Dataplot Vol 2 Vol 1

# RPOPDF

Name:
RPOPDF (LET)
Type:
Library Function
Purpose:
Compute the reflected power probability density function with shape parameter c.
Description:
The standard reflected power distribution has the following probability density function:

with c denoting the shape parameter.

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

The reflected power distribution is a special case of the beta distribution where the first shape parameter = 1 (the power distribution is a special case of the beta distribution where the second shape parameter = 1).

Syntax:
LET <y> = RPOPDF(<x>,<c>,<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 reflected power pdf value is stored;
<c> is a positive number, parameter, or variable that specifies the 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 = RPOPDF(0.3,0.2)
LET Y = RPOPDF(X,0.5,0,5)
PLOT RPOPDF(X,2.3) FOR X = 0 0.01 1
Note:
Reflected power random numbers, probability plots, and goodness of fit tests can be generated with the commands:

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

The following commands can be used to estimate the c shape parameter for the reflected power distribution:

LET C1 = <value>
LET C2 = <value>
REFLECTED POWER PPCC PLOT Y
REFLECTED POWER PPCC PLOT Y2 X2
REFLECTED POWER PPCC PLOT Y3 XLOW XHIGH
REFLECTED POWER KS PLOT Y
REFLECTED POWER KS PLOT Y2 X2
REFLECTED POWER KS PLOT Y3 XLOW XHIGH

The default values for C1 and C2 are 0.5 and 25.

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.

• 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.

The percent point function for the reflected power distribution is available in closed form, so this option is typically not needed.

• 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.

The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the ppcc plot, ks plot, and maximum likelihood estimates.

Default:
None
Synonyms:
None
Related Commands:
 RPOCDF = Compute the reflected power cumulative distribution function. RPOPPF = Compute the reflected power percent point function. RPOHAZ = Compute the reflected power hazard function. RPOCHAZ = Compute the reflected power cumulative hazard function. POWPDF = Compute the power probability density function. BETPDF = Compute the beta probability density function. TOPPDF = Compute the Topp and Leone probability density function. RGTPDF = Compute the generalized reflected Topp and Leone probability density function. GTLPDF = Compute the generalized Topp and Leone probability density function. TSPPDF = Compute the two-sided power probability density function. TRIPDF = Compute the triangular probability density function. TRAPDF = Compute the trapezoid probability density function. UNIPDF = Compute the uniform probability density function. JSBPDF = Compute the Johnson SB probability density function.
Reference:
Samuel Kotz and J. Rene Van Dorp 2004, "Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications", World Scientific, pp. 199-201.
Applications:
Distributional Modeling
Implementation Date:
2007/12
Program 1:
```
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR
.
LET C  = 0.5
TITLE C = ^c
PLOT RPOPDF(X,C) FOR X = 0  0.01  0.99
.
LET C  = 1
TITLE C = ^c
PLOT RPOPDF(X,C) FOR X = 0  0.01  1
.
LET C  = 1.5
TITLE C = ^c
PLOT RPOPDF(X,C) FOR X = 0  0.01  1
.
LET C  = 2
TITLE C = ^c
PLOT RPOPDF(X,C) FOR X = 0  0.01  1
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Reflected Power Probability Density Functions
```

Program 2:
```
let c = 2.2
let y = reflected power rand numb for i = 1 1 200
.
let csav = c
reflected power ppcc plot y
just center
move 50 5
let c = shape
text maxppcc = ^maxppcc, C = ^c
move 50 2
text Csav = ^csav
.
char x
line blank
reflected power prob plot y
move 50 5
text PPA0 = ^ppa0, PPA1 = ^ppa1
move 50 2
let upplim = ppa0 + ppa1
text Lower Limit = ^ppa0, Upper Limit = ^upplim
char blank
line solid
.
class lower 0
class upper 1
class width 0.05
relative hist y
line color blue
limits freeze
pre-erase off
let a = ppa0
let b = a + (ppa1 - ppa0)
plot rpopdf(x,c,a,b) for x = a 0.01 b
limits
pre-erase on
line color black all
.
let ksloc = ppa0
let ksscale = upplim
reflected power 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:            REFLECTED POWER
NUMBER OF OBSERVATIONS              =      200

TEST:
KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3483635E-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: 12/17/2007
Last updated: 12/17/2007