Sampling from Pareto Distribution

The srpareto_custom() function generates random samples from a Pareto distribution using the STORS algorithm. It employs an optimized proposal distribution around the mode and Inverse Transform (IT) method for the tails.

Usage

srpareto_custom(n = 1, x = NULL)

Arguments

n: Integer, length 1. Number of samples to draw.
x: (optional) Numeric vector of length $n$. If provided, this vector is overwritten in place to avoid any memory allocation.

Value

A numeric vector of length n containing random samples from the Pareto distribution. The shape and scale parameters are specified during the optimization process using srpareto_optimize().

NOTE: When the x parameter is specified, it is updated in-place with the simulation for performance reasons.

Details

The Pareto Distribution

The Pareto distribution has the probability density function (PDF): $$f(x | \alpha, \sigma) = \frac{\alpha \sigma^\alpha}{x^{\alpha + 1}}, \quad x \geq \sigma,$$ where:

$\alpha$: is the shape parameter ($\alpha > 0$), which determines the tail heaviness of the distribution.
$\sigma$: is the scale parameter ($\sigma > 0$), which determines the minimum possible value of $x$.

The Pareto distribution is widely used in modelling phenomena with heavy tails, such as wealth distribution, insurance losses, and natural events.

This function samples from a proposal constructed using srpareto_optimize, employing the STORS algorithm.

By default, srpareto_custom() samples from the standard Pareto distribution with shape = 1 and rate = 1. The proposal distribution is pre-optimized at package load time using srpareto_optimize() with steps = 4091, creating a scalable proposal centred around the mode.

Note

This function is not scalable. Therefore, only the srpareto_custom() version is available, which requires the proposal to be pre-optimized using srpareto_optimize() before calling this function.

Examples