18.3 Wiener First Passage Time Distribution
18.3.1 Probability Density Function
If \(\alpha \in \mathbb{R}^+\), \(\tau \in \mathbb{R}^+\), \(\beta \in [0, 1]\) and \(\delta \in \mathbb{R}\), then for \(y > \tau\), \[ \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right) \] where \(\phi(x)\) denotes the standard normal density function; see (Feller 1968), (Navarro and Fuss 2009).
18.3.2 Sampling Statement
y ~ wiener(alpha, tau, beta, delta)
Increment target log probability density with wiener_lpdf(y | alpha, tau, beta, delta)
dropping constant additive terms.
18.3.3 Stan Functions
real wiener_lpdf(reals y | reals alpha, reals tau, reals beta, reals delta)
The log of the Wiener first passage time density of y given boundary
separation alpha, non-decision time tau, a-priori bias beta and drift
rate delta
18.3.4 Boundaries
Stan returns the first passage time of the accumulation process over the upper boundary only. To get the result for the lower boundary, use \[ \text{wiener}(y | \alpha, \tau, 1 - \beta, - \delta) \] For more details, see the appendix of Vandekerckhove and Wabersich (2014).
References
Feller, William. 1968. An Introduction to Probability Theory and Its Applications. Vol. 1. 3. Wiley, New York.
Vandekerckhove, Joachim, and Dominik Wabersich. 2014. “The RWiener Package: An R Package Providing Distribution Functions for the Wiener Diffusion Model.” The R Journal 6/1. http://journal.r-project.org/archive/2014-1/vandekerckhove-wabersich.pdf.