Day 23: Inverse Gaussian Process

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 \(X(t)āˆ’X(s)\)Ā (forĀ t>s) are independent and follow the Inverse Gaussian distribution with meanĀ \(\mu(t-s)\) and scale parameterĀ \(\eta\),

šŸ”” Random Facts šŸ””#

  • This Inverse Gaussian distribution appears to have been first derived in 1900 byĀ Louis Bachelier as the time a stock reaches a certain price for the first time. In 1915 it was used independently byĀ Erwin Schrƶdinger andĀ Marian v. SmoluchowskiĀ 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 ā€œOn an Inverse Gaussian Process,ā€ published in theĀ Scandinavian Actuarial Journal. Wasan’s 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 šŸ“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(1–2), 69–96. 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), 188–197. https://doi.org/10.1198/TECH.2009.08197

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨