# First-hitting-time model

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First-hitting-time model

In statistics, first-hitting-time models are a sub-class of survival models. The first hitting time, also called first passage time, of a set A with respect to an instance of a stochastic process is the time until the stochastic process first enters A.

## Examples

A common example of a first-hitting-time model is a ruin problem, such as Gambler's ruin. In this example, an entity (often described as a gambler or an insurance company) has an amount of money which varies randomly with time, possibly with some drift. The model considers the event that the amount of money reaches 0, representing bankruptcy. The model can answer questions such as the probability that this occurs within finite time, or the mean time until which it occurs.

First-hitting-time models can be applied to expected lifetimes, of patients or mechanical devices. When the process reaches an adverse threshold state for the first time, the patient dies, or the device breaks down.

## Latent vs observable

In many real world applications, the process is latent, or unobservable. When first hitting time models are equipped with regression structures, accommodating covariate data, we call such regression structure Threshold regression. The threshold state, parameters of the process, and even time scale may depend on corresponding covariates.

A first-hitting-time (FHT) model has two underlying components: (1) a parent stochastic process $\{X(t)\}\,\,$, and (2) a threshold. The first hitting time is defined as the time when the stochastic process first reaches the threshold. It is very important to distinguish whether the sample path of the parent process is latent (i.e., unobservable) or observable, and such distinction is a characteristic of the FHT model. By far, latent processes are most common. To give an example, we can use a Wiener process $\{X(t), t\ge0\,\}\,$ as the parent stochastic process. Such Wiener process can be defined with the mean parameter ${\mu}\,\,$, the variance parameter ${\sigma^2}\,\,$, and the initial value $X(0)=x_0>0\,$.

## References

• Whitmore, G. A. (1986). "First passage time models for duration data regression structures and competing risks", The Statistician, 35, 207–219. JSTOR 2987525
• Whitmore, G. A. (1995). "Estimating degradation by a Wiener diffusion process subject to measurement error", Lifetime Data Analysis, 1, 307-319.
• Whitmore, G. A., M. J. Crowder and J. F. Lawless (1998). "Failure inference from a marker process based on a bivariate Wiener model", Lifetime Data Analysis, 4, 229-251.
• Redner, S (2001). A Guide to First-Passage Processes, Cambridge University Press. ISBN 0521652480

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