Stopped-Process Estimator

• Stopped-process estimator is defined as an estimator whose output depends on a statistically calibrated stopping event, bridging abstract stochastic analysis with sequential inference.
• It employs advanced techniques such as Daniell integrals and Clark-Ocone representations to extend classical estimation beyond right-continuous submartingales.
• Applications include Bayesian drift estimation, rare-event simulation, and early stopping in machine learning, demonstrating its versatility in handling diverse data-driven stopping rules.

In the cited literature, a stopped-process estimator is most naturally understood as an estimator whose definition, validity, or terminal output depends on a stopping event: evaluation of a stochastic process at a stopping time, inference from trajectories observed only up to first hitting or killing, or termination of an iterative estimation procedure by a statistically calibrated stopping rule. This perspective spans abstract stochastic analysis, sequential prediction, Bayesian filtering, rare-event simulation, technology-assisted review, and modern early stopping in machine learning (Grobler et al., 2020).

1. Formal scope and canonical constructions

The classical object is a stopped process $X_{t\wedge S}$, with stopped value $X_S$. In abstract settings this pointwise plug-in may be unavailable, and a more general definition is required. Grobler and Schwanke define the stopping element by a Daniell integral with respect to the spectral measure $\mu_\mathbb{S}$ of the stopping time: $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$ thereby extending the notion of a stopped value beyond right-continuous submartingales with the Doob-Meyer property to all suitable Daniell-integrable adapted processes (Grobler et al., 2020). Their generalized optional sampling theorem replaces earlier restrictions by proving $F_{S^+}X_T\geq X_S$ for right-uo-continuous submartingales under mild conditions.

A complementary construction appears for functionals measurable with respect to a stopped Wiener process. Dorogovtsev and Shcherbakov consider measures absolutely continuous with respect to the law of $w(\cdot\wedge T)$, where $T$ is the exit time from a domain, and derive an orthogonal expansion analogous to the Itô-Wiener decomposition for functionals such as $f=\varphi(w(T))$. Their Clark-Ocone-type representation,

$$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$

makes the stopped functional explicit in terms of the stopped path and a Girsanov-transformed Brownian motion $\tilde w$ (Riabov, 2015).

Setting	Stopped quantity or estimator	Representative formulation
Abstract stopped process	Stopping element	$X_S$0
Stationary binary series	Estimator at recurrence stopping times	$X_S$1
Bayesian drift estimation	Posterior mean stopped optimally	$X_S$2
Self-similar stopped observation	Density estimator of $X_S$3 from $X_S$4	Mellin-inversion estimator of $X_S$5
TAR stopping rule	Estimated total relevant documents	$X_S$6 after Poisson-rate fitting

These constructions show that the “stopped” aspect can enter either through the argument of the estimator, the sampling scheme that generates its data, or the rule that selects its final iterate.

2. Estimation at selected stopping times

Morvai’s work on stationary binary time series establishes a decisive negative and positive pair of results. Consistent forward prediction at all times is impossible for all stationary ergodic binary processes, but almost sure consistency becomes achievable at a sparse sequence of recurrence-based stopping times (0710.3760). The stopping times are defined recursively by

$X_S$7

and the estimator

$X_S$8

satisfies

$X_S$9

The price of universality is sparsity: if the process entropy $\mu_\mathbb{S}$0, then for small enough $\mu_\mathbb{S}$1, $\mu_\mathbb{S}$2 almost surely for large $\mu_\mathbb{S}$3, with $\mu_\mathbb{S}$4.

A biologically different but structurally related example is DNA unzipping, modeled as a stopped birth and death process with unknown transition probabilities. Andreoletti and Diel use repeated stopped trajectories to form sufficient statistics $\mu_\mathbb{S}$5, $\mu_\mathbb{S}$6, and, in continuous time, $\mu_\mathbb{S}$7, then build local information functions $\mu_\mathbb{S}$8 and define a maximum-likelihood estimator

$\mu_\mathbb{S}$9

The posterior probability of a candidate base is explicitly available, and the sitewise error probability decays exponentially,

$X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$0

with $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$1 determined by local passage statistics and information-theoretic contrasts in the stacking energies (Andreoletti et al., 2011). Here the stopped-process estimator is not merely evaluated at a stopping time; it is statistically identified from an ensemble of independently stopped paths.

These two examples correct a common misconception that stopping only truncates data. In both cases, stopping creates a structure—recurrence in one case, first-passage path statistics in the other—that makes otherwise unavailable inference possible.

3. Sequential estimation, optimal stopping, and early termination

In Bayesian sequential least-squares estimation for the drift of a Wiener process, the estimator is the posterior mean

$X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$2

and the stopping problem is to minimize

$X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$3

Ekström and Vaicenavicius show that this reduces to an optimal stopping problem for a diffusion in natural scale; the continuation region shrinks monotonically in time, and the shape of the stopping region depends on the prior. In particular, a natural criterion is to stop when the observation cost dominates the local variance reduction rate, informally when $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$4 (Ekström et al., 2019).

An industrial instance of stopping-time estimation appears in wood panel compression. Liebl, Munk, and coauthors model the near-infrared spectra as a functional time series of $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$5 curves, split the data into training and testing samples, compute integrated squared forecast errors under rolling one-step-ahead prediction, and then estimate the optimal stopping time by a one-break structural break method on the resulting univariate error series. In the application, the original sample yields estimated stopping time $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$6, while bootstrap modes are $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$7 and $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$8 (Shang et al., 2022). The procedure is explicitly conservative in simulation, rarely producing false-early stops.

Recent machine-learning formulations retain the same logic but replace stopping times by iteration indices. Early-stopped aggregation computes estimators only along a model-complexity ladder until the first increase of a unified energy: $X_\mathbb{S}=\int X_t\,d\mu_\mathbb{S}(t),$9 and then aggregates only the estimators up to $F_{S^+}X_T\geq X_S$0. The framework covers variational Bayes, variational empirical Bayes, and frequentist penalized aggregation, and is driven by a common data-fit plus complexity-control functional (Ohn et al., 15 Apr 2026). ScoreStop makes the stopping rule itself into a functional score test on validation data, with statistic

$F_{S^+}X_T\geq X_S$1

asymptotically $F_{S^+}X_T\geq X_S$2 under the null that the current predictor is the population risk minimizer. Because it uses gradients rather than loss values, the same construction extends to implicit losses such as LambdaRank and data-dependent losses such as Cox regression via influence functions (Hines et al., 1 Jun 2026).

A plausible implication is that stopped-process estimation has become a unifying language for both stochastic-process inference and iterative model selection: in each case, a random or data-adaptive termination rule is part of the estimator, not an external convenience.

4. Inference from stopped observations and rare-event trajectories

A different branch of the subject treats the stopping time itself as the latent quantity to be inferred. Mnatsakanov, Belomestny, and Schoenmakers consider a known one-dimensional self-similar process $F_{S^+}X_T\geq X_S$3 and an unknown random time $F_{S^+}X_T\geq X_S$4, independent of $F_{S^+}X_T\geq X_S$5, observed only through an i.i.d. sample of $F_{S^+}X_T\geq X_S$6. Self-similarity gives

$F_{S^+}X_T\geq X_S$7

turning the problem into multiplicative deconvolution. They estimate the Mellin transform of $F_{S^+}X_T\geq X_S$8 empirically and invert it to obtain a nonparametric estimator $F_{S^+}X_T\geq X_S$9 of the density $w(\cdot\wedge T)$0. For suitable smoothness classes, the estimator attains the rate

$w(\cdot\wedge T)$1

and for Bessel processes they further establish asymptotic normality (Schulmann, 2019).

Bayesian parameter inference for partially observed stopped or killed processes replaces transform inversion by path-space Monte Carlo. Jasra, Del Moral, and collaborators treat processes started from $w(\cdot\wedge T)$2 and stopped at first hitting of a known set $w(\cdot\wedge T)$3, with partial observations. Their method embeds multi-level sequential Monte Carlo inside particle MCMC, using nested level sets from $w(\cdot\wedge T)$4 to $w(\cdot\wedge T)$5 and resampling only when particles hit intermediate sets. Because the appropriate levels depend on the parameter proposal, they also introduce adaptive level-set selection inside PMCMC (Jasra et al., 2012).

For rare-event simulation, Whiteley, Heng, and Jasra construct proposals in reverse time. Their reverse-time transition kernel is based on Nagasawa’s formula,

$w(\cdot\wedge T)$6

where $w(\cdot\wedge T)$7 is the Green’s function up to the target set. A central simplification is that ratios of Green’s functions can often be approximated by low-dimensional conditional sampling distributions, substantially reducing proposal design complexity in high dimension (Koskela et al., 2016).

Numerical approximation of stopped dynamics raises a further issue: the estimator may fail unless stopping is built into the scheme itself. For singular Langevin dynamics with potentials including Lennard-Jones interaction with confinement, a stopped explicit splitting scheme with rejection accepts an update only when a Lyapunov function remains below a threshold. Under this rule, weak error expansions of Talay-Tubaro type hold both at finite time and for the invariant measure,

$w(\cdot\wedge T)$8

restoring high-order bias analysis for singular potentials (Journel, 2023).

5. Stopping criteria as estimators in document retrieval and review

In technology-assisted review and ranked retrieval, the stopped-process estimator often takes the form of an estimated total number of relevant documents or an estimated achieved recall. Callaghan and Müller-Hansen model the occurrence of relevant documents in a ranked list as an inhomogeneous Poisson process with exponentially decaying intensity

$w(\cdot\wedge T)$9

so that

$T$0

After fitting $T$1 and $T$2 from an initial reviewed prefix, they use the Poisson CDF over the full candidate set to obtain an upper bound $T$3 on the total number of relevant documents at a user-specified confidence level, then set the stopping threshold

$T$4

for target recall $T$5. On the CLEF 2017 eHealth Technology-Assisted Review task—$T$6 topics, about $T$7k documents, and $T$8 ranked runs—this Poisson-process method saved on average $T$9 of judgments, compared with $f=\varphi(w(T))$0 for tuned knee and $f=\varphi(w(T))$1 for target method, while maintaining reliability (Sneyd et al., 2019).

The 2023 point-process formulation generalizes this idea by comparing four rate functions—exponential, hyperbolic, power law, and AP-Prior—and by considering both inhomogeneous Poisson and Cox-process variants. It is also the first to explore stopping method robustness by reporting performance on a range of rankings of varying effectiveness across CLEF e-Health, TREC Total Recall, and TREC Legal. The method estimates the remaining relevant documents by integrating the fitted rate over the unreviewed suffix and stops when the observed relevant count exceeds the target implied by the estimated total (Stevenson et al., 2023).

A distinct prevalence-estimation approach uses Chao’s population size estimator. To create the repeated-capture structure needed by Chao in TAR without multiple human reviewers, the method deploys an ensemble-based Active Learning QUERY-BY-COMMITTEE strategy, tracks how often distinct committee members retrieve the same relevant documents, and estimates

$f=\varphi(w(T))$2

Stopping can be optimistic, using $f=\varphi(w(T))$3, or conservative, using the upper bound of the $f=\varphi(w(T))$4 confidence interval. In the reported simulation study, the conservative Chao variants at a $f=\varphi(w(T))$5 recall target achieved mean recall about $f=\varphi(w(T))$6 or $f=\varphi(w(T))$7, work saved over random sampling about $f=\varphi(w(T))$8 or $f=\varphi(w(T))$9, and triggered in $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$0 or $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$1 of runs, respectively (Bron et al., 2024).

Two recurrent misconceptions are explicitly addressed in this literature. First, guarantees obtained under random ordering do not automatically transfer to ranked retrieval; the target method provides guarantees only in random orderings, not arbitrary rankings (Sneyd et al., 2019). Second, heuristic rules such as knee detection may work empirically but do not by themselves provide probabilistic recall guarantees (Sneyd et al., 2019).

6. Dependence, filtration, and reliability of stopped estimators

The validity of a stopped-process estimator depends sharply on the filtration with respect to which stopping occurs. In sequential multiple testing, adaptively stopped e-processes are e-values only for stopping times in their own local filtrations. Ramdas and collaborators show that the stopped e-BH procedure fails in full generality if one stops with respect to a richer global filtration, because information can leak across streams. Under a simple causal condition—excluding unobserved confounding from the past—local e-processes are also global e-processes, and the stopped e-BH procedure controls the false discovery rate at arbitrary stopping times (Wang et al., 12 Feb 2025). This result makes precise that “anytime-valid” does not mean “valid under every information structure.”

An analogous preservation question arises for counting processes stopped at an independent random time $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$2. de Uña-Álvarez and Belzunce study the number of events $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$3 before $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$4 when $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$5 has the decreasing failure rate property. They show that if the interarrival times satisfy suitable association and stochastic ordering conditions, then $$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$6 is d-DFR. In particular,

$$\varphi(w(T)) = T\varphi(u) + \int_0^T \nabla T\varphi(w(s))\,d\tilde w(s),$$7

and sufficient conditions on the arrival epochs preserve the reliability class of the stopping time in the stopped count (Badía et al., 2012).

These results indicate that stopped-process estimation is never purely a matter of truncation. Its inferential content depends on dependence structure, admissible stopping rules, and the interaction between local and global information. A plausible implication is that the most technically delicate part of many stopped estimators is not the estimator formula itself but the proof that stopping does not invalidate it.