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_lpdf( y | y_min, alpha) dropping constant additive terms.

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_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 generated quantities block. For a description of argument and return types, see section 10.8.3.