CDFPLOT Statement: CAPABILITY Procedure

Dictionary of Options

The following entries provide detailed descriptions of the options specific to the CDFPLOT statement. The notes Traditional Graphics and Line Printer identify options that can be used only with traditional graphics and line printer plots, respectively. See Dictionary of Common Options: CAPABILITY Procedure for detailed descriptions of options common to all the plot statements.

ALPHA=value

specifies the shape parameter $\alpha$ for distribution functions requested with the BETA, GAMMA, PARETO, and POWER options. Enclose the ALPHA= option in parentheses after the distribution keyword. If you do not specify a value for $\alpha$ , the procedure calculates a maximum likelihood estimate. For examples, see the entries for the distribution options.

BETA<(beta-options)>

displays a fitted beta distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ I_{\frac{x - \theta }{\sigma }} (\alpha , \beta ) & \mbox{for $\theta < x < \theta + \sigma $} \\ 1 & \mbox{for $x \geq \sigma + \theta $} \end{array} \right.$

where $I_ y (\alpha , \beta )$ is the incomplete beta function, and

$\theta =$ lower threshold parameter (lower endpoint) $\sigma =$ scale parameter $(\sigma >0)$ $\alpha =$ shape parameter $(\alpha >0)$ $\beta =$ shape parameter $(\beta >0)$

The beta distribution is bounded below by the parameter $\theta$ and above by the value $\theta + \sigma$ . You can specify $\theta$ and $\sigma$ by using the THETA= and SIGMA= beta-options, as illustrated in the following statements, which fit a beta distribution bounded between 50 and 75. The default values for $\theta$ and $\sigma$ are 0 and 1, respectively.

proc capability;
   cdfplot / beta(theta=50 sigma=25);
run;

The beta distribution has two shape parameters, $\alpha$ and $\beta$ . If these parameters are known, you can specify their values with the ALPHA= and BETA= beta-options. If you do not specify values for $\alpha$ and $\beta$ , the procedure calculates maximum likelihood estimates.

The BETA option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the BETA distribution option.

BETA=value B=value

specifies the second shape parameter $\beta$ for beta distribution functions requested by the BETA option. Enclose the BETA= option in parentheses after the BETA keyword. If you do not specify a value for $\beta$ , the procedure calculates a maximum likelihood estimate. For examples, see the preceding entry for the BETA option.

C=value

specifies the shape parameter c for Weibull distribution functions requested with the WEIBULL option. Enclose the C= option in parentheses after the WEIBULL keyword. If you do not specify a value for c, the procedure calculates a maximum likelihood estimate. You can specify the SHAPE= option as an alias for the C= option.

CDFSYMBOL='character'

Line Printer specifies the character used to plot the points on cdf line printer plots. The default is the plus sign (+). Use the SYMBOL statement to control the plotting symbol in graphical output.

EXPONENTIAL<(exponential-options)> EXP<(exponential-options )>

displays a fitted exponential distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ 1 - \exp \left(-\frac{x - \theta }{\sigma } \right) & \mbox{for $x > \theta $ } \end{array} \right.$

where $\theta =$ threshold parameter $\sigma =$ scale parameter $(\sigma >0)$

The parameter $\theta$ must be less than or equal to the minimum data value. You can specify $\theta$ with the THETA= exponential-option. The default value for $\theta$ is 0. You can specify $\sigma$ with the SIGMA= exponential-option. By default, a maximum likelihood estimate is computed for $\sigma$ . For example, the following statements fit an exponential distribution with $\theta = 10$ and a maximum likelihood estimate for $\sigma$ :

proc capability;
   cdfplot / exponential(theta=10 l=2 color=green);
run;

The exponential curve is green and has a line type of 2.

The EXPONENTIAL option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the EXPONENTIAL option.

GAMMA<(gamma-options)>

displays a fitted gamma distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ \frac{1}{\Gamma (\alpha ) \sigma } \int _{\theta }^ x \left(\frac{t-\theta }{\sigma } \right)^{\alpha - 1} \exp \left( -\frac{t - \theta }{\sigma } \right) dt & \mbox{for $x > \theta $ } \end{array} \right.$

where $\theta =$ threshold parameter $\sigma =$ scale parameter $(\sigma >0)$ $\alpha =$ shape parameter $(\alpha >0)$

The parameter $\theta$ for the gamma distribution must be less than the minimum data value. You can specify $\theta$ with the THETA= gamma-option. The default value for $\theta$ is 0. In addition, the gamma distribution has a shape parameter $\alpha$ and a scale parameter $\sigma$ . You can specify these parameters with the ALPHA= and SIGMA= gamma-options. By default, maximum likelihood estimates are computed for $\alpha$ and $\sigma$ . For example, the following statements fit a gamma distribution function with $\theta =4$ and maximum likelihood estimates for $\alpha$ and $\sigma$ :

proc capability;
   cdfplot / gamma(theta=4);
run;

Note that the maximum likelihood estimate of $\alpha$ is calculated iteratively using the Newton-Raphson approximation. The gamma-options ALPHADELTA=, ALPHAINITIAL=, and MAXITER= control the approximation.

The GAMMA option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the GAMMA option.

GUMBEL<(Gumbel-options)>

displays a fitted Gumbel distribution (also known as Type 1 extreme value distribution) function on the cdf plot. The equation of the fitted cdf is

$F(x) = \exp \left( -e^{-(x - \mu )/\sigma } \right)$

where $\mu =$ location parameter $\sigma =$ scale parameter $(\sigma >0)$

You can specify known values for $\mu$ and $\sigma$ with the MU= and SIGMA= Gumbel-options. By default, maximum likelihood estimates are computed for $\mu$ and $\sigma$ .

The GUMBEL option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the GUMBEL option.

IGAUSS<(iGauss-options)>

displays a fitted inverse Gaussian distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \Phi \left\{ \sqrt {\frac{\lambda }{x}} \left( \frac{x}{\mu } - 1 \right) \right\} + e^{2\lambda /\mu } \Phi \left\{ -\sqrt {\frac{\lambda }{x}} \left( \frac{x}{\mu } + 1 \right) \right\}$

where $\Phi (\cdot )$ is the standard normal cumulative distribution function, and $\mu =$ mean parameter $(\mu > 0)$ $\lambda =$ shape parameter $(\lambda >0)$

You can specify known values for $\mu$ and $\lambda$ with the MU= and LAMBDA= iGauss-options. By default, maximum likelihood estimates are computed for $\mu$ and $\lambda$ .

The IGAUSS option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the IGAUSS option.

LAMBDA=value

specifies the shape parameter $\lambda$ for distribution functions requested with the IGAUSS option. Enclose the LAMBDA= option in parentheses after the IGAUSS distribution keyword. If you do not specify a value for $\lambda$ , the procedure calculates a maximum likelihood estimate.

LEGEND=name | NONE

Traditional Graphics specifies the name of a LEGEND statement describing the legend for specification limit reference lines and superimposed distribution functions. Specifying LEGEND=NONE, which suppresses all legend information, is equivalent to specifying the NOLEGEND option.

LOGNORMAL<(lognormal-options)>

displays a fitted lognormal distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ \Phi \left( \frac{\log (x-\theta )-\zeta }{\sigma } \right) & \mbox{for $x > \theta $ } \end{array} \right.$

where $\Phi (\cdot )$ is the standard normal cumulative distribution function, and $\theta =$ threshold parameter $\zeta =$ scale parameter $\sigma =$ shape parameter $(\sigma >0)$

The parameter $\theta$ for the lognormal distribution must be less than the minimum data value. You can specify $\theta$ with the THETA= lognormal-option. The default value for $\theta$ is 0. In addition, the lognormal distribution has a shape parameter $\sigma$ and a scale parameter $\zeta$ . You can specify these parameters with the SIGMA= and ZETA= lognormal-options. By default, maximum likelihood estimates are computed for $\sigma$ and $\zeta$ . For example, the following statements fit a lognormal distribution function with $\theta =10$ and maximum likelihood estimates for $\sigma$ and $\zeta$ :

proc capability;
   cdfplot / lognormal(theta = 10);
run;

The LOGNORMAL option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the LOGNORMAL option.

MU=value

specifies the parameter $\mu$ for distribution functions requested with the GUMBEL, IGAUSS, and NORMAL options. Enclose the MU= option in parentheses after the distribution keyword. For the normal and inverse Gaussian distributions, the default value of $\mu$ is the sample mean. If you do not specify a value for $\mu$ for the Gumbel distribution, the procedure calculates a maximum likelihood estimate.

NOCDFLEGEND

suppresses the legend for the superimposed theoretical cumulative distribution function.

NOECDF

suppresses the observed distribution function (the empirical cumulative distribution function) of the variable, which is drawn by default. This option enables you to create theoretical cdf plots without displaying the data distribution. The NOECDF option can be used only with a theoretical distribution (such as the NORMAL option).

NOLEGEND

suppresses legends for specification limits, theoretical distribution functions, and hidden observations. Specifying the NOLEGEND option is equivalent to specifying LEGEND=NONE.

NORMAL<(normal-options)>

displays a fitted normal distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left.\begin{array}{ll} \Phi \left( \frac{x - \mu }{\sigma } \right) & \mbox{for $-\infty < x < \infty $} \end{array} \right.$

where $\Phi (\cdot )$ is the standard normal cumulative distribution function, and $\mu =$ mean $\sigma =$ standard deviation $(\sigma >0)$

You can specify known values for $\mu$ and $\sigma$ with the MU= and SIGMA= normal-options, as shown in the following statements:

proc capability;
   cdfplot / normal(mu=14 sigma=.05);
run;

By default, the sample mean and sample standard deviation are calculated for $\mu$ and $\sigma$ . The NORMAL option can appear only once in a CDFPLOT statement. For an example, see Output 5.4.1. See Table 5.10 for a list of secondary options you can specify with the NORMAL option.

NOSPECLEGEND NOSPECL

suppresses the portion of the legend for specification limit reference lines.

PARETO<(Pareto-options)>

displays a fitted generalized Pareto distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = 1 - { \left( 1 - \frac{\alpha (x - \theta )}{\sigma } \right) }^\frac {1}{\alpha }$

where $\theta =$ threshold parameter $\sigma =$ scale parameter $(\sigma >0)$ $\alpha =$ shape parameter

The parameter $\theta$ for the generalized Pareto distribution must be less than the minimum data value. You can specify $\theta$ with the THETA= Pareto-option. The default value for $\theta$ is 0. In addition, the generalized Pareto distribution has a shape parameter $\alpha$ and a scale parameter $\sigma$ . You can specify these parameters with the ALPHA= and SIGMA= Pareto-options. By default, maximum likelihood estimates are computed for $\alpha$ and $\sigma$ .

The PARETO option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the PARETO option.

POWER<(power-options)>

displays a fitted power function distribution on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ {\left( \frac{x - \theta }{\sigma } \right)}^{\alpha } & \mbox{for $\theta < x < \theta + \sigma $} \\ 1 & \mbox{for $x \geq \theta + \sigma $} \end{array} \right.$

where $\theta =$ lower threshold parameter (lower endpoint) $\sigma =$ scale parameter $(\sigma > 0)$ $\alpha =$ shape parameter $(\alpha > 0)$

The power function distribution is bounded below by the parameter $\theta$ and above by the value $\theta + \sigma$ . You can specify $\theta$ and $\sigma$ by using the THETA= and SIGMA= power-options. The default values for $\theta$ and $\sigma$ are 0 and 1, respectively.

You can specify a value for the shape parameter, $\alpha$ , with the ALPHA= power-option. If you do not specify a value for $\alpha$ , the procedure calculates a maximum likelihood estimate.

The power function distribution is a special case of the beta distribution with its second shape parameter, $\beta = 1$ .

The POWER option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the POWER option.

RAYLEIGH<(Rayleigh-options)>

displays a fitted Rayleigh distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = 1 - e^{-(x - \theta )^2/(2\sigma ^2)}$

where $\theta =$ threshold parameter $\sigma =$ scale parameter $(\sigma >0)$

The parameter $\theta$ for the Rayleigh distribution must be less than the minimum data value. You can specify $\theta$ with the THETA= Rayleigh-option. The default value for $\theta$ is 0. You can specify $\sigma$ with the SIGMA= Rayleigh-option. By default, a maximum likelihood estimate is computed for $\sigma$ .

The RAYLEIGH option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the RAYLEIGH option.

SIGMA=value

specifies the parameter $\sigma$ for distribution functions requested by the BETA, EXPONENTIAL, GAMMA, GUMBEL, LOGNORMAL, NORMAL, PARETO, POWER, RAYLEIGH, and WEIBULL options. Enclose the SIGMA= option in parentheses after the distribution keyword. The following table summarizes the use of the SIGMA= option:

Distribution Option	SIGMA= Specifies	Default Value	Alias
BETA	scale parameter $\sigma$	1	SCALE=
POWER
EXPONENTIAL	scale parameter $\sigma$	maximum likelihood estimate	SCALE=
GAMMA
WEIBULL
GUMBEL	scale parameter $\sigma$	maximum likelihood estimate
PARETO
RAYLEIGH
LOGNORMAL	shape parameter $\sigma$	maximum likelihood estimate	SHAPE=
NORMAL	scale parameter $\sigma$	standard deviation

SYMBOL='character'

Line Printerspecifies the character used to plot the theoretical distribution function on line printer plots. Enclose the SYMBOL= option in parentheses after the distribution option. The default character is the first letter of the distribution option keyword.

THETA=value THRESHOLD=value

specifies the lower threshold parameter $\theta$ for theoretical cumulative distribution functions requested with the BETA, EXPONENTIAL, GAMMA, LOGNORMAL, PARETO, POWER, RAYLEIGH, and WEIBULL options. Enclose the THETA= option in parentheses after the distribution keyword. The default value is 0.

VSCALE=PERCENT | PROPORTION

specifies the scale of the vertical axis. The value PERCENT scales the data in units of percent of observations per data unit. The value PROPORTION scales the data in units of proportion of observations per data unit. The default is PERCENT.

WEIBULL<(Weibull-options)>

displays a fitted Weibull distribution function on the cdf plot. The equation of the fitted cdf is

$F(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $x \leq \theta $} \\ 1 - \exp \left( - \left( \frac{x - \theta }{\sigma } \right)^ c \right) & \mbox{for $x > \theta $ } \end{array} \right.$

where $\theta =$ threshold parameter $\sigma =$ scale parameter $(\sigma >0)$ $\mi {c} =$ shape parameter $(\mi {c} >0)$

The parameter $\theta$ must be less than the minimum data value. You can specify $\theta$ with the THETA= Weibull-option. The default value for $\theta$ is 0. In addition, the Weibull distribution has a shape parameter c and a scale parameter $\sigma$ . You can specify these parameters with the SIGMA= and C= Weibull-options. By default, maximum likelihood estimates are computed for c and $\sigma$ . For example, the following statements fit a Weibull distribution function with $\theta =15$ and maximum likelihood estimates for $\sigma$ and c:

proc capability;
   cdfplot / weibull(theta=15);
run;

Note that the maximum likelihood estimate of c is calculated iteratively using the Newton-Raphson approximation. The Weibull-options CDELTA=, CINITIAL=, and MAXITER= control the approximation.

The WEIBULL option can appear only once in a CDFPLOT statement. See Table 5.10 for a list of secondary options you can specify with the WEIBULL option.

ZETA=value

specifies a value for the scale parameter $\zeta$ for a lognormal distribution function requested with the LOGNORMAL option. Enclose the ZETA= option in parentheses after the LOGNORMAL keyword. If you do not specify a value for $\zeta$ , a maximum likelihood estimate is computed. You can specify the SCALE= option as an alias for the ZETA= option.