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 \(y \geq y_{\text{min}}\), \[\begin{equation*} \text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}}. \end{equation*}\]
Distribution 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 \(\mu=0\).
Distribution 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 \(\alpha \in \mathbb{R}^+\), \(\tau \in \mathbb{R}^+\), \(\beta \in (0, 1)\), \(\delta \in \mathbb{R}\), \(s_{\delta} \in \mathbb{R}^{\geq 0}\), \(s_{\beta} \in [0, 1)\), and \(s_{\tau} \in \mathbb{R}^{\geq 0}\) then for \(y > \tau\),
\[\begin{equation*} \begin{split} &\text{Wiener}(y\mid \alpha,\tau,\beta,\delta,s_{\delta},s_{\beta},s_{\tau}) = \\ &\frac{1}{s_{\tau}}\int_{\tau}^{\tau+s_{\tau}}\frac{1}{s_{\beta}}\int_{\beta-\frac{1}{2}s_{\beta}}^{\beta+\frac{1}{2}s_{\beta}}\int_{-\infty}^{\infty} p_3(y-{\tau_0}\mid \alpha,\nu,\omega) \\ &\times \frac{1}{\sqrt{2\pi s_{\delta}^2}}\exp\Bigl(-\frac{(\nu-\delta)^2}{2s_{\delta}^2}\Bigr) \,d\nu \,d\omega \,d{\tau_0}= \\ &\frac{1}{s_{\tau}}\int_{\tau}^{\tau+s_{\tau}}\frac{1}{s_{\beta}}\int_{\beta-\frac{1}{2}s_{\beta}}^{\beta+\frac{1}{2}s_{\beta}} M\times p_3(y-{\tau_0}\mid \alpha,\nu,\omega) \,d\omega \,d{\tau_0}, \end{split} \end{equation*}\]
where \(p()\) denotes the density function, and \(M\) and \(p_3()\) are defined, by using \(t:=y-{\tau_0}\), as
\[\begin{equation*} M \coloneqq \frac{1}{\sqrt{1+s_{\delta}^2t}}\exp\Bigl(\alpha{\delta}\omega+\frac{\delta^2t}{2}+\frac{s_{\delta}^2\alpha^2\omega^2-2\alpha{\delta}\omega-\delta^2t}{2(1+s_{\delta}^2t)}\Bigr)\text{ and} \end{equation*}\]
\[\begin{equation*} p_3(t\mid \alpha,\delta,\beta) \coloneqq \frac{1}{\alpha^2}\exp\Bigl(-\alpha\delta\beta-\frac{\delta^2t}{2}\Bigr)f(\frac{t}{\alpha^2}\mid 0,1,\beta), \end{equation*}\]
where \(f(t^*=\frac{t}{\alpha^2}\mid0,1,\beta)\) can be specified in two ways:
\[\begin{equation*} f_l(t^*\mid 0,1,\beta) = \sum_{k=1}^\infty k\pi \exp\Bigl(-\frac{k^2\pi^2t^*}{2}\Bigr)\sin(k\pi \beta)\text{ and} \end{equation*}\]
\[\begin{equation*} f_s(t^*\mid0,1,\beta) = \sum_{k=-\infty}^\infty \frac{1}{\sqrt{2\pi(t^*)^3}}(\beta+2k) \exp\Bigl(-\frac{(\beta+2k)^2}{2t^*}\Bigr). \end{equation*}\]
Which of these is used in the computations depends on which expression requires the smaller number of components \(k\) to guarantee a pre-specified precision
In the case where \(s_{\delta}\), \(s_{\beta}\), and \(s_{\tau}\) are all \(0\), this simplifies to \[\begin{equation*} \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right) \end{equation*}\] where \(\phi(x)\) denotes the standard normal density function; see (Feller 1968), (Navarro and Fuss 2009).
Distribution statement
y ~ wiener(alpha, tau, beta, delta)
Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta).
y ~ wiener(alpha, tau, beta, delta, var_delta) Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta, var_delta).
y ~ wiener(alpha, tau, beta, delta, var_delta, var_beta, var_tau) Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta, var_delta, var_beta, var_tau).
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_lpdf(real y | real alpha, real tau, real beta, real delta, real var_delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta, drift rate delta, and inter-trial drift rate variability var_delta.
Setting var_delta to 0 recovers the 4-parameter signature above.
real wiener_lpdf(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau.
Setting var_delta, var_beta, and var_tau to 0 recovers the 4-parameter signature above.
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
real wiener_lupdf(real y | real alpha, real tau, real beta, real delta, real var_delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta, drift rate delta, and inter-trial drift rate variability var_delta, dropping constant additive terms.
Setting var_delta to 0 recovers the 4-parameter signature above.
real wiener_lupdf(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau, dropping constant additive terms.
Setting var_delta, var_beta, and var_tau to 0 recovers the 4-parameter signature above.
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).