holds under restrictions: proportional hazards, no left truncation, individual (ungrouped) penalties, a fixed sparsity level $s$, and — for the adaptive lasso — an assumed zero-consistent initial estimator whose existence is unproven. The paper explicitly poses the removal of each restriction as an open problem, stating verbatim that investigating the consistency of the lasso in high-dimensional Cox models with interval-censored data remains open.

Background

Survival analysis models the time to an event as a function of covariates. The Cox proportional hazards model assumes the hazard (instantaneous event rate) at time $t$ for a subject with covariate vector $\mathbf{Z}$ is $\lambda_0(t)\exp(\boldsymbol{\beta}^{\top}\mathbf{Z})$, with $\lambda_0$ an unspecified baseline hazard and $\boldsymbol{\beta}$ the regression coefficients. Estimation commonly faces censoring: exact event times are unobserved. Under right censoring one observes either the event time or a time after which the event had not yet occurred; the Cox partial likelihood then yields tractable inference. Interval censoring is harder: the event time $T$ is only known to lie in an interval $(L_i, R_i]$, as happens when subjects are inspected at a finite sequence of visit times (for example in dental or Alzheimer's studies). In that case there is no partial likelihood; one maximizes the full likelihood jointly over $\boldsymbol{\beta}$ and the baseline cumulative hazard function $\Lambda$, treating $\Lambda$ nonparametrically. The nonparametric maximum likelihood estimator (NPMLE) of $\Lambda$ is a step function whose jumps are confined to a data-determined set of disjoint "maximal intersections", and the likelihood is non-smooth in $\Lambda$, so $\hat{\Lambda}$ converges at the non-standard rate $n^{1/3}$ rather than $n^{1/2}$. Computation proceeds via an expectation-maximization (EM) algorithm that treats the unobserved exact event times as missing data.

In modern biomedical applications such as genome-wide association studies the covariate dimension $p$ can exceed the sample size $n$ by orders of magnitude, so variable selection is essential. Penalized methods add a penalty on the coefficients to the likelihood: the lasso ($L_1$ penalty) shrinks coefficients exactly to zero but introduces bias; the adaptive lasso reweights the $L_1$ penalty by inverse magnitudes of a preliminary "initial estimator" to remove that bias; and folded-concave penalties such as SCAD (smoothly clipped absolute deviation) and MCP (minimax concave penalty) combine unbiasedness for large coefficients with sparsity. A selection method has the oracle property when, with probability tending to one, it recovers exactly the set of nonzero coefficients, and its estimates of those coefficients have the same asymptotic distribution as the maximum likelihood estimator one could compute if that true set were known in advance.

For right-censored Cox data, oracle properties under "NP-dimensionality" ($p$ growing non-polynomially in $n$) were established by Bradic, Fan, and Jiang (2011) for folded-concave penalties, and Huang, Sun, Ying, Yu, and Zhang (2013) proved oracle inequalities for the lasso partial likelihood. For interval-censored data the theory lagged: the adaptive lasso of Li, Pak, and Todem (2020) — which also allows left truncation, meaning subjects who experienced the event before their study entry time are never observed — has the oracle property only in fixed dimension; the broken adaptive ridge regression of Zhao, Wu, Li, and Sun (2020) and the group-penalized sieve maximum likelihood method of Wu, Zhao, and Sun (2023) cover settings where $p$ diverges slower than $n$; and the high-dimensional methods of Wu et al. (2020) and Tian and Sun (2023), as characterized in the source paper, lack theoretical guarantees.

The source paper (Pak, Zhang, Wu, Weng, and Li, arXiv:2405.01275, published 2026) closed this gap: it developed penalized NPMLE methods with lasso, adaptive lasso, SCAD, and MCP penalties, computed by an EM algorithm, and proved the oracle property for folded-concave penalties and the adaptive lasso in the ultra-high-dimensional regime $\log p = O(n^{\delta})$, $\delta \in (0,1)$ — the first such results for interval-censored data. The theory, however, rests on restrictions that the authors themselves flag. Condition 5 fixes the effective dimensionality $s$ (the number of nonzero coefficients) and the coefficient values as $n$ grows, which is stronger than the counterparts for generalized linear models and right-censored Cox models. The adaptive-lasso theorems (Theorems 3 and 4 of the paper) are conditional on an initial estimator $\tilde{\boldsymbol{\beta}}$ being "zero-consistent" with some rate $r_n$ — meaning $r_n \max_{j \notin \mathcal{M}_*} |\tilde{\beta}_j| = O_P(1)$ over the null coefficients (with $r_n \to \infty$) while all nonzero coefficients stay bounded away from zero — following the notion of Huang, Ma, and Zhang (2008). The theorems require $\theta_n r_n \gg n^{\max\{-1/3,(\delta-1)/2\}}$ (with $\theta_n$ the tuning parameter), and Theorem 4's asymptotic normality additionally requires $\theta_n = o(n^{-1/2})$, which holds if the initial estimator's $L_2$-rate is faster than $n^{\min\{-1/6,-\delta/2\}}$. In all simulations and the genome-wide association application the lasso estimator served as the initial estimator, but the authors could not prove it satisfies these conditions: unlike right censoring, the Cox likelihood under interval censoring has no closed-form profile likelihood, and the best available uniform bound on the profile score vector is of order $n^{-1/3}$, inherited from the rate of $\hat{\Lambda}$. The paper states that, to the authors' knowledge, investigating the consistency of the lasso in high-dimensional Cox models with interval-censored data remains an open problem.

Problem Statement

Within the ultra-high-dimensional framework of the source paper — penalized nonparametric maximum likelihood estimation for the Cox model with interval-censored data, where $\log p = O(n^{\delta})$ for some $\delta \in (0,1)$ and the tuning, sparsity, and signal conditions follow the source's Conditions 1–6 — remove the restrictions under which the oracle property is currently proved. The common objective is to extend the only existing ultra-high-dimensional oracle-variable-selection theory for interval-censored Cox data to the settings the methodology's own applications demand; the five components are the specific theoretical gaps, each independently well-defined and jointly exhausting the source's open-problem list:

1. Initial-estimator zero-consistency (load-bearing for the adaptive lasso as implemented). Prove that the lasso estimator in this framework is zero-consistent with an explicit rate $r_n$ satisfying $\theta_n r_n \gg n^{\max\{-1/3,(\delta-1)/2\}}$ while $\theta_n = O(n^{-1/3})$ (and, for the asymptotic-normality component, $\theta_n = o(n^{-1/2})$, attainable from an $L_2$-rate faster than $n^{\min\{-1/6,-\delta/2\}}$), so that Theorems 3–4 of the source apply unconditionally to the procedure actually used; or exhibit another computable initial estimator that is zero-consistent at such a rate.

2. Diverging sparsity. Extend the oracle-property theory to $s = s_n$ growing with $n$ (with nonzero coefficients allowed to change with $n$), replacing the fixed-$s$ Condition 5.

3. Grouped selection. Develop grouped penalized variable selection (group lasso for group selection, or group bridge for bi-level selection; other group penalties such as group SCAD and group MCP are also admissible) for interval-censored data with asymptotic theory — selection consistency at the group level and asymptotic normality of the selected coefficients — valid in the ultra-high-dimensional regime, going beyond the slower-than-$n$ growth of $p$ covered by Wu, Zhao, and Sun (2023).

4. Semiparametric transformation models. Establish the oracle property for penalized variable selection under semiparametric transformation models (which nest the Cox model and other hazard link functions) with interval-censored data in the ultra-high-dimensional regime (the source notes that the computation extends straightforwardly via the EM algorithm of Zeng, Mao, and Lin (2016)) — transferring, or refuting, the source's conjecture that its proofs generalize.

5. Left truncation. Derive the asymptotic theory (selection consistency and estimation asymptotics) of the penalized methods for data subject to both interval censoring and left truncation, where the likelihood involves the study-entry times and the maximal-intersection structure of the NPMLE is modified; the source gives the likelihood and the EM modification, but no theory.

References

1. Pak, D., Zhang, J., Wu, D., Weng, H., and Li, C. (2024). Variable Selection in Ultra-high Dimensional Feature Space for the Cox Model with Interval-Censored Data. arXiv:2405.01275, DOI 10.48550/arXiv.2405.01275, https://arxiv.org/abs/2405.01275 (source paper)
2. Pak, D., Zhang, J., Wu, D., Weng, H., and Li, C. (2026). Variable selection in ultra-high dimensional feature space for the Cox model with interval-censored data. Journal of Statistical Planning and Inference 245, article 106423, DOI 10.1016/j.jspi.2026.106423, https://doi.org/10.1016/j.jspi.2026.106423 (published version)
3. Bradic, J., Fan, J., and Jiang, J. (2011). Regularization for Cox's proportional hazards model with NP-dimensionality. The Annals of Statistics 39(6), 3092–3120, DOI 10.1214/11-AOS911, https://doi.org/10.1214/11-AOS911
4. Huang, J., Sun, T., Ying, Z., Yu, Y., and Zhang, C.-H. (2013). Oracle inequalities for the lasso in the Cox model. The Annals of Statistics 41(3), 1142–1165, DOI 10.1214/13-AOS1098, https://doi.org/10.1214/13-AOS1098
5. Li, C., Pak, D., and Todem, D. (2020). Adaptive lasso for the Cox regression with interval censored and possibly left truncated data. Statistical Methods in Medical Research 29(4), 1243–1255, DOI 10.1177/0962280219856238, https://doi.org/10.1177/0962280219856238
6. Zhao, H., Wu, Q., Li, G., and Sun, J. (2020). Simultaneous Estimation and Variable Selection for Interval-Censored Data With Broken Adaptive Ridge Regression. Journal of the American Statistical Association 115(529), 204–216, DOI 10.1080/01621459.2018.1537922, https://doi.org/10.1080/01621459.2018.1537922
7. Wu, Y., Zhao, H., and Sun, J. (2023). Group Variable Selection for the Cox Model with Interval-Censored Failure Time Data. Biometrics 79(4), 3082–3095, DOI 10.1111/biom.13879, https://doi.org/10.1111/biom.13879
8. Zeng, D., Mao, L., and Lin, D. Y. (2016). Maximum likelihood estimation for semiparametric transformation models with interval-censored data. Biometrika 103(2), 253–271, DOI 10.1093/biomet/asw013, https://doi.org/10.1093/biomet/asw013
9. Huang, J., Ma, S., and Zhang, C.-H. (2008). Adaptive Lasso for sparse high-dimensional regression models. Statistica Sinica 18(4), 1603–1618, Zbl 1255.62198, open-archive PDF https://www3.stat.sinica.edu.tw/statistica/oldpdf/A18n420.pdf
10. Zhang, J., Li, C., and Weng, H. (2025). Post-selection inference for the Cox model with interval-censored data. Scandinavian Journal of Statistics 52(2), 710–735, DOI 10.1111/sjos.12768, https://doi.org/10.1111/sjos.12768
11. Reeder, H. T., Haneuse, S., and Lee, K. H. (2024). Group lasso priors for Bayesian accelerated failure time models with left-truncated and interval-censored data. Statistical Methods in Medical Research 33(8), 1412–1423, DOI 10.1177/09622802241262523, https://doi.org/10.1177/09622802241262523
12. Mahmoudi, S., Li, C., Cai, J., and Lu, S. (2026). Variable selection with broken adaptive ridge regression for interval-censored competing risks data. Lifetime Data Analysis 32(2), DOI 10.1007/s10985-026-09711-3, https://doi.org/10.1007/s10985-026-09711-3
13. Bercu, A., Guilloux, A., Proust-Lima, C., and Jacqmin-Gadda, H. (2025). A regularized multi-state model for covariate selection with interval-censored survival data. arXiv:2508.16256, https://arxiv.org/abs/2508.16256
14. Li, H., Cao, Z., Sun, J., and Tang, N. (2027, online). New Feature Screening Methods for Massive Interval-censored Failure Time Data. Statistica Sinica (advance publication), DOI 10.5705/ss.202023.0309, https://doi.org/10.5705/ss.202023.0309
15. Wu, Q., Zhao, H., Zhu, L., and Sun, J. (2020). Variable selection for high-dimensional partly linear additive Cox model with application to Alzheimer's disease. Statistics in Medicine 39(23), 3120–3134, DOI 10.1002/sim.8594, https://doi.org/10.1002/sim.8594
16. Tian, T., and Sun, J. (2023). Variable selection for nonparametric additive Cox model with interval-censored data. Biometrical Journal 65(1), 2100310, DOI 10.1002/bimj.202100310, https://doi.org/10.1002/bimj.202100310