Dataplot Vol 2 Vol 1

# KUMPDF

Name:
KUMPDF (LET)
Type:
Library Function
Purpose:
Compute the Kumaraswamy probability density function with shape parameters and .
Description:
The standard Kumaraswamy distribution has the following probability density 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> = KUMPDF(<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 pdf 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 = KUMPDF(0.3,0.2,1.2)
LET Y = KUMPDF(X,0.5,2)
PLOT KUMPDF(X,2,3) FOR X = 0 0.01 1
Note:
Kumaraswamy random numbers, probability plots, and goodness of fit tests can be generated with the commands:

LET ALPHA = <value>
LET BETA = <value>
LET A = <value>
LET B = <value>
LET Y = KUMARASWAMY RANDOM NUMBERS FOR I = 1 1 N
KUMARASWAMY PROBABILITY PLOT Y
KUMARASWAMY PROBABILITY PLOT Y2 X2
KUMARASWAMY PROBABILITY PLOT Y3 XLOW XHIGH
KUMARASWAMY KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
KUMARASWAMY CHI-SQUARE GOODNESS OF FIT Y2 X2
KUMARASWAMY CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH

The following commands can be used to estimate the alpha and beta shape parameters for the Kumaraswamy distribution:

LET ALPHA1 = <value>
LET ALPHA2 = <value>
LET BETA1 = <value>
LET BETA2 = <value>
KUMARASWAMY PPCC PLOT Y
KUMARASWAMY PPCC PLOT Y2 X2
KUMARASWAMY PPCC PLOT Y3 XLOW XHIGH
KUMARASWAMY KS PLOT Y
KUMARASWAMY KS PLOT Y2 X2
KUMARASWAMY KS PLOT Y3 XLOW XHIGH

The default values for ALPHA1 and ALPHA2 are 0.5 and 10. The default values for BETA1 and BETA2 are 0.5 and 10.

The probability plot can then be used to estimate the lower and upper limits (lower limit = PPA0, upper limit = PPA0 + PPA1).

The following options may be useful for these commands.

1. Instead of generating the PPCC plot or KS plot on the original data, we can generate them on selected percentiles of the data. For example, if we have 1,000 points, we can choose to generate the plots on 100 evenly spaced percentiles with the command

SET PPCC PLOT DATA POINTS 100

This can be used to speed up the generation of the plot for larger data sets.

Note that since the Kumaraswamy percent point function exists in closed form, this option is typically not needed.

2. For the KS plot, we can fix the location and scale. This is equivalent to assuming that the lower and upper limits are known (e.g., we could use the data minimum and maximum as the lower and upper limit values). Given that the lower and upper limits are LOWLIM and UPPLIM, enter the commands

LET KSLOC = LOWLIM
LET KSSCALE = UPPLIM

The PPCC plot is invariant to location and scale, so we cannot fix the lower and upper limits.

Default:
None
Synonyms:
None
Related Commands:
 KUMCDF = Compute the Kumaraswamy cumulative distribution 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 1:
```
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 KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 1.5
LET BETA  = 6
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 1.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 1.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 0.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 0.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 0.5
LET BETA  = 0.75
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 0.5
LET BETA  = 0.25
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
LET ALPHA = 1
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMPDF(X,ALPHA,BETA) FOR X = 0.01  0.01  0.99
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Kumaraswamy Probability Density Functions
```

Program 2:
```
let alpha = 1.2
let beta = 2.6
let alphasv = alpha
let betasv = beta
.
let y = kuaraswamy rand numb for i = 1 1 200
.
kumaraswamy ppcc plot y
justification center
move 50 6
let alpha = shape1
let beta = shape2
text Alpha = ^alpha, Beta = ^beta
move 50 3
text Alphasav = ^alphsav, Betasav = ^betasav
.
char x
line blank
kumaraswamy probability plot y
move 50 6
let lowlim = ppa0
let upplim = ppa0 + ppa1
text Lower Limit = ^lowlim, Upper Limit = ^upplim
move 50 3
text PPCC = ^ppcc
char blank
line solid
let ksscale = upplim
kumaraswamy kolm smir goodness of fit y
```

```                   KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST

NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
DISTRIBUTION:            KUMARASWAMY
NUMBER OF OBSERVATIONS              =      200

TEST:
KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3479910E-01

ALPHA LEVEL         CUTOFF              CONCLUSION
10%       0.086*              ACCEPT H0
0.085**
5%       0.096*              ACCEPT H0
0.095**
1%       0.115*              ACCEPT H0
0.114**

*  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )
```

Date created: 11/27/2007
Last updated: 11/27/2007