Summary of Theoretical Distributions :: SAS/QC(R) 13.1 User's Guide

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 5.56: Distributions and Parameters

			Parameters
Family	Distribution Function $F(x)$	Range	Location	Scale	Shape
Beta	$\int _{\theta }^{x} \frac{(t-\theta )^{\alpha -1}(\theta +\sigma -t)^{\beta -1}}{B(\alpha ,\beta )\sigma ^{(\alpha +\beta -1)}} dt$	$\theta < x <\theta +\sigma$	$\theta$	$\sigma$	$\alpha$ , $\beta$
Exponential	$1-\exp \left(-\frac{x-\theta }{\sigma }\right)$	$x \geq \theta$	$\theta$	$\sigma$
Gamma	$\int _{\theta }^{x} \frac{1}{\sigma \Gamma (\alpha )} \left(\frac{t-\theta }{\sigma }\right)^{\alpha -1} \exp \left(-\frac{t-\theta }{\sigma }\right) dt$	$x>\theta$	$\theta$	$\sigma$	$\alpha$
Gumbel	$\exp \left(-e^{(x-\mu )/\sigma }\right)$	all x	$\mu$	$\sigma$
Inverse Gaussian	$\Phi \left\{ \sqrt {\frac{\lambda }{x}} \left(\frac{x}{\mu } - 1\right)\right\} +$	x > 0	$\mu$		$\lambda$
	$e^{2\lambda /\mu } \Phi \left\{ -\sqrt {\frac{\lambda }{x}} \left(\frac{x}{\mu } + 1\right)\right\}$
Lognormal	$\int _{\theta }^{x} \frac{1}{\sigma \sqrt {2\pi }(t-\theta )} \exp \left(-\frac{(\log (t-\theta )-\zeta )^{2}}{2\sigma ^{2}}\right) dt$	$x>\theta$	$\theta$	$\zeta$	$\sigma$
Normal	$\int _{-\infty }^{x} \frac{1}{\sigma \sqrt {2\pi }} \exp \left(-\frac{(t-\mu )^2}{2\sigma ^{2}}\right) dt$	all x	$\mu$	$\sigma$
Generalized Pareto	$1 - {\left(1 - \frac{\alpha (x - \theta )}{\sigma }\right)}^{1/\alpha }$	all x	$\theta$	$\sigma$	$\alpha$
Power Function	${\left( \frac{x - \theta }{\sigma } \right)}^{\alpha }$	$\theta < x <\theta +\sigma$	$\theta$	$\sigma$	$\alpha$
Rayleigh	$1 - e^{-(x - \theta )^2 / \left(2\sigma ^2\right)}$	$x \geq \theta$	$\theta$	$\sigma$
Weibull	$1-\exp \left(-\left(\frac{x-\theta }{\sigma }\right)^{c}\right)$	$x>\theta$	$\theta$	$\sigma$	c

You can request these distributions with the BETA, EXPONENTIAL, GAMMA, GUMBEL, IGAUSS, NORMAL, LOGNORMAL, PARETO, POWER, RAYLEIGH, 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 5.57: 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 $\alpha$
Gumbel	None	maximum likelihood estimates for $\mu$ and $\sigma$
Inverse Gaussian	None	sample estimate for $\mu$ , maximum likelihood estimate for $\lambda$
Lognormal	$\theta =0$	maximum likelihood estimates for $\sigma$ and $\zeta$
Normal	None	sample estimates for $\mu$ and $\sigma$
Generalized Pareto	$\theta =0$	maximum likelihood estimates for $\sigma$ and $\alpha$
Power Function	$\theta =0$ , $\sigma =1$	maximum likelihood estimate for $\alpha$
Rayleigh	$\theta =0$	maximum likelihood estimate for $\sigma$
Weibull	$\theta =0$	maximum likelihood estimates for $\sigma$ and c