Sampling from Gamma Distribution

The srgamma_custom() function generates random samples from a Gamma distribution using the STORS algorithm. It employs an optimized proposal distribution around the mode and Adaptive Rejection Sampling (ARS) for the tails.

Usage

srgamma_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 Gamma distribution. The shape and rate parameters are specified during the optimization process using srgamma_optimize().

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

Details

The Gamma Distribution

The Gamma distribution has the probability density function (PDF): $$f(x | \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} \exp(-\beta x), \quad x \geq 0,$$ where:

$\alpha$: is the shape parameter ($\alpha > 0$), which determines the shape of the distribution.
$\beta$: is the rate parameter ($\beta > 0$), which determines the rate of decay.

The Gamma distribution is widely used in statistics, particularly in Bayesian inference and modelling waiting times.

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

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

Note

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

Examples