18.2 Pareto Type 2 Distribution
18.2.1 Probability Density Function
If \(\mu \in \mathbb{R}\), \(\lambda \in \mathbb{R}^+\), and \(\alpha \in \mathbb{R}^+\), then for \(y \geq \mu\), \[ \mathrm{Pareto\_Type\_2}(y|\mu,\lambda,\alpha) = \ \frac{\alpha}{\lambda} \, \left( 1+\frac{y-\mu}{\lambda} \right)^{-(\alpha+1)} \! . \]
Note that the Lomax distribution is a Pareto Type 2 distribution with \(\mu=0\).
18.2.2 Sampling Statement
y ~ pareto_type_2(mu, lambda, alpha)
Increment target log probability density with pareto_type_2_lpdf( y | mu, lambda, alpha)
dropping constant additive terms.
18.2.3 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_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.