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Introduction
First-hitting-time Models are a sub-class of survival models in statistics.

First hitting times, sometimes called first passage times, can be found in many types of stochastic processes. When it comes to lifetime context, the state of the underlying process is naturally used to represent the health of an individual or the strength of an item. As soon as the process reaches an adverse threshold state, for the first time, the individual would experience a clinical end, or the item would terminate operation. 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.

Basically, 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 $$X(0)=x_0>0\,$$.