18.1 Pareto Distribution
18.1.1 Probability Density Function
If \(y_{\text{min}} \in \mathbb{R}^+\) and \(\alpha \in \mathbb{R}^+\), then for \(y \in \mathbb{R}^+\) with \(y \geq y_{\text{min}}\), \[ \text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}}. \]
18.1.2 Sampling Statement
y ~ pareto(y_min, alpha)
Increment target log probability density with pareto_lupdf(y | y_min, alpha).
18.1.3 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.