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).

Available since 2.0

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

Available since 2.12

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

Available since 2.25

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

Available since 2.0

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

Available since 2.12

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

Available since 2.12

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.

Available since 2.18

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).

Available since 2.5

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

Available since 2.18

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

Available since 2.25

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

Available since 2.5

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

Available since 2.18

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

Available since 2.18

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.

Available since 2.18

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 , .

Distribution statement

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

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

Available since 2.7

y ~ wiener(alpha, tau, beta, delta, var_delta) Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta, var_delta).

Available since 2.35

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).

Available since 2.35

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_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.

Available since 2.35

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.

Available since 2.35

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

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.

Available since 2.35

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.

Available since 2.35

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).