Is the process always Brownian motion?

No. Random walks, Markov chains and other stochastic processes can define first-hitting times.

Is crossing by time t the same as being above the threshold at time t?

No; earlier crossings followed by returns can make them different.

What if a unit has not crossed by study end?

Its event time may be censored, requiring an appropriate likelihood contribution.

AI Glossary / Statistics and probability / Advanced

First-hitting-time model

Definition

A first-hitting-time model describes an event time as the first moment a stochastic process reaches a specified boundary or set. The underlying process might represent degradation, accumulated evidence or another changing quantity. The event occurs when its path first meets the defined condition, not simply when its value is inspected at a later date. The resulting event-time distribution depends on the process, starting state and boundary. First passage time is a closely related term, with exact conventions depending on the model.

What it means

Rather than selecting a lifetime distribution without an explicit process story, this approach connects duration to an evolving mechanism. A component might reach a degradation threshold after a sequence of random changes, or a monitored statistic might first cross a control limit. The process can be observed directly, partially observed or latent. A plausible interpretation can help specify covariates and explain behavior, but it does not prove that the assumed process is the true mechanism generating the data.

How it works

Specify a stochastic process X(t), an initial state and a boundary set B. Define T as the infimum of times t for which X(t) enters B, allowing an infinite value if the boundary is never reached. Derive or numerically approximate the distribution of T using the process assumptions. For particular Wiener-process models with drift toward a fixed boundary, first-hitting times have inverse Gaussian distributions; other processes or boundaries produce different laws. Fit parameters to observed event times and any process measurements, treating censoring appropriately. Threshold regression links model parameters, such as initial distance or drift, to explanatory variables.

Three practical examples

Equipment degradation: model the duration until a monitored condition first reaches a defined failure threshold.
Quality monitoring: study the time until a process statistic first crosses an intervention boundary.
Decision-process research: represent response time through accumulated noisy evidence reaching a decision threshold.

Think of it this way

Think of following a moving marker until it touches a line for the first time. The full route matters. A marker may cross the line and later return, so looking only at its final position can miss an earlier event. The model turns that route-dependent question into a probability distribution.

Worked example

Consider a hypothetical discrete random walk starting at zero. At each step it independently moves up one or down one with equal probability, and the event is first reaching level two. At step two, only the path up-up reaches the boundary, so P(T = 2) = 1/4. By step four there are 16 equally likely four-step paths. Four start with up-up and have already hit at step two. Two additional paths first hit at step four: up-down-up-up and down-up-up-up. Thus P(T = 4) = 2/16 = 1/8 and P(T less than or equal to 4) = 6/16 = 3/8. In contrast, only five paths finish at level two or above, giving endpoint probability 5/16. The path up-up-down-down illustrates the difference: it already hit the boundary but finishes at zero. These exact enumerated probabilities concern this toy walk, not a continuous degradation model.

Illustrative case study

Fictional case: Juniper Monitoring models when an equipment indicator first reaches an inspection threshold. Engineers distinguish an actual first crossing from a high reading at the final inspection and document the observation schedule. Some units have not crossed by study end, so their times are right-censored rather than treated as failures on that date. Analysts compare predicted event-time behavior with held-out records and consider alternative process assumptions. This invented case illustrates defining the event and handling incomplete follow-up, without certifying the physical meaning of a fitted latent process.

Common misconceptions and mistakes

Do not equate a first-crossing probability with the probability of being beyond the boundary at one final time. A path can cross and return. Not every first-hitting-time model has an inverse Gaussian distribution. Depending on drift and boundaries, some paths may never hit, producing a distribution with probability mass at infinity. Sparse observation can miss crossings between measurements. Treating the final observation time as the event time biases censored data analysis. An appealing mechanistic label does not remove parameter-identification problems or justify causal interpretations of covariates.

Business use

Businesses can use first-hitting-time models when a meaningful threshold defines an operational event and the evolution toward it can be modeled credibly. Specify whether the event concerns a physical failure, a warning condition or a management action, since these are different targets. Record inspection intervals and interventions that may alter the process. Evaluate sensitivity to boundaries, drift and observation noise. The approach can support duration forecasts and monitoring analysis, but decisions should reflect uncertainty in both the event-time calculation and the underlying process representation.

Check your understanding

Why does the worked example give six crossing paths but only five high-ending paths?

Answer: One path crosses at step two and returns to zero by step four, so first passage records an event that the final-position check misses.

Related terms

Stochastic process
Survival function
Censoring
first passage time
absorbing boundary
threshold regression
inverse Gaussian distribution

Learn next

derive a survival function from first-crossing probabilities, then compare continuous observation with periodic inspection and examine how censoring changes parameter estimation.

Sources

Sources: Lee and Whitmore, Threshold Regression for Survival Analysis: Modeling Event Times by a Stochastic Process Reaching a Boundary: https://arxiv.org/abs/0708.0346 ; Lee and Whitmore, First Hitting Time Models for Lifetime Data, publisher summary: https://www.sciencedirect.com/science/article/pii/S0169716103230303 ; Whitmore, First-Passage-Time Models for Duration Data: Regression Structures and Competing Risks: https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2987525 .

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