Day 23: Inverse Gaussian Process#
TheĀ Inverse Gaussian ProcessĀ is a stochastic process whose increments follow theĀ Inverse Gaussian distribution, a continuous probability distribution often used to model positive, skewed data. This process is widely applied in fields such as finance, reliability engineering, and queueing theory.
Definition#
AnĀ Inverse Gaussian processĀ is a stochastic processĀ \(\{X(t),t\geq 0\}\) where:
\(X(0)=0\),
The incrementsĀ \(X(t)āX(s)\)Ā (forĀ t>s) are independent and follow the Inverse Gaussian distribution with meanĀ \(\mu(t-s)\) and scale parameterĀ \(\eta\),
š Random Facts š#
This Inverse Gaussian distribution appears to have been first derived in 1900 by Louis Bachelier as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödinger and Marian v. Smoluchowski as the time to first passage of a Brownian motion.
The name inverse Gaussian was proposed by British medical physicist and statisticianĀ Maurice TweedieĀ in 1945
In 1968, M. T. Wasan introduced the concept of theĀ Inverse Gaussian processĀ in his paper āOn an Inverse Gaussian Process,ā published in theĀ Scandinavian Actuarial Journal. Wasanās work laid the foundation for subsequent research into the Inverse Gaussian process, influencing studies in areas such as bivariate distributions and first-passage time distributions in stochastic processes.
The Inverse Gaussian process is used in various fields to model cumulative or first-passage phenomena where skewed, positive increments are observed
More to Read š#
TWEEDIE, M. Inverse Statistical Variates.Ā NatureĀ 155, 453 (1945). https://doi.org/10.1038/155453a0
Wasan, M. T. (1968). On an inverse Gaussian process.Ā Scandinavian Actuarial Journal,Ā 1968(1ā2), 69ā96. https://doi.org/10.1080/03461238.1968.10413264
Wang, X., & Xu, D. (2010). An Inverse Gaussian Process Model for Degradation Data.Ā Technometrics,Ā 52(2), 188ā197. https://doi.org/10.1198/TECH.2009.08197
P.s. If you are curious about probability distributions visit the Advent Calendar 2023 āØ