Dataplot Vol 2 Vol 1

# KUMCDF

Name:
KUMCDF (LET)
Type:
Library Function
Purpose:
Compute the Kumaraswamy cumulative distribution function with shape parameters and .
Description:
The standard Kumaraswamy distribution has the following cumulative distribution function:

with and denoting the shape parameters.

This distribution can be extended with lower and upper bound parameters. If a and b denote the lower and upper bounds, respectively, then the location and scale parameters are:

location = a
scale = b - a

The general form of the distribution can then be found by using the relation

This distribution has been proposed as a more tractable alternative to the beta distribution.

Syntax:
LET <y> = KUMCDF(<x>,<alpha>,<beta>,<a>,<b>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable containing values in the interval (a,b);
<y> is a variable or a parameter (depending on what <x> is) where the computed Kumaraswamy cdf value is stored;
<alpha> is a positive number, parameter, or variable that specifies the first shape parameter;
<beta> is a positive number, parameter, or variable that specifies the second shape parameter;
<a> is a number, parameter, or variable that specifies the lower limit;
<b> is a number, parameter, or variable that specifies the upper limit;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If <a> and <b> are omitted, they default to 0 and 1, respectively.

Examples:
LET A = KUMCDF(0.3,0.2,1.2)
LET Y = KUMCDF(X,0.5,2)
PLOT KUMCDF(X,2,3) FOR X = 0 0.01 1
Default:
None
Synonyms:
None
Related Commands:
 KUMPDF = Compute the Kumaraswamy probability density function. KUMPPF = Compute the Kumaraswamy percent point function. RGTPDF = Compute the reflected generalized Topp and Leone probability density function. GTLPDF = Compute the generalized Topp and Leone probability density function. TOPPDF = Compute the Topp and Leone probability density function. TSPPDF = Compute the two-sided power probability density function. BETPDF = Compute the beta probability density function. TRIPDF = Compute the triangular probability density function. TRAPDF = Compute the trapezoid probability density function. UNIPDF = Compute the uniform probability density function. POWPDF = Compute the power probability density function. JSBPDF = Compute the Johnson SB probability density function.
Reference:
Kumaraswamy (1980), "A Generalized Probability Density Function for Double-Bounded Random Processes", Journal of Hydrology, 46: 79-88.
Applications:
Distributional Modeling
Implementation Date:
2007/11
Program:
```
CASE ASIS
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 3 3
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 3
.
LET ALPHA = 2
LET BETA  = 3
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 6
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 0.75
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 0.25
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Kumaraswamy Cumulative Distribution Functions
```

Date created: 11/27/2007
Last updated: 11/27/2007
Please email comments on this WWW page to alan.heckert@nist.gov.