positive_lower-bounded_distributions.qmd

---
pagetitle: Positive Lower-Bounded Distributions
---

# 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)`.
{{< since 2.0 >}}
<!-- real; pareto ~; -->
\index{{\tt \bfseries pareto }!sampling statement|hyperpage}

### Stan functions

<!-- real; pareto_lpdf; (reals y | reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_lpdf }!{\tt (reals y \textbar\ reals y\_min, reals alpha): real}|hyperpage}

`real` **`pareto_lpdf`**`(reals y | reals y_min, reals alpha)`<br>\newline
The log of the Pareto density of y given positive minimum value y_min
and shape alpha
{{< since 2.12 >}}

<!-- real; pareto_lupdf; (reals y | reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_lupdf }!{\tt (reals y \textbar\ reals y\_min, reals alpha): real}|hyperpage}

`real` **`pareto_lupdf`**`(reals y | reals y_min, reals alpha)`<br>\newline
The log of the Pareto density of y given positive minimum value y_min
and shape alpha dropping constant additive terms
{{< since 2.25 >}}

<!-- real; pareto_cdf; (reals y | reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_cdf }!{\tt (reals y \textbar\ reals y\_min, reals alpha): real}|hyperpage}

`real` **`pareto_cdf`**`(reals y | reals y_min, reals alpha)`<br>\newline
The Pareto cumulative distribution function of y given positive
minimum value y_min and shape alpha
{{< since 2.0 >}}

<!-- real; pareto_lcdf; (reals y | reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_lcdf }!{\tt (reals y \textbar\ reals y\_min, reals alpha): real}|hyperpage}

`real` **`pareto_lcdf`**`(reals y | reals y_min, reals alpha)`<br>\newline
The log of the Pareto cumulative distribution function of y given
positive minimum value y_min and shape alpha
{{< since 2.12 >}}

<!-- real; pareto_lccdf; (reals y | reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_lccdf }!{\tt (reals y \textbar\ reals y\_min, reals alpha): real}|hyperpage}

`real` **`pareto_lccdf`**`(reals y | reals y_min, reals alpha)`<br>\newline
The log of the Pareto complementary cumulative distribution function
of y given positive minimum value y_min and shape alpha
{{< since 2.12 >}}

<!-- R; pareto_rng; (reals y_min, reals alpha); -->
\index{{\tt \bfseries pareto\_rng }!{\tt (reals y\_min, reals alpha): R}|hyperpage}

`R` **`pareto_rng`**`(reals y_min, reals alpha)`<br>\newline
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](conventions_for_probability_functions.qmd#prng-vectorization).
{{< 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)`.
{{< since 2.5 >}}
<!-- real; pareto_type_2 ~; -->
\index{{\tt \bfseries pareto\_type\_2 }!sampling statement|hyperpage}

### Stan functions

<!-- real; pareto_type_2_lpdf; (reals y | reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_lpdf }!{\tt (reals y \textbar\ reals mu, reals lambda, reals alpha): real}|hyperpage}

`real` **`pareto_type_2_lpdf`**`(reals y | reals mu, reals lambda, reals alpha)`<br>\newline
The log of the Pareto Type 2 density of y given location mu, scale
lambda, and shape alpha
{{< since 2.18 >}}

<!-- real; pareto_type_2_lupdf; (reals y | reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_lupdf }!{\tt (reals y \textbar\ reals mu, reals lambda, reals alpha): real}|hyperpage}

`real` **`pareto_type_2_lupdf`**`(reals y | reals mu, reals lambda, reals alpha)`<br>\newline
The log of the Pareto Type 2 density of y given location mu, scale
lambda, and shape alpha dropping constant additive terms
{{< since 2.25 >}}

<!-- real; pareto_type_2_cdf; (reals y | reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_cdf }!{\tt (reals y \textbar\ reals mu, reals lambda, reals alpha): real}|hyperpage}

`real` **`pareto_type_2_cdf`**`(reals y | reals mu, reals lambda, reals alpha)`<br>\newline
The Pareto Type 2 cumulative distribution function of y given location
mu, scale lambda, and shape alpha
{{< since 2.5 >}}

<!-- real; pareto_type_2_lcdf; (reals y | reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_lcdf }!{\tt (reals y \textbar\ reals mu, reals lambda, reals alpha): real}|hyperpage}

`real` **`pareto_type_2_lcdf`**`(reals y | reals mu, reals lambda, reals alpha)`<br>\newline
The log of the Pareto Type 2 cumulative distribution function of y
given location mu, scale lambda, and shape alpha
{{< since 2.18 >}}

<!-- real; pareto_type_2_lccdf; (reals y | reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_lccdf }!{\tt (reals y \textbar\ reals mu, reals lambda, reals alpha): real}|hyperpage}

`real` **`pareto_type_2_lccdf`**`(reals y | reals mu, reals lambda, reals alpha)`<br>\newline
The log of the Pareto Type 2 complementary cumulative distribution
function of y given location mu, scale lambda, and shape alpha
{{< since 2.18 >}}

<!-- R; pareto_type_2_rng; (reals mu, reals lambda, reals alpha); -->
\index{{\tt \bfseries pareto\_type\_2\_rng }!{\tt (reals mu, reals lambda, reals alpha): R}|hyperpage}

`R` **`pareto_type_2_rng`**`(reals mu, reals lambda, reals alpha)`<br>\newline
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](conventions_for_probability_functions.qmd#prng-vectorization).
{{< since 2.18 >}}

## Wiener First Passage Time Distribution

For an extended explanation of how to use the `wiener_lpdf` and
`wiener_l[c]cdf_unnorm` functions, see @Henrich2024.

### 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 one representation that converges fast for small reaction-time values ("small time expansion"):
\begin{equation*}
\text{Wiener}(y|\alpha, \tau, \beta, \delta) =
\frac{\alpha}{(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 + \beta)\alpha }{\sqrt{y - \tau}}\right),
\end{equation*}
where $\phi(x)$ denotes the standard normal density function,
and one representation that converges fast for large reaction-time values ("large time expansion"):
\begin{equation*}
\text{Wiener}(y|\alpha, \tau, \beta, \delta) =
\frac{\pi}{\alpha^2} \exp \! \left(- \delta \alpha \beta -
\frac{\delta^2(y-\tau)}{2}\right) \sum_{k = 1}^{\infty} k \exp \! \left(-\frac{k^2\pi^2(y-\tau)};
{2\alpha^2}\right) \sin \!(k\pi\beta)
\end{equation*}
see [@Feller1968], [@NavarroFuss2009].

### Cumulative distribution function

For the cumulative distribution function (cdf) there also exist two expressions
depending on the reaction time.

Let $\alpha$, $\tau$, $\beta$, $\delta$, $s_{\delta}$, $s_{\beta}$, $s_{\tau}$
and $y$ be as above.

The formula for the large-time cdf of decision times (excluding the additive
reaction time components summarized in $\tau$ for the time being) at the upper
boundary is stated as follows:

\begin{equation}
F(y|\alpha, \beta, \delta) = P(\alpha, \beta, \delta) -
    \exp\left(\delta\alpha(1-\beta)-\frac{\delta^2 y}{2}\right)F_l(y|\alpha,\beta,\delta),
\end{equation}
where $P(\alpha,\beta,\delta)$ is the probability to hit the upper boundary, defined as

\begin{equation}
P(\alpha, \beta, \delta) =
\begin{cases}
    \frac{1-\exp(2\delta \alpha \beta)}{\exp(-2\delta \alpha(1-\beta)) - \exp(2\delta \alpha \beta)}, & \text{for } \delta\neq 0 \\
    \beta, & \text{for } \delta=0,
\end{cases}
\end{equation}

and

\begin{equation}
F_l(y|\alpha, \beta, \delta) =
    \frac{2\pi}{\alpha^2}\sum_{k=1}^{\infty}{\frac{k\sin{k\pi(1-\beta)}}{\delta^2+(k\pi)^2/\alpha^2}}\exp(-\frac{k^2\pi^2y}{2\alpha^2}).
\end{equation}


The formula for the small-time cdf at the upper boundary is stated as follows:

\begin{equation}
F(y|\alpha,\beta,\delta) = \exp\left(\delta \alpha(1-\beta) -\frac{\delta^2y}{2}\right)F_s(y|\alpha, \beta,\delta),
\end{equation} where

\begin{equation}
F_s(y|\alpha,\beta,\delta) = \sum_{k=0}^{\infty}(-1)^k\phi\left(\frac{\alpha(k+\beta^{*}_k)}
    {\sqrt{y}} \right) \times \left( R \left(\frac{\alpha(k+\beta^{*}_k)+\delta y}{\sqrt{y}} \right)
        + R \left(\frac{\alpha(k+\beta^*_k)-\delta y}{\sqrt{y}} \right)\right),
\end{equation}

where $\beta^*_k=(1-\beta)$ for $k$ even, $\beta^*_k=\beta$ for $k$ odd, and $R$ is Mill's ratio.

The cdf for the lower boundary is $F(y|\alpha,1-\beta,-\delta)$

### Distribution statement

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

Increment target log probability density with `wiener_lupdf(y | alpha, tau, beta, delta)`.
{{< 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)`.
{{< 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)`.
{{< since 2.35 >}}

<!-- real; wiener ~; -->
\index{{\tt \bfseries wiener }!sampling statement|hyperpage}

### Stan functions

<!-- real; wiener_lpdf; (reals y | reals alpha, reals tau, reals beta, reals delta); -->
\index{{\tt \bfseries wiener\_lpdf }!{\tt (reals y \textbar\ reals alpha, reals tau, reals beta, reals delta): real}|hyperpage}

`real` **`wiener_lpdf`**`(reals y | reals alpha, reals tau, reals beta, reals delta)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `beta`, and drift
rate `delta`.
{{< since 2.18 >}}

<!-- real; wiener_lpdf; (real y | real alpha, real tau, real beta, real delta, real var_delta); -->
\index{{\tt \bfseries wiener\_lpdf }!{\tt (real y \textbar\ real alpha, real tau, real beta, real delta, real var\_delta): real}|hyperpage}

`real` **`wiener_lpdf`**`(real y | real alpha, real tau, real beta, real delta, real var_delta)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `beta`, drift
rate `delta`, and inter-trial drift rate variability `var_delta`.

Setting `var_delta` to `0` recovers the 4-parameter signature above.
{{< 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); -->
\index{{\tt \bfseries wiener\_lpdf }!{\tt (real y \textbar\ real alpha, real tau, real beta, real delta, real var\_delta, real var\_beta, real var\_tau): real}|hyperpage}

`real` **`wiener_lpdf`**`(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `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.
{{< since 2.35 >}}

<!-- real; wiener_lupdf; (reals y | reals alpha, reals tau, reals beta, reals delta); -->
\index{{\tt \bfseries wiener\_lupdf }!{\tt (reals y \textbar\ reals alpha, reals tau, reals beta, reals delta): real}|hyperpage}

`real` **`wiener_lupdf`**`(reals y | reals alpha, reals tau, reals beta, reals delta)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `beta`, and drift
rate `delta`, dropping constant additive terms
{{< since 2.25 >}}


<!-- real; wiener_lupdf; (real y | real alpha, real tau, real beta, real delta, real var_delta); -->
\index{{\tt \bfseries wiener\_lupdf }!{\tt (real y \textbar\ real alpha, real tau, real beta, real delta, real var\_delta): real}|hyperpage}

`real` **`wiener_lupdf`**`(real y | real alpha, real tau, real beta, real delta, real var_delta)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `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.
{{< 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); -->
\index{{\tt \bfseries wiener\_lupdf }!{\tt (real y \textbar\ real alpha, real tau, real beta, real delta, real var\_delta, real var\_beta, real var\_tau): real}|hyperpage}

`real` **`wiener_lupdf`**`(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)`<br>\newline
The log of the Wiener first passage time density of `y` given boundary
separation `alpha`, non-decision time `tau`, starting point `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.
{{< since 2.35 >}}

---

**Note:** The `lcdf` and `lccdf` functions for the `wiener` distribution are
conditional and unnormalized, meaning that the cdf does not asymptote at 1,
but rather at the probability to hit the upper [boundary](#boundaries).

Similarly, the ccdf is defined as the probability to hit the upper boundary less
the value of the cdf, as opposed to the more typical $1 - \textrm{cdf}$.


<!-- real; wiener_lcdf_unnorm; (real y, real alpha, real tau, real beta, real delta); -->
\index{{\tt \bfseries wiener\_lcdf\_unnorm }!{\tt (real y, real alpha, real tau, real beta, real delta): real}|hyperpage}

`real` **`wiener_lcdf_unnorm`**`(real y, real alpha, real tau, real beta, real delta)`<br>\newline

The log of the cumulative distribution function (cdf) of the Wiener distribution
of `y` given boundary separation `alpha`, non-decision time `tau`, starting point
`beta`, and drift rate `delta`.

{{< since 2.38 >}}

<!-- real; wiener_lccdf_unnorm; (real y, real alpha, real tau, real beta, real delta); -->
\index{{\tt \bfseries wiener\_lccdf\_unnorm }!{\tt (real y, real alpha, real tau, real beta, real delta): real}|hyperpage}

`real` **`wiener_lccdf_unnorm`**`(real y, real alpha, real tau, real beta, real delta)`<br>\newline

The log of the complementary cumulative distribution function (ccdf) of the
Wiener distribution of `y` given boundary separation `alpha`, non-decision time
`tau`, starting point `beta`, and drift rate `delta`.

{{< since 2.38 >}}

<!-- real; wiener_lcdf_unnorm; (real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau); -->
\index{{\tt \bfseries wiener\_lcdf\_unnorm }!{\tt (real y, real alpha, real tau, real beta, real delta, real var\_delta, real var\_beta, real var\_tau): real}|hyperpage}

`real` **`wiener_lcdf_unnorm`**`(real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)`<br>\newline

The log of the cumulative distribution function (cdf) of the Wiener distribution
of `y` given boundary separation `alpha`, non-decision time `tau`, starting point
`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`.

{{< since 2.38 >}}


<!-- real; wiener_lccdf_unnorm; (real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau); -->
\index{{\tt \bfseries wiener\_lccdf\_unnorm }!{\tt (real y, real alpha, real tau, real beta, real delta, real var\_delta, real var\_beta, real var\_tau): real}|hyperpage}

`real` **`wiener_lccdf_unnorm`**`(real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)`<br>\newline

The log of the complementary cumulative distribution function (ccdf) of the
Wiener distribution of `y` given boundary separation `alpha`, non-decision time
`tau`, starting point `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`.

{{< since 2.38 >}}

### 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-Wabersich:2014.

### Vectorization

The 5- and 7-argument forms of the `wiener` distribution functions (listed above
as recieving only `real`) are implemented in such a way where they can be
fully [vectorized](conventions_for_probability_functions.qmd#prng-vectorization),
but currently only versions that accept all `real` and all `vector` arguments
are exposed by Stan. If there are additional signatures that would prove useful,
please request them by [opening an issue](https://github.com/stan-dev/stanc3/issues/new).

### Tolerance tuning

The 5- and 7-argument forms of the `wiener` distribution functions can also
accept an additional `data real` argument controlling the required precision of
the gradient calculation of the function. If omitted, this defaults to `1e-4`
for the density and `1e-8` for the cdf functions.