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## 19.3 Wiener First Passage Time Distribution

### 19.3.1 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 , .

### 19.3.2 Sampling statement

y ~ wiener(alpha, tau, beta, delta)

Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta).
Available since 2.7

### 19.3.3 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
Available since 2.18

real wiener_lupdf(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 dropping constant additive terms
Available since 2.25

### 19.3.4 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.
Navarro, Danielle J, and Ian G Fuss. 2009. “Fast and Accurate Calculations for First-Passage Times in Wiener Diffusion Models.” Journal of Mathematical Psychology 53 (4): 222–30.
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.