ERRCDF

ERRCDF (LET)

Library Function

Compute the standard error cumulative distribution 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).

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 x > 0, Dataplot computes the error cdf function with the following formula:

with denoting the gamma function (HELP GAMMA for details), denoting the incomplete gamma function (HELP GAMMAI for details), and denoting the shape parameter. Negative values of x are computed by using the above formula for positive x and taking advantage of the symmetry of the error distribution.

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.

LET <y> = ERRCDF(<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 cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Note that the location and scale parameters are optional.

LET A = ERRCDF(3,2)
LET X2 = ERRCDF(X1,A)
PLOT ERRCDF(X,1.5) FOR X = -3 0.01 3

None

None

ERRPDF	= Compute the error probability density 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.

"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.

Distributional Modeling

2003/5

 
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 ERRCDF(X,1) FOR X = -5  0.01  5
PLOT ERRCDF(X,1.5) FOR X = -5  0.01  5
PLOT ERRCDF(X,2) FOR X = -5  0.01  5
PLOT ERRCDF(X,4) FOR X = -5  0.01  5
END OF MULTIPLOT