Positive Lower-Bounded Distributions
The positive lower-bounded probabilities have support on real values above some positive minimum value.
Pareto distribution
Probability density function
If \(y_{\text{min}} \in \mathbb{R}^+\) and \(\alpha \in \mathbb{R}^+\), then for \(y \in \mathbb{R}^+\) with \(y \geq y_{\text{min}}\), \[\begin{equation*} \text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}}. \end{equation*}\]
Sampling statement
y ~ pareto(y_min, alpha)
Increment target log probability density with pareto_lupdf(y | y_min, alpha).
Stan functions
real pareto_lpdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha
real pareto_lupdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha dropping constant additive terms
real pareto_cdf(reals y | reals y_min, reals alpha)
The Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha
real pareto_lcdf(reals y | reals y_min, reals alpha)
The log of the Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha
real pareto_lccdf(reals y | reals y_min, reals alpha)
The log of the Pareto complementary cumulative distribution function of y given positive minimum value y_min and shape alpha
R pareto_rng(reals y_min, reals alpha)
Generate a Pareto variate with positive minimum value y_min and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Pareto type 2 distribution
Probability density function
If \(\mu \in \mathbb{R}\), \(\lambda \in \mathbb{R}^+\), and \(\alpha \in \mathbb{R}^+\), then for \(y \geq \mu\), \[\begin{equation*} \mathrm{Pareto\_Type\_2}(y|\mu,\lambda,\alpha) = \ \frac{\alpha}{\lambda} \, \left( 1+\frac{y-\mu}{\lambda} \right)^{-(\alpha+1)} \! . \end{equation*}\]
Note that the Lomax distribution is a Pareto Type 2 distribution with \(\mu=0\).
Sampling statement
y ~ pareto_type_2(mu, lambda, alpha)
Increment target log probability density with pareto_type_2_lupdf(y | mu, lambda, alpha).
Stan functions
real pareto_type_2_lpdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 density of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lupdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 density of y given location mu, scale lambda, and shape alpha dropping constant additive terms
real pareto_type_2_cdf(reals y | reals mu, reals lambda, reals alpha)
The Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lcdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lccdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 complementary cumulative distribution function of y given location mu, scale lambda, and shape alpha
R pareto_type_2_rng(reals mu, reals lambda, reals alpha)
Generate a Pareto Type 2 variate with location mu, scale lambda, and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Wiener First Passage Time Distribution
Probability density function
If \(\alpha \in \mathbb{R}^+\), \(\tau \in \mathbb{R}^+\), \(\beta \in [0, 1]\) and \(\delta \in \mathbb{R}\), then for \(y > \tau\), \[\begin{equation*} \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right) \end{equation*}\] where \(\phi(x)\) denotes the standard normal density function; see (Feller 1968), (Navarro and Fuss 2009).
Sampling statement
y ~ wiener(alpha, tau, beta, delta)
Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta).
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
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
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 \[\begin{equation*} \text{Wiener}(y | \alpha, \tau, 1 - \beta, - \delta) \end{equation*}\] For more details, see the appendix of Vandekerckhove and Wabersich (2014).