First-hitting-time model

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.

Contents

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\,.

See also

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

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Model aircraft — A die cast Boeing 747 400 model. Model aircraft are flying or non flying models of existing or imaginary aircraft using a variety of materials including plastic, diecast metal, polystyrene, balsa wood, foam and fibreglass. Flying designs range… …   Wikipedia

  • Time travel — This article details time travel itself. For other uses, see Time Traveler. Time travel is the concept of moving between different moments in time in a manner analogous to moving between different points in space, either sending objects (or in… …   Wikipedia

  • model — 01. She worked as a [model] in Milan for a couple of years, and then went into fashion design herself. 02. My son and I enjoy playing with his [model] rockets on the weekends. 03. The boys raced their [model] cars across the school field. 04.… …   Grammatical examples in English

  • New Zealand's Next Top Model, Cycle 2 — New Zealand s Next Top Model Cycle 2 Promotional photograph of the cast of Cycle 2 of New Zealand s Next Top Model Format Reality television Created by Tyr …   Wikipedia

  • New Zealand's Next Top Model, Cycle 3 — New Zealand s Next Top Model Cycle 3 Promotional photograph of the cast of Cycle 3 of New Zealand s Next Top Model Format Reality television Created by Tyr …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • List of one-time The Simpsons characters — Further information: List of characters in The Simpsons and List of recurring characters in The Simpsons The following is a list of one time characters from the American animated television comedy series The Simpsons. Some of the… …   Wikipedia

  • Markov chain — A simple two state Markov chain. A Markov chain, named for Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized …   Wikipedia

  • List of One Tree Hill characters — The original five from the first season; Upper row (L R) Lucas, Peyton and Haley. Lower row (L R) Brooke and Nathan. One Tree Hill is an American television series created by Mark Schwahn for The WB in 2003. After its third season, The WB and UPN …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”