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Base SAS Procedures Guide: Statistical Procedures
 

The UNIVARIATE Procedure

CDFPLOT Statement

  • CDFPLOT <variables> </ options>;

The CDFPLOT statement plots the observed cumulative distribution function (CDF) of a variable The CDF is defined as

StartLayout 1st Row 1st Column upper F Subscript upper N Baseline left-parenthesis x right-parenthesis 2nd Column equals 3rd Column percent of nonmissing values less-than-or-equal-to x 2nd Row 1st Column Blank 2nd Column equals 3rd Column StartFraction number of values less-than-or-equal-to x Over upper N EndFraction times 100 percent-sign EndLayout

where N is the number of nonmissing observations. The CDF is an increasing step function that has a vertical jump of StartFraction 1 Over upper N EndFraction at each value of x equal to an observed value. The CDF is also referred to as the empirical cumulative distribution function (ECDF).

You can use any number of CDFPLOT statements in the UNIVARIATE procedure. The components of the CDFPLOT statement are as follows:

variables

specify variables for which to create CDF plots. If you specify a VAR statement, the variables must also be listed in the VAR statement. Otherwise, the variables can be any numeric variables in the input data set. If you do not specify any variables, then by default the procedure creates a CDF plot for each variable listed in the VAR statement, or for each numeric variable in the DATA= data set if you do not specify a VAR statement.

For example, suppose a data set named Steel contains exactly three numeric variables: Length, Width, and Height. The following statements create a CDF plot for each of the three variables: 

proc univariate data=Steel;
   cdfplot;
run;

The following statements create a CDF plot for Length and a CDF plot for Width: 

proc univariate data=Steel;
   var Length Width;
   cdfplot;
run;

The following statements create a CDF plot for Width: 

proc univariate data=Steel;
   var Length Width;
   cdfplot Width;
run;
options

specify the theoretical distribution for the plot or add features to the plot. If you specify more than one variable, the options apply equally to each variable. Specify all options after the slash (/) in the CDFPLOT statement. In each CDFPLOT statement, you can specify only one option that names a distribution, but you can specify any number of other options. The distributions available are listed in Table 1. By default, the procedure produces a plot for the normal distribution.

Table 1 through Table 3 list the CDFPLOT options by function. For complete descriptions, see the sections Dictionary of Options and Dictionary of Common Options. The options can be any of the following:

  • primary options

  • secondary options

  • general options

Distribution Options

Table 1 lists primary options for requesting a theoretical distribution.

Table 1: Primary Options for Theoretical Distribution

Option Description
BETA(beta-options) Plots two-parameter beta distribution function, parameters theta and sigma assumed known
EXPONENTIAL(exponential-options) Plots one-parameter exponential distribution function, parameter theta assumed known
GAMMA(gamma-options) Plots two-parameter gamma distribution function, parameter theta assumed known
GUMBEL(Gumbel-options) Plots Gumbel distribution with location parameter mu and scale parameter sigma
IGAUSS(iGauss-options) Plots inverse Gaussian distribution with mean mu and shape parameter lamda
LOGNORMAL(lognormal-options) Plots two-parameter lognormal distribution function, parameter theta assumed known
NORMAL(normal-options) Plots normal distribution function
PARETO(Pareto-options) Plots generalized Pareto distribution with threshold parameter theta, scale parameter sigma, and shape parameter alpha
POWER(power-options) Plots power function distribution with threshold parameter theta, scale parameter sigma, and shape parameter alpha
RAYLEIGH(Rayleigh-options) Plots Rayleigh distribution with threshold parameter theta and scale parameter sigma
WEIBULL(Weibull-options) Plots two-parameter Weibull distribution function, parameter theta assumed known


Table 2 lists secondary options that specify distribution parameters and control the display of a theoretical distribution function. Specify these options in parentheses after the distribution keyword. For example, you can request a normal probability plot with a distribution reference line by specifying the NORMAL option as follows: 

proc univariate;
   cdfplot / normal(mu=10 sigma=0.5);
run;

The MU= and SIGMA= options specify the parameters mu equals 10 and sigma equals 0.5 for the distribution function. If you do not specify these parameters, maximum likelihood estimates are computed.

Table 2: Secondary Distribution Options

Option Description
Traditional Graphics Options Used with All Distributions
COLOR= Specifies color of theoretical distribution function
L= Specifies line type of theoretical distribution function
W= Specifies width of theoretical distribution function
Beta-Options
ALPHA= Specifies first shape parameter alpha for beta distribution function
BETA= Specifies second shape parameter beta for beta distribution function
SIGMA= Specifies scale parameter sigma for beta distribution function
THETA= Specifies lower threshold parameter theta for beta distribution function
Exponential-Options
SIGMA= Specifies scale parameter sigma for exponential distribution function
THETA= Specifies threshold parameter theta for exponential distribution function
Gamma-Options
ALPHA= Specifies shape parameter alpha for gamma distribution function
ALPHADELTA= Specifies change in successive estimates of alpha at which the Newton-Raphson approximation of ModifyingAbove alpha With caret terminates
ALPHAINITIAL= Specifies initial value for alpha in the Newton-Raphson approximation of ModifyingAbove alpha With caret
MAXITER= Specifies maximum number of iterations in the Newton-Raphson approximation of ModifyingAbove alpha With caret
SIGMA= Specifies scale parameter sigma for gamma distribution function
THETA= Specifies threshold parameter theta for gamma distribution function
Gumbel-Options
MU= Specifies location parameter mu for Gumbel distribution function
SIGMA= Specifies scale parameter sigma for Gumbel distribution function
IGauss-Options
LAMBDA= Specifies shape parameter lamda for inverse Gaussian distribution function
MU= Specifies mean mu for inverse Gaussian distribution function
Lognormal-Options
SIGMA= Specifies shape parameter sigma for lognormal distribution function
THETA= Specifies threshold parameter theta for lognormal distribution function
ZETA= Specifies scale parameter zeta for lognormal distribution function
Normal-Options
MU= Specifies mean mu for normal distribution function
SIGMA= Specifies standard deviation sigma for normal distribution function
Pareto-Options
ALPHA= Specifies shape parameter alpha for generalized Pareto distribution function
SIGMA= Specifies scale parameter sigma for generalized Pareto distribution function
THETA= Specifies threshold parameter theta for generalized Pareto distribution function
Power-Options
ALPHA= Specifies shape parameter alpha for power function distribution
SIGMA= Specifies scale parameter sigma for power function distribution
THETA= Specifies threshold parameter theta for power function distribution
Rayleigh-Options
SIGMA= Specifies scale parameter sigma for Rayleigh distribution function
THETA= Specifies threshold parameter theta for Rayleigh distribution function
Secondary Weibull-Options
C= Specifies shape parameter c for Weibull distribution function
ITPRINT Requests table of iteration history and optimizer details
MAXITER= Specifies maximum number of iterations in the Newton-Raphson approximation of ModifyingAbove c With caret
SIGMA= Specifies scale parameter sigma for Weibull distribution function
THETA= Specifies threshold parameter theta for Weibull distribution function


General Options

Table 3 summarizes general options for enhancing CDF plots.

Table 3: General CDFPLOT Statement Options

Option Description
General Graphics Options
HREF= Specifies reference lines perpendicular to the horizontal axis
HREFLABELS= Specifies labels for HREF= lines
HREFLABPOS= Specifies position for HREF= line labels
NOECDF Suppresses plot of empirical (observed) distribution function
NOHLABEL Suppresses label for horizontal axis
NOVLABEL Suppresses label for vertical axis
NOVTICK Suppresses tick marks and tick mark labels for vertical axis
STATREF= Specifies reference lines at values of summary statistics
STATREFLABELS= Specifies labels for STATREF= lines
STATREFSUBCHAR= Specifies substitution character for displaying statistic values in STATREFLABELS= labels
VAXISLABEL= Specifies label for vertical axis
VREF= Specifies reference lines perpendicular to the vertical axis
VREFLABELS= Specifies labels for VREF= lines
VREFLABPOS= Specifies position for VREF= line labels
VSCALE= Specifies scale for vertical axis
Options for Traditional Graphics Output
ANNOTATE= Specifies annotate data set
CAXIS= Specifies color for axis
CFRAME= Specifies color for frame
CHREF= Specifies colors of references lines requested using the HREF= option
CSTATREF= Specifies colors of references lines requested using the STATREF= option
CTEXT= Specifies color for text
CVREF= Specifies colors of references lines requested using the VREF= option
DESCRIPTION= Specifies description for graphics catalog member
FONT= Specifies text font
HAXIS= Specifies AXIS statement for horizontal axis
HEIGHT= Specifies height of text used outside framed areas
HMINOR= Specifies number of horizontal axis minor tick marks
INFONT= Specifies software font for text inside framed areas
INHEIGHT= Specifies height of text inside framed areas
LHREF= Specifies types of reference lines requested using the HREF= option
LSTATREF= Specifies types of reference lines requested using the STATREF= option
LVREF= Specifies types of reference lines requested using the VREF= option
NAME= Specifies name of plot in graphics catalog
NOFRAME Suppresses frame around plotting area
TURNVLABELS Turns and vertically strings out characters in labels for vertical axis
VAXIS= Specifies AXIS statement for vertical axis
VMINOR= Specifies number of vertical axis minor tick marks
WAXIS= Specifies line thickness for axes and frame
WHREF= Specifies thicknesses of reference lines requested using the HREF= option
WSTATREF= Specifies thicknesses of reference lines requested using the STATREF= option
WVREF= Specifies thicknesses of reference lines requested using the VREF= option
Options for ODS Graphics Output
NOCDFLEGEND Suppresses legend for superimposed theoretical CDF
ODSFOOTNOTE= Specifies footnote displayed on plot
ODSFOOTNOTE2= Specifies secondary footnote displayed on plot
ODSTITLE= Specifies title displayed on plot
ODSTITLE2= Specifies secondary title displayed on plot
OVERLAY Overlays plots for different class levels
Options for Comparative Plots
ANNOKEY Applies annotation requested in ANNOTATE= data set to key cell only
CFRAMESIDE= Specifies color for filling row label frames
CFRAMETOP= Specifies color for filling column label frames
CPROP= Specifies color for proportion of frequency bar
CTEXTSIDE= Specifies color for row labels
CTEXTTOP= Specifies color for column labels
INTERTILE= Specifies distance between tiles in comparative plot
NCOLS= Specifies number of columns in comparative plot
NROWS= Specifies number of rows in comparative plot
Miscellaneous Options
CONTENTS= Specifies table of contents entry for CDF plot grouping


Dictionary of Options

The following entries provide detailed descriptions of the options specific to the CDFPLOT statement. For detailed descriptions of options common to all plot statements, see the section Dictionary of Common Options.

ALPHA=value

specifies the shape parameter alpha for distribution functions that are requested by 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 BETA and GAMMA options.

BETA<(beta-options )>

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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column upper I Subscript StartFraction x minus theta Over sigma EndFraction Baseline left-parenthesis alpha comma beta right-parenthesis 2nd Column for theta less-than x less-than theta plus sigma 3rd Row 1st Column 1 2nd Column for x greater-than-or-equal-to sigma plus theta EndLayout

where upper I Subscript y Baseline left-parenthesis alpha comma beta right-parenthesis is the incomplete beta function and

  • theta equals lower threshold parameter (lower endpoint)

  • sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

  • alpha equals shape parameter left-parenthesis alpha greater-than 0 right-parenthesis

  • beta equals shape parameter left-parenthesis beta greater-than 0 right-parenthesis

The beta distribution is bounded below by the parameter theta and above by the value theta plus 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 univariate;
   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 in 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. Table 2 lists options you can specify with the BETA distribution option.

BETA=value
B=value

specifies the second shape parameter beta for beta distribution functions that are 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
SHAPE=value

specifies the shape parameter c for Weibull distribution functions that are requested by the WEIBULL option. Enclose the C= option in parentheses after the WEIBULL keyword. If you do not specify this option, the procedure calculates a maximum likelihood estimate.

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

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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column 1 minus exp left-parenthesis minus StartFraction x minus theta Over sigma EndFraction right-parenthesis 2nd Column for x greater-than theta EndLayout

where

  • theta equals threshold parameter

  • sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

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 equals 10 and a maximum likelihood estimate for sigma: 

proc univariate;
   cdfplot / exponential(theta=10);
run;

The EXPONENTIAL option can appear only once in a CDFPLOT statement. Table 2 lists the 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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column StartFraction 1 Over normal upper Gamma left-parenthesis alpha right-parenthesis sigma EndFraction integral Subscript theta Superscript x Baseline left-parenthesis StartFraction t minus theta Over sigma EndFraction right-parenthesis Superscript alpha minus 1 Baseline exp left-parenthesis minus StartFraction t minus theta Over sigma EndFraction right-parenthesis d t 2nd Column for x greater-than theta EndLayout

where

  • theta equals threshold parameter

  • sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

  • alpha equals shape parameter left-parenthesis alpha greater-than 0 right-parenthesis

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 equals 4 and maximum likelihood estimates for alpha and sigma: 

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

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. Table 2 lists the 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

upper F left-parenthesis x right-parenthesis equals exp left-parenthesis minus e Superscript minus left-parenthesis x minus mu right-parenthesis slash sigma Baseline right-parenthesis

where mu equals location parameter sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

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. Table 2 lists 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

upper F left-parenthesis x right-parenthesis equals normal upper Phi left-brace StartRoot StartFraction lamda Over x EndFraction EndRoot left-parenthesis StartFraction x Over mu EndFraction minus 1 right-parenthesis right-brace plus e Superscript 2 lamda slash mu Baseline normal upper Phi left-brace minus StartRoot StartFraction lamda Over x EndFraction EndRoot left-parenthesis StartFraction x Over mu EndFraction plus 1 right-parenthesis right-brace

where normal upper Phi left-parenthesis dot right-parenthesis is the standard normal cumulative distribution function, and mu equals mean parameter left-parenthesis mu greater-than 0 right-parenthesis lamda equals shape parameter left-parenthesis lamda greater-than 0 right-parenthesis

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

The IGAUSS option can appear only once in a CDFPLOT statement. Table 2 lists secondary options you can specify with the IGAUSS option.

LAMBDA=value

specifies the shape parameter lamda for distribution functions that are requested by the IGAUSS option. Enclose the LAMBDA= option in parentheses after the IGAUSS distribution keyword. If you do not specify a value for lamda, the procedure calculates a maximum likelihood estimate.

LOGNORMAL<(lognormal-options)>

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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column normal upper Phi left-parenthesis StartFraction log left-parenthesis x minus theta right-parenthesis minus zeta Over sigma EndFraction right-parenthesis 2nd Column for x greater-than theta EndLayout

where normal upper Phi left-parenthesis dot right-parenthesis is the standard normal cumulative distribution function and

  • theta equals threshold parameter

  • zeta equals scale parameter

  • sigma equals shape parameter left-parenthesis sigma greater-than 0 right-parenthesis

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, estimates of sigma and zeta are computed as described in the section Lognormal Distribution.

For example, the following statements fit a lognormal distribution function with theta equals 10 and estimates for sigma and zeta: 

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

The LOGNORMAL option can appear only once in a CDFPLOT statement.

MU=value

specifies the parameter mu for theoretical cumulative distribution functions that are requested by the GUMBEL, IGAUSS, and NORMAL option. Enclose the MU= option in parentheses after the distribution keyword. For the inverse Gaussian and normal distributions, the default value is the sample mean. If you do not specify a value for mu for the Gumbel distribution, the procedure calculates a maximum likelihood estimate. For an example, see the entry for the NORMAL option.

NOCDFLEGEND
NOLEGEND

suppresses the legend for the superimposed theoretical cumulative distribution function. The NOCDFLEGEND option applies only to ODS Graphics output.

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).

NORMAL<(normal-options)>

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

upper F left-parenthesis x right-parenthesis equals StartLayout 1st Row 1st Column normal upper Phi left-parenthesis StartFraction x minus mu Over sigma EndFraction right-parenthesis 2nd Column for negative normal infinity less-than x less-than normal infinity EndLayout

where normal upper Phi left-parenthesis dot right-parenthesis is the standard normal cumulative distribution function and

  • mu equals mean

  • sigma equals standard deviation left-parenthesis sigma greater-than 0 right-parenthesis

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

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

By default, the sample mean and sample standard deviation are calculated for mu and sigma, respectively. The NORMAL option can appear only once in a CDFPLOT statement. Table 2 lists options that you can specify with the NORMAL option.

PARETO<(Pareto-options)>

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

upper F left-parenthesis x right-parenthesis equals 1 minus left-parenthesis 1 minus StartFraction alpha left-parenthesis x minus theta right-parenthesis Over sigma EndFraction right-parenthesis Superscript StartFraction 1 Over alpha EndFraction

where theta equals threshold parameter sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis alpha equals shape parameter

The parameter theta for the generalized Pareto distribution must be less than the minimum data value. You can specify theta in 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. Table 2 lists options that 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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column left-parenthesis StartFraction x minus theta Over sigma EndFraction right-parenthesis Superscript alpha Baseline 2nd Column for theta less-than x less-than theta plus sigma 3rd Row 1st Column 1 2nd Column for x greater-than-or-equal-to theta plus sigma EndLayout

where theta equals lower threshold parameter (lower endpoint) sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis alpha equals shape parameter left-parenthesis alpha greater-than 0 right-parenthesis

The power function distribution is bounded below by the parameter theta and above by the value theta plus 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 equals 1.

The POWER option can appear only once in a CDFPLOT statement. Table 2 lists options that 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

upper F left-parenthesis x right-parenthesis equals 1 minus e Superscript minus left-parenthesis x minus theta right-parenthesis squared slash left-parenthesis 2 sigma squared right-parenthesis

where theta equals threshold parameter sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

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. Table 2 lists options that you can specify with the RAYLEIGH option.

SIGMA=value |EST

specifies the parameter sigma for the distribution functions whose display is requested by one of the distributions in Table 4. Enclose the SIGMA= option in parentheses after the distribution keyword.

Table 4: Distributions for SIGMA= Option

Distribution Option SIGMA= Specifies Default Value Alias
BETA Scale parameter sigma 1 SCALE=
EXPONENTIAL Scale parameter sigma Maximum likelihood estimate SCALE=
GAMMA Scale parameter sigma Maximum likelihood estimate SCALE=
GUMBEL Scale parameter sigma Maximum likelihood estimate
LOGNORMAL Shape parameter sigma Estimate calculated as described in the section Lognormal Distribution SHAPE=
NORMAL Scale parameter sigma standard deviation
PARETO Scale parameter sigma Maximum likelihood estimate
POWER Scale parameter sigma 1
RAYLEIGH Scale parameter sigma Maximum likelihood estimate
WEIBULL Scale parameter sigma Maximum likelihood estimate SCALE=


THETA=value |EST
THRESHOLD=value |EST

specifies the lower threshold parameter theta for theoretical cumulative distribution functions that are requested by 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. You can specify the following values:

PERCENT

scales the data in units of percent of observations per data unit.

PROPORTION

scales the data in units of proportion of observations per data unit.

By default, VSCALE=PERCENT.

WEIBULL<(Weibull-options)>

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

upper F left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to theta 2nd Row 1st Column 1 minus exp left-parenthesis minus left-parenthesis StartFraction x minus theta Over sigma EndFraction right-parenthesis Superscript c Baseline right-parenthesis 2nd Column for x greater-than theta EndLayout

where

  • theta equals threshold parameter

  • sigma equals scale parameter left-parenthesis sigma greater-than 0 right-parenthesis

  • c equals shape parameter left-parenthesis c greater-than 0 right-parenthesis

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 equals 15 and maximum likelihood estimates for sigma and c: 

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

The WEIBULL option can appear only once in a CDFPLOT statement. Table 2 lists options that you can specify with the WEIBULL option.

ZETA=value
SCALE=value

specifies a value for the scale parameter zeta for a lognormal distribution function that is requested by 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.

Last updated: April 10, 2023