generateInverseGamma(alpha, beta)

Overview

The generateInverseGamma(alpha, beta) function generates a random number that follows an Inverse Gamma distribution using the Marsaglia and Tsang method for Gamma distribution and then taking the reciprocal.

Parameters

Returns

Example

local x = StatBook.generateInverseGamma(2, 1)
print(x)  -- Output will vary

Mathematical Background

The function uses the Marsaglia and Tsang method to generate a random number $x$ that follows a Gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$. It then takes the reciprocal of this number to generate a value from an Inverse Gamma distribution.

For $\alpha < 1$, the function uses a recursive approach to find $x$ for the Gamma distribution:

where $u$ is a uniformly distributed random number between 0 and 1.

For $\alpha \geq 1$, $x$ for the Gamma distribution is calculated using the formula:

where $d = \alpha - \frac{1}{3}$, $v = (1 + c \times Z)^{3}$, and $c = \frac{1}{\sqrt{9 \times d}}$.

The reciprocal of $x$ from the Gamma distribution is then taken to generate the Inverse Gamma distributed number.