Pearson Type VI :: SAS(R) Simulation Studio 12.3: User's Guide

Pearson Type VI

The density function of the Pearson Type VI distribution is

$f(x) = \frac{\left(\frac{x}{\beta }\right)^{\alpha _1 - 1}}{\beta G(\alpha _1, \alpha _2)\left[1 + \left(\frac{x}{\beta }\right)\right]^{\alpha _1 + \alpha _2}}$

where and

$G(\alpha _1, \alpha _2) = \frac{\Gamma (\alpha _1)\Gamma (\alpha _2)}{\Gamma (\alpha _1 + \alpha _2)}$

The function $\Gamma (z)$ is defined in the section Beta.

Parameters:

$\alpha _1$	is a shape parameter, $\alpha _1 > 0$ .
$\alpha _2$	is a shape parameter, $\alpha _2 > 0$ .
$\beta$	is a scale parameter, $\beta > 0$ .

If and are independent random variables with $X_1 \sim \mbox{Gamma}(\alpha _1, \beta )$ and $X_2 \sim \mbox{Gamma}(\alpha _2, 1)$ , then $Y = \frac{X_1}{X_2} \sim \mbox{PearsonTypeVI}(\alpha _1, \alpha _2, \beta )$ .

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