Dataplot Vol 2 Vol 1

# ERRPDF

Name:
ERRPDF (LET)
Type:
Library Function
Purpose:
Compute the standard error probability density function. This is also referred to as the exponential power distribution, the Subbotin distribution, or the general error distribution.

Dataplot supports another distribution that is also called the exponential power distribution. This is a different distribution than the one described here (enter HELP PEXPDF for details).

Description:
Note that there are several different parameterizations of the error distribution in the literature. We will use the parameterization of Radikamalla (see the Reference section below).

The standard error distribution has the following probability density function:

with and denoting the gamma function (HELP GAMMA for details) and the shape parameter, respectively.

For = 1 and = 2, the error distribution is equivalent to the double exponential and normal distribution, respectively. As goes to infinity, the error distribution approximates a uniform distribution. For applications, values of between 1 and 6 are typically of most interest. The error distribution is sometimes used in simulation studies because it provides a symmetric distribution with widely varying kurtosis.

The error distribution can be generalized with location and scale parameters in the usual way.

Syntax:
LET <y> = ERRPDF(<x>,<alpha>,<loc>,<scale>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a variable, a number, or a parameter;
<alpha> is a number or parameter specifying the shape parameter;
<loc> is a number or parameter that specifies the value of the location parameter;
<scale> is a number or parameter that specifies the value of the scale parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed error pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Note that the location and scale parameters are optional.

Examples:
LET A = ERRPDF(3,2)
LET X2 = ERRPDF(X1,A)
PLOT ERRPDF(X,1.5) FOR X = -3 0.01 3
Note:
You can generate random numbers, probability plots, Kolmogorov-Smirnov and chi-square goodness of fit tests for an error distribution with the following commands:

LET ALPHA = <value>
LET Y = ERROR RANDOM NUMBERS FOR I = 1 1 N
ERROR PROBABILITY PLOT Y
ERROR KOLMOGOROV-SMIRNOV GOODNESS OF FIT TEST Y
ERROR CHI-SQUARE GOODNESS OF FIT TEST Y

You can estimate the parameters of an error distribution by generating a ppcc or Kolmogorov-Smirnov plot:

LET ALPHA1 = <value>
LET ALPHA2 = <value>
ERROR PPCC PLOT Y
ERROR KS PLOT Y

The default values of ALPHA1 and ALPHA2 are 1 and 5, respectively.

Default:
None
Synonyms:
None
Related Commands:
 ERRCDF = Compute the error cumulative distribution function. ERRPPF = Compute the error percent point function. NORPDF = Compute the normal probability density function. DEXPDF = Compute the double exponential probability density function. RANDOM NUMBERS = Generate random numbers from 90+ univariate distributions. PROBABILITY PLOT = Generate a probability plot. PPCC PLOT = Generate a ppcc plot. KS PLOT = Generate a Kolmogorov-Smirnov plot.
Reference:
"Random Sampling From the Exponential Power Distribution", Pandu R. Radikamalla, Journal of the American Statistical Association, September, 1980, pp. 683-686.

"Statistical Distributions", Third Edition, Evans, Hastings, and Peacock, Wiley, 2000.

"Continuous Univariate Distributions--Volume 2, Second Edition", Johnson, Kotz, and Balakrishnan, Wiley, 1994.

Applications:
Distributional Modeling
Implementation Date:
2003/5
Program:
```
Y1LABEL Probability
X1LABEL X
LABEL CASE ASIS
X1LABEL DISPLACEMENT 12
Y1LABEL DISPLACEMENT 12
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 100
MULTIPLOT SCALE FACTOR 2
TITLE AUTOMATIC
PLOT ERRPDF(X,1) FOR X = -5  0.01  5
PLOT ERRPDF(X,1.5) FOR X = -5  0.01  5
PLOT ERRPDF(X,2) FOR X = -5  0.01  5
PLOT ERRPDF(X,4) FOR X = -5  0.01  5
END OF MULTIPLOT
```

Date created: 7/7/2004
Last updated: 7/7/2004