Summary of Theoretical Distributions

PPPLOT Statement

Summary of Theoretical Distributions

You can use the PPPLOT statement to request P-P plots based on the theoretical distributions summarized in the following table:

Table 8.12: Distributions and Parameters

			Parameters
Family	Distribution Function F(x)	Range	Location	Scale	Shape

Beta	$\int_{\theta}^x \frac{(t-\theta )^{\alpha-1}(\theta+\sigma-t)^{\beta-1}} {B(\alpha,\beta)\sigma^{(\alpha+\beta-1)}} dt$	$\theta\lt x \lt\theta+\sigma$	$\theta$	$\sigma$	$\alpha$ , $\beta$
Exponential	$1-\exp(-\frac{x-\theta}{\sigma})$	$x \geq \theta$	$\theta$	$\sigma$
Gamma	$\int_{\theta}^x \frac{1}{\sigma\Gamma(\alpha)} (\frac{t-\theta}{\sigma})^{\alpha-1} \exp(-\frac{t-\theta}{\sigma}) dt$	$x\gt\theta$	$\theta$	$\sigma$	$\alpha$
Lognormal	$\int_{\theta}^x \frac{1}{\sigma\sqrt{2\pi}(t-\theta)} \exp(-\frac{(\log(t-\theta)-\zeta)^2}{2\sigma^2}) dt$	$x\gt\theta$	$\theta$	$\zeta$	$\sigma$
Normal	$\int_{-\infty}^x \frac{1}{\sigma\sqrt{2\pi}} \exp(-\frac{(t-\mu)^2}{2\sigma^2}) dt$	all x	$\mu$	$\sigma$
Weibull	$1-\exp(-(\frac{x-\theta}{\sigma})^c)$	$x\gt\theta$	$\theta$	$\sigma$	c

You can request these distributions with the BETA, EXPONENTIAL, GAMMA, LOGNORMAL, NORMAL, and WEIBULL options, respectively. If you do not specify a distribution option, a normal P-P plot is created.

To create a P-P plot, you must provide all of the parameters for the theoretical distribution. If you do not specify parameters, then default values or estimates are substituted, as summarized by the following table:

Table 8.13: Defaults for Parameters

Family	Default Values	Estimated Values
Beta	$\theta=0$ , $\sigma=1$	maximum likelihood estimates for $\alpha$ and $\beta$
Exponential	$\theta=0$	maximum likelihood estimate for $\sigma$
Gamma	$\theta=0$	maximum likelihood estimates for $\sigma$ and $\alpha$
Lognormal	$\theta=0$	maximum likelihood estimates for $\sigma$ and $\zeta$
Normal	None	sample estimates for $\mu$ and $\sigma$
Weibull	$\theta=0$	maximum likelihood estimates for $\sigma$ and c

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