{"id":"4f957e1e-9b28-4e52-99cc-c9ca48871276","arxiv_id":"2507.15696","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A renewable online estimator for censored quantile regression is built from a quadratic surrogate of the martingale objective and shown asymptotically equivalent to the full-data oracle estimator.","lead":"This paper designs an online estimator for censored quantile regression that updates with each new data batch using only compact summary statistics, not the full historical raw data. It reports that the streaming estimator achieves the same asymptotic efficiency as the full-data oracle estimator while using less memory and, for many batches, less computation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stated Proposition 2 is too weak to support Theorem 2: its o_p(1) remainder is larger than the O(N_B^{-1}) local objective terms that drive the oracle-efficiency claim.","rationale":"The reader's CONDITIONAL verdict is appropriate, and the same proposition is at issue; however, the load-bearing weakness is sharper than 'the expansion might fail.' As stated, Proposition 2's remainder is o_p(1), which is too coarse for the √N_B asymptotics: at the local neighborhood relevant to Theorem 2, both sides of the proposed quadratic identity are O_p(1/N_B), so the stated result does not control the error at the needed order. This is a proof-gap in the posted version, not necessarily a false claim; the supplementary file may contain a stronger pointwise or local expansion. But since the supplement is not included, the central equivalence theorem rests on an unverifiable stronger statement. The proposed test (re-derive the local expansion from Knight's identity or inspect the supplement) would settle whether the gap is cosmetic or substantive. I keep the CONDITIONAL verdict rather than moving to REJECT because the method is plausible, the simulations are supportive, and the missing ingredient is a sharpened lemma rather than a demonstrated contradiction. I do not fully agree with the reader's framing because the stated concern about poor estimation of the weight matrix is secondary: consistency of bΓ is not the bottleneck if Proposition 2's local remainder is also uncontrolled; conversely, a rate for bΓ would be needed only within the same local expansion. The verdict remains conditional on the authors providing the supplementary proof and the sharper remainder order.","tokens_in":18843,"tokens_out":15032,"duration_ms":177685,"concrete_test":"Obtain the supplementary proof of Proposition 2 and check the local-scale expansion: for β = bβ_{N1}(τ)+u/√N1, the proof must show n1[l_{kn1}(β)-l_{kn1}(bβ_{N1})] - (1/2)u^T Γ_{n1}u = o_p(1) (up to the paper's factor convention), uniformly over bounded u. If the proof only yields the unnormalized o_p(1) remainder, Theorem 2 is unsupported. Numerically, simulate one batch at n = 10^3, 10^4, 10^5, compute bβ_n, fix u, and plot n times the objective difference against (1/2)u^T bΓ u; the ratio should approach 1. If the normalized remainder does not vanish, the surrogate in (5) is not N_B^{-1}-accurate and the oracle-efficiency claim fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 2 (Section 2.2) states l_{kn1}(β)-l_{kn1}(bβ_{N1}) = (β-bβ_{N1})^T Γ_{n1}(τ)(β-bβ_{N1}) + o_p(1). The online loss G_kB in (5) is built on this expansion, and Theorem 2 claims √N_B oracle equivalence. The relevant perturbation for that theorem is local: β = bβ_{N_B-1}(τ)+u/√N_B. At this scale the quadratic term in Proposition 2 is O_p(1/N_B); after the historical weight (N_B-1)/N_B in G_kB it is still O(1/N_B), and the current-batch term is the same order. A remainder o_p(1) is a factor N_B larger, so the stated result says only that both sides converge to zero; it does not identify the leading term at the scale where the minimizer fluctuates. Theorem 2 follows only if the supplementary proof establishes a locally uniform expansion with remainder o_p(∥β-bβ_{N1}∥^2 + n^{-1/2}∥β-bβ_{N1}∥), i.e., an N_B-normalized quadratic approximation with o_p(1) remainder. The cited Lemma S2 only gives consistency of bΓ_NB, not a rate at this local scale, and the supplemental proof is not included with the posting. As stated, Proposition 2 is too weak to carry the oracle-efficiency claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes an online updating method for censored quantile regression under right censoring, building on the martingale-based estimating equations of Peng and Huang (2008). The key idea is to replace the historical-data contribution to the unsmooth objective by a quadratic surrogate involving a resampling-based weight matrix, yielding the renewable loss G_kB in Eq. (5); the authors then develop a majorize-minimize algorithm with closed-form parameter updates and give a computational complexity analysis. The main theoretical claim, stated as Theorem 2 under conditions C1-C5, is that the online estimator achieves the same sqrt(N_B) convergence rate and the same limiting Gaussian process as the oracle estimator computed from all raw data. The paper reports simulations across normal, skewed extreme-value, and heteroscedastic error settings and applies the method to a SEER colon cancer data set.","tokens_in":19140,"tokens_out":6024,"duration_ms":62224,"significance":"If the theoretical claims hold, this is a useful contribution to online survival analysis: it extends renewable estimation to a nonsmooth censored quantile regression setting, relaxes the quantile-grid-size condition from o(N_B^{-1/2}) to o(tilde-n^{-1/2}), and provides a storage-bounded online procedure supported by a detailed computational complexity analysis. The simulation study is extensive, with multiple batch sizes and error distributions, and the reported SD/ASE ratios and coverage probabilities are mostly close to nominal; the SEER application demonstrates feasibility on a large data set. However, the theoretical core is not yet verifiable from the posted manuscript because the proofs are deferred to an unavailable supplement and, as discussed below, one stated proposition is too weak for the oracle-efficiency conclusion. The strengths of the paper are the explicit online objective, the resampling-based weight matrix estimator, and the practical emphasis on memory and time savings; the main weakness is the gap between the stated propositions and the asymptotic theorem.","major_comments":[{"comment":"As stated, Proposition 2 gives l_{kn1}(β) - l_{kn1}(bβ_N1) = (β - bβ_N1)^T Γ_{n1}(τ)(β - bβ_N1) + o_p(1). This remainder is not small enough for the local analysis needed in Theorem 2. At the local scale β = bβ_{N_B-1}(τ) + u/sqrt(N_B), the quadratic term is O_p(1/N_B), and the current-batch term in G_kB is also O_p(1/N_B), whereas the stated o_p(1) remainder is larger by a factor of N_B. To obtain the claimed sqrt(N_B) oracle equivalence, the proof must supply a locally uniform expansion with remainder o_p(||β - bβ_N1||^2 + n_1^{-1/2}||β - bβ_N1||), and it also needs a rate for bΓ_N1 - Γ_N1 at this local scale. Neither is stated or proved in the posted text; Lemma S2, cited for consistency of bΓ_NB, is not available and in any case consistency alone does not provide the required local rate. This gap is load-bearing for Theorem 2.","section":"Section 2.2, Proposition 2; Section 4.1, Theorem 2"},{"comment":"Proposition 1 imposes the condition ||β(τ_k)|| ≤ c(N_B^{-1} Σ_{b=1}^B sqrt(n_b))^{-1} on the point at which the oracle loss is approximated by the weighted batch losses. Since this approximation is subsequently used to define the global objective G_kB, the condition must hold on the domain over which G_kB is minimized, at least with high probability; otherwise the decomposition (4) is not justified. As written, the condition is not derived from the model or from consistency of bβ_NB, and its right-hand side depends on the batch-size configuration in a way that is not interpreted. The authors should either prove the condition for the relevant domain or replace it with a standard boundedness/uniformity condition.","section":"Section 2.2, Proposition 1"},{"comment":"The asymptotic regime is not fully specified. Conditions C5 only say n_b → ∞ for each b = 1,...,B and m = o(tilde-n^{-1/2}), but Theorem 2's statement that sqrt(N_B){bβ_NB(τ) - β0(τ)} converges weakly needs a precise statement about whether B is fixed or grows with the n_b, and if B grows, what relation between B and the batch sizes is required. This matters for the claim that the online estimator maintains oracle efficiency as the stream continues; otherwise the result covers only a single final cumulative point, not the streaming asymptotics suggested by the paper.","section":"Section 4.1, Conditions C5 and Theorem 2"},{"comment":"The variance derivation contains a dimensionally inconsistent display: the text reads bβ_NB(τ) = β0(τ)/sqrt(N_B) + Γ_NB^{-1}(τ) (1/N_B) Σ Φ{...} + o_p(1/sqrt(N_B)), which would imply bβ_NB(τ) tends to 0 rather than β0(τ). Presumably the left-hand side should be sqrt(N_B){bβ_NB(τ) - β0(τ)}. Since the resampling covariance estimator is built on this formula, the authors should correct the display and verify the algebra in the subsequent lines.","section":"Section 4.2, variance estimation"},{"comment":"Propositions 1 and 2 and Theorems 1 and 2 are all proved in a Supplementary Material that is not included with the arXiv posting; Sections A and B are referenced but absent. The central claims of the paper, especially Theorem 2's oracle-efficiency statement, therefore cannot be verified from the posted manuscript. The supplement should be provided to reviewers before a final recommendation is made.","section":"General: supplementary material"}],"minor_comments":[{"comment":"The phrase 'independent of the steaming data' should read 'independent of the streaming data'.","section":"Section 3.1"},{"comment":"The phrase 'time-dependent covaraites' contains a typo; it should be 'time-dependent covariates'.","section":"Section 1"},{"comment":"The word 'quadartic' in 'a simple quadartic surrogate function' should be 'quadratic'.","section":"Section 3.2"},{"comment":"The reference list contains duplicate entries for Chu and Sit with different years but the same title, and the Hjort and Pollard entry has a typo in 'Asmptotics'.","section":"References"},{"comment":"At B = 40 the oracle coverage probabilities in Tables 1 and 2 are sometimes substantially below 95% (e.g., 0.887 and 0.890), and the text's statement that coverage probabilities are all around the nominal level does not address this; the authors should comment on these oracle CPs when comparing the online and oracle methods.","section":"Section 5, Tables 1-2"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim rests on proofs in a supplement that is not included with the posting; the editor should require the supplement before further review. The relation to existing renewable quantile regression work by Jiang and Yu (2022) and Sun et al. (2023) should also be clarified, as the novelty in the censored survival setting should be stated more sharply relative to those papers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first renewable/online estimator for censored quantile regression, and it is a sensible extension of Peng and Huang. But the main text states a quadratic approximation whose remainder is too large to imply the oracle-efficiency theorem, and the proof is only in a supplement we do not have. Worth a serious referee, but the authors need to show the stronger local expansion and share the supplement.\n\nWhat is new: they take the martingale-based estimating equation of Peng and Huang (2008), approximate the historical objective with a quadratic form using a weight matrix estimated by resampling least squares, and produce a closed-form update via MM. They also weaken the grid-size assumption from o(N^{-1/2}) to o(tilde n^{-1/2}), which matters when batches are small relative to the total. The writing is clear, the simulations cover three error settings and compare honestly against the oracle, and the application to SEER shows a real speedup at 16 batches. The citation pattern is appropriate: Jiang and Yu, Chen and Yuan, Sun et al., and Sit and Xing are all cited, and the claim of first-online is credible.\n\nThe soft spots, in proportion. First, the stress-test note is right. Proposition 2 gives l_{kn1}(beta) - l_{kn1}(beta_hat_{N1}) = (beta - beta_hat)^T Gamma (beta - beta_hat) + o_p(1). At the local scale beta = beta_hat + u/sqrt(N_B), the quadratic term is O_p(1/N_B), while an o_p(1) remainder is a factor sqrt(N_B) larger. Theorem 2's sqrt(N_B) equivalence cannot follow from that statement alone. The supplement must prove a locally uniform expansion with remainder o_p(||beta - beta_hat||^2 + n^{-1/2}||beta - beta_hat||), and that is not what is written. This is a gap between the stated proposition and the claimed theorem, not necessarily a fatal flaw, but it has to be fixed in revision. Second, no supplement, code, or data link is included in the arXiv posting, so the key approximations and the consistency of the ReLS weight matrix are unverifiable from the preprint. Third, small point: the speed advantage only appears at B=40; at B=5 and 20 the online method is slower. That is fine and honestly reported, but it tempers the computational-advantage rhetoric.\n\nWho this is for: statisticians working on online inference for survival data, especially quantile regression people. It deserves review, but with the supplement and a precise statement of the local expansion.","headline":"First online censored quantile regression, but the stated quadratic expansion is too weak for the efficiency theorem; send to review with the supplement required.","tokens_in":19661,"tokens_out":2889,"would_cite":false,"duration_ms":31191,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G08","62N01"],"pacs":[],"model":"deepseek-v4-flash","headline":"An online updating scheme for censored quantile regression matches the asymptotic efficiency of refitting all raw data once.","keywords":["Censored quantile regression","Online updating","Renewable estimators","Streaming data","Survival analysis","Majorize-minimize algorithm","Resampling least squares"],"falsifier":"Simulate a known accelerated-failure-time model with right censoring, stream it in batches satisfying $m=o(\\tilde n^{-1/2})$, and compare the online estimator with the full-data oracle estimator over many replications. If the online confidence intervals undercover while the oracle intervals are calibrated, or if the standardized difference $\\sqrt{N_B}(\\hat\\beta_{N_B}-\\beta_0)$ is visibly non-Gaussian or biased when the oracle is not, then Proposition 2's quadratic approximation or the consistency of $\\hat\\Gamma$ is the point to re-examine.","tokens_in":18629,"feed_emoji":"📈","tokens_out":9230,"duration_ms":100204,"temperature":0.7,"pith_summary":"Quantile regression for censored survival times asks how covariates shift different quantiles of the event-time distribution, but the standard estimation objective is unsmooth and, when data arrive in streams, cannot be updated without revisiting every past record. This paper constructs an online version of that estimator: it replaces the historical part of the unsmooth loss with a quadratic surrogate built from a second-order expansion, so that only the current batch plus a small set of summary statistics are needed. The central claim is that this online estimator is asymptotically equivalent to the oracle estimator computed from all raw data, in the sense of the same convergence rate and the same limiting Gaussian process. If true, large registry or cohort data that arrive in batches could be analyzed continuously at a fraction of the memory and computation time, with no loss of statistical efficiency.","feed_headline":"Streaming survival quantiles keep full-data accuracy","feed_subtitle":"Estimator updates from the current batch plus compact summaries, matching oracle efficiency with far less memory.","key_machinery":"The load-bearing mechanism is a local quadratic expansion of the unsmooth check-function loss: Proposition 2 states that for a historical batch, $l_{kn1}(\\beta(\\tau))-l_{kn1}(\\hat\\beta_{N1}(\\tau)) = \\{\\beta(\\tau)-\\hat\\beta_{N1}(\\tau)\\}^T \\Gamma_{n1}(\\tau)\\{\\beta(\\tau)-\\hat\\beta_{N1}(\\tau)\\}+o_p(1)$, where $\\Gamma_{nb}(\\tau)$ is a conditional-density-weighted second-moment matrix of the covariates. Plugging this surrogate into the accumulated objective turns the online problem into minimizing a convex function of the current batch loss plus a quadratic term, which admits a closed-form update through a majorize-minimize algorithm with an isotropic quadratic majorizer. The weight matrix $\\Gamma$ is estimated without density estimation by a resampling least-squares procedure, and the variance estimate is updated recursively from per-batch covariance matrices via the same $\\Gamma$ matrices.","core_discovery":"Under regularity conditions C1–C5, the paper establishes that $\\sqrt{N_B}\\{\\hat\\beta_{N_B}(\\tau)-\\beta_0(\\tau)\\}$ converges weakly to a mean-zero Gaussian process $A(\\beta_0(\\tau))^{-1}\\Phi(\\Omega(\\tau))$, the same limiting process as the oracle estimator obtained by analyzing the entire accumulated data once. This equivalence is achieved through the online objective $G_{kB}(\\beta(\\tau_k))$, which combines the current batch's quantile-regression loss with a quadratic penalty $\\frac{N_{B-1}}{N_B}\\{\\beta(\\tau_k)-\\hat\\beta_{N_{B-1}}(\\tau_k)\\}^T\\hat\\Gamma_{N_{B-1}}(\\tau_k)\\{\\beta(\\tau_k)-\\hat\\beta_{N_{B-1}}(\\tau_k)\\}$ inherited from earlier batches. The paper also shows its quantile-grid condition is weaker than the oracle's: $m=o(\\tilde n^{-1/2})$ rather than $m=o(N_B^{-1/2})$, because the local quadratic approximation only needs each batch to be large. Consequently the online estimates are claimed to match the oracle in convergence rate and efficiency while storing only the current batch and $O(K)$ summary matrices.","pith_inferences":["Beyond the paper: the same quadratic-surrogate pattern could convert other non-smooth estimating equations with martingale structure into online form, such as median regression for censored cost data or competing-risks quantiles, provided the required second-order expansion holds.","Beyond the paper: the resampling least-squares estimate of $\\hat\\Gamma$ is reused both for updates and for variance; a simulation with known conditional density could isolate whether ReLS error or the quadratic approximation is the dominant finite-sample source of coverage loss.","Beyond the paper: the weakened grid-size condition suggests that in a streaming pipeline one should set the quantile grid from the smallest expected batch size, not from the final sample size, and a formal rule for choosing the grid as batches grow could be developed."],"forward_implications":["After each new data batch arrives, quantile effect estimates and standard errors can be refreshed from the new batch plus stored summary statistics, without reprocessing earlier patient-level records.","The online estimator carries the same asymptotic distribution as the oracle estimator, so confidence bands and hypothesis tests built on that distribution retain their nominal properties.","Because the grid size assumption is relaxed from $o(N_B^{-1/2})$ to $o(\\tilde n^{-1/2})$, practitioners can choose quantile grids according to the batch size rather than the total accumulated sample, lowering the number of parameters tracked.","The recursive variance formula means the extra cost of producing confidence intervals is a running sum of per-batch sandwich terms, so inference remains available at every accumulation point.","Storage drops from $O(N_B)$ patient records to $O(n_B+K)$ summary quantities, which keeps the method workable when memory is much smaller than total sample size."],"supporting_citations":[{"why":"Defines the martingale-based estimating equation and oracle estimator whose convergence rate and limiting process the online method is engineered to match.","marker":"Peng and Huang (2008)"},{"why":"Supplies the resampling least squares procedure used to estimate the weight matrix without conditional density estimation.","marker":"Zeng and Lin (2008)"},{"why":"Provides the L1 minimization routine used to compute batch-level censored quantile regression estimates within each update.","marker":"Barrodale and Roberts (1974)"}],"fun_headline_variants":["Streaming survival quantiles match batch efficiency","Online quantile regression lowers memory, keeps accuracy","Survival quantiles stream with oracle-level precision","Low-storage online quantiles rival full-data fits","Fast quadratic updates for streaming survival quantiles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction leans on the assumption that the historical loss difference is exactly captured by a quadratic term with a consistently estimable weight matrix; if that second-order expansion fails or the resampling estimate of the weight matrix drifts, the claimed statistical equivalence to the oracle estimator breaks.","fun_headline_variants_meta":{"raw":{"variants":["Streaming survival quantiles match batch efficiency","Online quantile regression lowers memory, keeps accuracy","Survival quantiles stream with oracle-level precision","Low-storage online quantiles rival full-data fits","Fast quadratic updates for streaming survival quantiles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1289,"prompt_tokens":946,"completion_tokens":343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":273}},"tokens_in":562,"tokens_out":343,"duration_ms":4745,"temperature":1.0,"reasoning_tokens":273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:25:19.378466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a known accelerated-failure-time model with right censoring, stream it in batches satisfying $m=o(\\tilde n^{-1/2})$, and compare the online estimator with the full-data oracle estimator over many replications. If the online confidence intervals undercover while the oracle intervals are calibrated, or if the standardized difference $\\sqrt{N_B}(\\hat\\beta_{N_B}-\\beta_0)$ is visibly non-Gaussian or biased when the oracle is not, then Proposition 2's quadratic approximation or the consistency of $\\hat\\Gamma$ is the point to re-examine.","supporting_citations":[{"cited_title":"M., Huang, Y","cited_arxiv_id":null,"evidence_quote":"Defines the martingale-based estimating equation and oracle estimator whose convergence rate and limiting process the online method is engineered to match."},{"cited_title":"L., Lin, D","cited_arxiv_id":null,"evidence_quote":"Supplies the resampling least squares procedure used to estimate the weight matrix without conditional density estimation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the L1 minimization routine used to compute batch-level censored quantile regression estimates within each update."}],"review_version":1}