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Model-free Methods for Event History Analysis and Efficient Adjustment (PhD Thesis)

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Pith's one-line read This thesis establishes that a model-free parameter called the Local Covariance Measure can be estimated by double machine learning so that, under conditional local independence, the estimator converges uniformly at a $\sqrt{n}$ rate to a…

desk verdict The X-LCT in Chapter 2 is a genuine advance in nonparametric event-history testing; read it for that result, but note that the rate conditions for the recommended nuisance estimators remain unverified. read the letter →

arxiv 2502.07906 v1 pith:XKH6OEMV submitted 2025-02-11 stat.ME math.STstat.MLstat.TH

Alexander Mangulad Christgau This is my paper · ORCID
classification stat.MEmath.STstat.MLstat.TH MSC 62G1062G0562G2062N0162N02
keywords conditionallocalindependenceCovarianceMeasuredoublemachinelearningcountingprocessnonparametrictestingcovariateadjustmentefficiencybounddoublyrobustestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis argues that many statistical targets can be defined model-free, meaning they have a clear meaning for any data-generating distribution instead of only inside a parametric model, and can still be estimated at parametric speed by combining machine-learning nuisance estimates with double machine learning. Its main contribution is the Local Covariance Measure, a time-indexed covariance between a residual process and the compensated counting-process martingale, which is identically zero under the hypothesis of conditional local independence. The thesis proves that a cross-fitted estimator of this measure converges uniformly at a $\sqrt{n}$ rate to a mean-zero Gaussian martingale under the null, provided the nuisance functions are consistent with modest rates, and uses this to build the Local Covariance Test with uniform asymptotic level and power against local alternatives. The same model-free logic yields the Debiased Outcome-adapted Propensity Estimator for efficient covariate adjustment and the Aalen Covariance Measure for assumption-lean regression in event-history settings.

What carries the argument

The load-bearing object is the residual process $G_t$, for example the additive residual $X_t - E[X_t\mid\mathcal{F}_{t-}]$, together with the stochastic integral $I_t=\int_0^t G_s\,dM_s$. The Local Covariance Measure is $\gamma_t=E[I_t]$, and its defining property is that residualization makes the integrand orthogonal to the past, giving the estimator a Neyman orthogonal structure: substituting slow, nonparametric nuisance estimates does not create a first-order bias. Under the alternative, $\gamma_t=\int_0^t \mathrm{Cov}(G_s,\lambda_s-\underline{\lambda}_s)\,ds$, so the LCM is nonzero exactly when the two intensities differ in a way the residual can detect. The proof machinery combines a uniform extension of the functional martingale central limit theorem, chaining arguments for stochastic equicontinuity, sample splitting and cross-fitting to remove dependence, and the empirical variance estimator $\hat V_n(t)=|J_n|^{-1}\sum_j\int_0^t(\hat G_{j,s})^2\,dN_{j,s}$.

What would settle it

Simulate data from the Cox-type model of Section 2.6 under the null with the historical functional linear model and compute the $L^2$ error product $\sqrt{|J_n|}\,g(n)h(n)$ for the kernel-based estimates of $\lambda$ and $\Pi$; if this product fails to converge to zero, the uniform level and power of the X-LCT are not guaranteed.

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Extended reading notes

Core claim

Under the hypothesis that a counting process $N$ is conditionally locally independent of a c\`adl\`ag process $X$ given a filtration $(\mathcal{F}_t)$, the Local Covariance Measure $\gamma_t = E[\int_0^t G_s\,dM_s]$ is the zero function, where $G_t$ is a residual process satisfying $E[G_t\mid\mathcal{F}_{t-}]=0$ and $M_t=N_t-\int_0^t\lambda_s\,ds$ is the compensated martingale. The thesis shows that estimating $\gamma$ by plugging machine-learning estimates of the intensity $\lambda$ and of the residual map into the stochastic integral, with sample splitting or cross-fitting, yields a process that is asymptotically indistinguishable from a Gaussian martingale: under Assumptions 2.4.1 and 2.4.2, $\sqrt{|J_n|}(\hat\gamma^{(\cdot)}-\gamma)$ converges uniformly to a mean-zero Gaussian martingale, and the resulting test statistic converges to the supremum of a standard Brownian motion. For covariate adjustment, the thesis identifies the optimal subset of covariates and shows that DOPE attains the semiparametric efficiency bound; for event-history association, it shows that the Aalen Covariance Measure is estimable in a doubly robust way at a $\sqrt{n}$ rate under modest nuisance rates.

Load-bearing premise

The load-bearing premise is that the machine-learning estimates of the conditional intensity and the residual process are accurate enough that the product of their $L^2$ errors shrinks faster than $n^{-1/2}$; for the historical functional linear model used in the simulations, the thesis states this rate is only conjectured, not rigorously established.

Editorial extensions

If this is right

  • Researchers can test whether a time-varying exposure directly drives an event process without committing to a parametric hazard model, as long as the nuisance functions can be estimated at the required rates.
  • The X-LCT critical values are distribution-free: under the null the statistic converges to $\sup_{0\le t\le 1}|B_t|$ for a standard Brownian motion, so no simulation of the null distribution is needed.
  • The uniform asymptotic level means the test can serve as a subroutine in constraint-based procedures for learning local independence graphs, replacing an oracle test with a practical one.
  • For covariate adjustment, DOPE provides a single estimator that attains the efficiency bound no matter which subset of covariates is used, and it identifies the optimal subset.
  • The Aalen Covariance Measure gives an assumption-lean replacement for the exposure coefficient in an Aalen additive hazards model, so effect estimates retain a clear interpretation when the model is misspecified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the uniform asymptotic theory holds for counting processes, the same proof pattern should extend to tests for local independence in general semimartingale systems, provided nuisance estimation rates can be established; the discrete-time observation issue would be the main obstacle.
  • The residual process is a user choice: transformations, time shifts, and linear or nonlinear filters of $X$ can be inserted, which the paper notes should shift power toward the corresponding departure from the null; a systematic power comparison across such filters is a natural next step.
  • The rate condition $\sqrt{|J_n|}g(n)h(n)\to 0$ means a fast rate for one nuisance can compensate for a slow rate for the other; a practical benchmark would be to verify the conjectured kernel-estimation rates for the historical functional linear model used in the simulations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This PhD thesis (arXiv:2502.07906) contains a methodological introduction and three stand-alone papers. Chapter 2 defines the Local Covariance Measure (LCM), a time-indexed functional parameter that quantifies deviations from conditional local independence of a counting process, and proposes estimation by double machine learning with sample splitting or cross-fitting, leading to the (X-)LCT test. The paper proves, under Assumptions 2.4.1, 2.4.2, and 2.5.1, uniform weak convergence of the estimator to a mean zero Gaussian martingale under the null, uniform asymptotic level, and power against sqrt(n) local alternatives for the additive residual process. Chapter 3 presents the Debiased Outcome-adapted Propensity Estimator (DOPE) for efficient covariate adjustment, and Chapter 4 introduces the Aalen Covariance Measure (ACM) with double robustness. The full text supplied for review covers Chapter 1 and Chapter 2 in detail; my detailed assessment focuses on those chapters.

Significance. If the results hold as stated, Chapter 2 is a substantial contribution: it is the first nonparametric test of conditional local independence with explicit uniform asymptotic guarantees, and it extends double machine learning from scalar parameters to a time-indexed functional parameter. The proof machinery, including a uniform version of Rebolledo's martingale CLT and uniform chaining arguments in Section 2.B, is of independent interest and is presented carefully. Chapter 1 gives a clear, self-contained introduction to Neyman orthogonality and rate double robustness. The theorems are carefully stated as conditional results, and the thesis openly discusses the limitations of the available rate theory; however, the central gap between the rate assumptions in the theorems and the practical estimators used in the simulation study needs to be addressed in the thesis version.

major comments (3)
  1. [§2.D and §2.6.1, Assumption 2.4.2] Assumption 2.4.2 is load-bearing for Theorem 2.4.6, Theorem 2.5.1, and Theorem 2.5.4, because the remainder R_3^{(n)} in (2.4.21) is controlled only by the product rate sqrt(|J_n|) g(n) h(n) -> 0. Yet Section 2.D states that for the full historical functional linear model no published rate results are available, and Section 2.6.1 states that sufficient rate results 'should be possible but have not yet been established rigorously.' The simulation study in Section 2.6.2 uses exactly this model to estimate both rho_X and rho_Y (see (2.6.36) and the four kernels), so the implemented X-LCT in Algorithm 2 is not verified to satisfy the assumptions of the theorems. If the true rates are slower than n^{-1/4+epsilon}, the uniform level and power conclusions do not apply to the recommended implementation. The thesis should either establish the required rates under explicit regularity conditions for the historical functional linear model, or present the uniform guarantees as conditional on Assumption 2.4.2 and describe the simulations as illustrative rather than confirmatory.
  2. [§2.5.1, §2.5.2, Theorems 2.5.2 and 2.5.4] The uniform power result Theorem 2.5.2 is proved only for the sample-split LCT with the additive residual process, while the cross-fitted X-LCT of Definition 2.5.3, which the paper recommends for practical use, has only the uniform level statement in Theorem 2.5.4. No theorem gives uniform power for the cross-fitted test Psi^K_n; its power is supported only by the simulation study in Section 2.6.4. If the proof extends to cross-fitting, the extension should be stated; otherwise the abstract and Section 2.7 should specify that the uniform power guarantee applies to the sample-split version and that the X-LCT's power is an empirical finding.
  3. [§2.D, §2.B, Assumption 2.4.2] Assumption 2.4.2 requires convergence uniformly over the parameter set Theta. The rate survey in Section 2.D gives only pointwise rates: for example, the n^{-(1+m)/(2m+3)} heuristic for the historical functional linear model is derived from a fixed-t prediction error bound in Cai and Yuan (2012), and the convolution-model rate from Manrique (2016) is likewise stated for a fixed model. No argument in Section 2.D establishes uniformity over theta in Theta or over t in [0,1]. The simulation settings vary kernels and beta_2 over finite grids in Section 2.6.2, which does not fill this gap. The manuscript should state which additional regularity conditions on Theta, such as smoothness classes with uniform constants, make the uniform convergence in Assumption 2.4.2 hold.
minor comments (5)
  1. [Abstract and Sammenfatning] The text contains a rendering error: '?n-consistency' appears where '\sqrt{n}-consistency' is intended, both in the English abstract and the Danish summary.
  2. [§2.6.2, Assumption 2.4.1] The simulation uses unbounded Gaussian innovations, while Assumption 2.4.1 requires uniformly bounded processes; the text says the reported results were generated without caps. Please either report a capped version or state explicitly that the simulations are intended as an approximation to, rather than a verification of, Assumption 2.4.1.
  3. [§2.5.2 and §2.6.4] The main text does not state the number of folds K used for the X-LCT in the simulation study; please give K in the main text or confirm the value given in Section 2.G.1.
  4. [§2.3.2] The statement that the statistic in (2.3.16) is 'closely related' to the partial copula could be made precise; it would be useful to specify the exact transformation and whether the asymptotic results of Petersen and Hansen (2021) apply directly to this version.
  5. [Chapter 1, Example 1.0.1] The notation uses P both for a single distribution and for a collection of distributions in the same section (e.g., 'P in \mathcal{P}'); a brief notation table or a change of symbol for the collection would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the X-LCT null distribution follows from a martingale CLT, not from the fitted nuisance estimators; the unverified rate condition is an honest limitation, not a circular reduction.

full rationale

The central derivation chain is self-contained. The LCM is defined as a functional of the true distribution, and the estimator is analyzed through the explicit decomposition (2.4.17), where the leading term U^(n) uses the true residual process and the true martingale, while the nuisance-dependent remainders R_1, R_2, R_3 and D_2 are shown to vanish under Assumptions 2.4.1 and 2.4.2. The null distribution in Theorems 2.5.1 and 2.5.4 follows from Rebolledo's martingale CLT together with the Brownian time-change and scale invariance property, so the test is not constructed to have its null distribution by definition. The paper explicitly flags, in Section 2.D and Section 2.7, that rigorous rate results for the historical functional linear model used in the simulations have not yet been established: 'For the full historical functional linear model we are not aware of any published rate results' and 'we regard it is as an independent research project to establish rates for general historical regression methods.' This is a stated limitation and a correctness/robustness gap for the recommended practical implementation, but it is not a circular reduction of the theoretical claim: the theorems are conditional on Assumption 2.4.2, not derived from the conjecture that the assumption holds. Self-citations to the published version [Christgau et al., 2023b] and to the supplement [Christgau et al., 2023c] are bibliographic references to the same results included in the thesis, and no load-bearing argument reduces to an unverified self-citation. The proofs and technical lemmas are contained in the included supplement, and the paper builds on external, established results for the martingale CLT and double machine learning. Therefore no circularity is present; the main risk is the unverified rate condition, which belongs to correctness assessment rather than to circularity analysis.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The main assumptions are the i.i.d. data-generating process, the martingale definition of local independence, boundedness and rate conditions on nuisance estimators, and usual causal assumptions for DOPE and ACM. No free parameters fitted to data appear in the derivations; hyperparameters like the number of cross-fitting folds K are user choices and do not affect the asymptotic claims. No new physical or ontological entities are proposed; the new statistical functionals (LCM, ACM, DOPE) are defined directly in terms of observable distributions.

assumptions (6)
  • domain assumption Observations are i.i.d. according to an unknown distribution P.
    Chapter 1 states 'it will be assumed throughout this thesis' (Section 1). This is standard for the statistical theory developed.
  • domain assumption Local independence is defined via the martingale property: N_t - Lambda_t remains a martingale after enlarging the filtration.
    Definition 2.2.1 in Chapter 2. This is the formal mathematical meaning of the target concept.
  • ad hoc to paper Assumption 2.4.1: The intensity process and the residual process are uniformly bounded almost surely on [0,1].
    This strong boundedness assumption is introduced to make the martingale CLT and remainder bounds tractable. The authors note it can be relaxed at the cost of more complex proofs (Section 2.7).
  • ad hoc to paper Assumption 2.4.2: Nuisance estimators satisfy g(n), h(n) -> 0 and sqrt(n) g(n) h(n) -> 0 uniformly.
    This is the key rate double robustness assumption. It is not verified for the historical functional linear model estimators used in the simulation; Section 2.D says such rate results are scarce.
  • ad hoc to paper Assumption 2.5.1: The asymptotic variance function V(t) is bounded away from zero at t=1 uniformly over the parameter set.
    Needed to avoid a degenerate null distribution for the supremum-based test statistic.
  • domain assumption For DOPE and ACM: Standard causal identification conditions such as positivity, consistency, and no unmeasured confounding.
    These are not stated in the provided text but are typical for adjustment estimators. We cannot verify their exact form from the abstract only.

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Pith. "Pith review of Model-free Methods for Event History Analysis and Efficient Adjustment (PhD Thesis)." pith.science (2026). https://pith.science/paper/XKH6OEMV

@misc{pith2026250207906,
  author       = {Pith},
  title        = {Pith review of: Model-free Methods for Event History Analysis and Efficient Adjustment (PhD Thesis)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKH6OEMV}},
  note         = {Machine review of arXiv:2502.07906}
}
abstract

This thesis contains a series of independent contributions to statistics, unified by a model-free perspective. The first chapter elaborates on how a model-free perspective can be used to formulate flexible methods that leverage prediction techniques from machine learning. Mathematical insights are obtained from concrete examples, and these insights are generalized to principles that permeate the rest of the thesis. The second chapter studies the concept of local independence, which describes whether the evolution of one stochastic process is directly influenced by another. To test local independence, we define a model-free parameter called the Local Covariance Measure (LCM). We formulate an estimator for the LCM, from which a test of local independence is proposed. We discuss how the size and power of the proposed test can be controlled uniformly and investigate the test in a simulation study. The third chapter focuses on covariate adjustment, a method used to estimate the effect of a treatment by accounting for observed confounding. We formulate a general framework that facilitates adjustment for any subset of covariate information. We identify the optimal covariate information for adjustment and, based on this, introduce the Debiased Outcome-adapted Propensity Estimator (DOPE) for efficient estimation of treatment effects. An instance of DOPE is implemented using neural networks, and we demonstrate its performance on simulated and real data. The fourth and final chapter introduces a model-free measure of the conditional association between an exposure and a time-to-event, which we call the Aalen Covariance Measure (ACM). We develop a model-free estimation method and show that it is doubly robust, ensuring $\sqrt{n}$-consistency provided that the nuisance functions can be estimated with modest rates. A simulation study demonstrates the use of our estimator in several settings.

Figures

Figures reproduced from arXiv: 2502.07906 by the authors.

Figure 2.1
Figure 2.1. 1: Local independence graph illustrating a dependence structure among the three processes X, Z and N. Here N is the indicator of death for an individual, X is their cumulative pension savings and Z is a covariate process. All nodes in this graph have implicit self-loops. There is no edge from X to N, which indicates that death is not directly influenced by pension savings. This can be formalized as N being condition… view at source ↗
Figure 2.2
Figure 2.2. 2: Local independence graphs illustrating how the three processes X, Y , and Z could affect each other and time of death in the Cox example. There is no direct influence of X (pension savings) on time of death in either of the two graphs, but in the left graph the death indicator is furthermore conditionally locally independent of X given the history of Z and N. In the right graph, Z and N does not block all paths f… view at source ↗
Figure 2.2
Figure 2.2. 3: Histograms of the distributions of three different estimators of γ1. Each histogram contains 1000 estimates fitted to samples of size n “ 500. The samples were sam￾pled from a model that satisfies the hypothesis of conditional local independence and hence the ground truth is γ1 “ 0. See Section 2.6.2 for further details of the data generating process. However, we cannot expect the plug-in estimator to have a ? n-… view at source ↗
Figures from the paper (16 more)
Figure 2.2
Figure 2.2. Figure 2.2: 4: A time dependent extension of [PITH_FULL_IMAGE:figures/full_fig_p041_2_2.png]
Figure 2.6
Figure 2.6. Figure 2.6: 5: Empirical cumulative distribution functions of simulated p-values for the cross-fitted local covariance test and the hazard ratio test. The simulated data satisfies the hypothesis of conditional local independence, so the p￾values are supposed to be uniformly dist…
Figure 2.6
Figure 2.6. Figure 2.6: 6: For each ρ0 P t0, 5, 10u, the lines show the average rejection rates of our proposed test X-LCT (blue) and the hazard ratio test (orange) as functions of sample size, with each average taken over 8 different settings. For each setting, the rejection rate is comput…
Figure 2
Figure 2. Figure 2: G.1: Empirical distribution functions of p-values for the three different condi￾tional local independence tests considered, simulated under the sampling scheme described in Section 2.6. The dotted line shows y “ x correspond￾ing to a uniform distribution. 91 [PITH_FUL…
Figure 2
Figure 2. Figure 2: G.2: Sample paths of ˇγ K,p500q fitted on data sampled from three different alter￾natives as described in Section 2.6.4. Here pX, Y, Zq are sampled from the scheme described in Section 2.6, with both ρX and ρY being the constant kernel and with β “ ´1. For each alterna…
Figure 2
Figure 2. Figure 2: G.4: The plots show the average rejection rate of the double machine learning tests based on the supremum statistic (blue) and the endpoint statistic (red). n “ 2000. This is different from the previous settings, and can be explained by a slower convergence of the inte…
Figure 3.1
Figure 3.1. Figure 3.1: 1: The covariate W can have a complex data structure, even if the information it represents is structured and can be categorized into components that influence treatment and outcome separately. confounding. If the underlying confounding mechanisms are captured by a s…
Figure 3.3
Figure 3.3. Figure 3.3: 2: The σ-algebra Q given in Definition 3.3.2 as a description of W. Definition 3.3.2. Define the σ-algebras Q – ł PPP QP , QP – σpFpy |t,W; Pq; y P R, t P Tq, R – ł PPP RP , RP – σpbtpW; Pq; t P Tq. ♣ Note that QP , Q, RP and R are all descriptions of W, see [PITH_F…
Figure 3.5
Figure 3.5. Figure 3.5: 3: Root mean square errors for various estimators of χ1 plotted against sample size. Each data point is an average over 900 datasets, 300 for each choice of d P t4, 12, 36u. The bands around each line correspond to ˘1 standard error. For this plot, the outcome regres…
Figure 3.5
Figure 3.5. Figure 3.5: 4: Root mean square errors for various estimators of χ1 plotted against sample size. Each data point is an average over 900 datasets, 300 for each choice of d P t4, 12, 36u. The bands around each line correspond to ˘1 standard error. For this plot, the outcome regres…
Figure 3.5
Figure 3.5. Figure 3.5: 5: Coverage of asymptotic confidence intervals of the adjusted mean χ1, ag￾gregated over d P t4, 12, 36u and with the true β given in (3.5.19). 120 [PITH_FULL_IMAGE:figures/full_fig_p132_3_5.png]
Figure 3.5
Figure 3.5. Figure 3.5: 6: Distribution of estimated propensity scores based on the full covariate set W, the single-index WJ ˆθ, and the outcome predictions gpp1,Wq, respec￾tively. we see that the full covariate set yields propensity scores that essentially violate positiv￾ity, with this e…
Figure 3
Figure 3. Figure 3: B.1: Neural network architecture for single-index model. 3.B. Details of simulation study Our experiments were conducted in Python [Van Rossum et al., 1995]. The linear and logistic regression was imported from the scikit-learn package [Pedregosa et al., 2011], and the…
Figure 3
Figure 3. Figure 3: B.2: Comparison of DOPE estimators with and without sample splitting. Estimator Estimate BS se BS CI Regr. (NN) 0.020 0.008 (0.002, 0.035) Regr. (Logistic) 0.027 0.009 (0.012, 0.048) DOPE-BCL (Logistic) 0.024 0.010 (0.004, 0.040) Naive contrast 0.388 0.010 (0.369, 0.40…
Figure 3
Figure 3. Figure 3: B.3: Coverage of asymptotic confidence intervals of the adjusted mean χ1, ag￾gregated over d P t4, 12, 36u and with the true β given in (3.5.19). The numbers in the parentheses denote the corresponding average width of the confidence interval. 138 [PITH_FULL_IMAGE:fig…
Figure 4.5
Figure 4.5. Figure 4.5: 1: Scaled RMSE for various estimators with respect to the cumulative direct effect. 161 [PITH_FULL_IMAGE:figures/full_fig_p173_4_5.png]

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Reviewed August 8, 2026 · model on record in the stance chip above.