The CAPABILITY Procedure

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