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Dataplot Vol 2 Vol 1

ADECDF

Name:
    ADECDF (LET)
Type:
    Library Function
Purpose:
    Compute the asymmetric double exponential cumulative distribution function.
Description:
    The asymmetric double exponential distribution has the following cumulative distribution function:

      F(x,kappa) = 1 - (1/(1+kappa**2))*EXP[-SQRT(2)*kappa*ABS(x)] x >= 0; = (kappa**2/(1+kappa**2))*EXP[(-SQRT(2)/kappa)*ABS(x)] x < 0; kappa > 0

    with kappa denoting the shape parameter.

    The standard asymmetric double exponential distribution can be generalized with a location parameter, theta, and a scale parameter sigma. Simply replace x with

      (x-theta)/sigma

    in the above formula.

    If kappa = 1 and the scale parameter = SQRT(2), the asymmetric double exponential distribution reduces to the symmetric double exponential distribution.

    The asymmetric double exponential distribution is also known as the asymmetric Laplace distribution.

Syntax:
    LET <y> = ADECDF(<x>,<k>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable or a parameter;
                <k> is a positive number of parameter that specifies the value of the shape parameter;
                <loc> is an optional number or parameter that specifies the value of the location parameter;
                <scale> is an optional positive 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 asymmetric double exponential cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET Y = ADECDF(3,1)
    LET Y = ADECDF(X1,K)
    PLOT ADECDF(X,K) FOR X = -5 0.01 5
Note:
    Kotz, Kozubowski, and Podgorski define an alternate parameterization based on the parameter mu. These parameterizations are related by:

      mu = (1/SQRT(2))*((1/k) - k)

      k = SQRT(2)/(mu + SQRT(2 + mu**2))

    The mu parameter can be any real number. A value of mu = 0 reduces to the symmetric double exponential (i.e., kappa = 1).

    By default, Dataplot uses the kappa parameterization. To use the mu parameterization, enter the command

      SET ASYMMETRIC DOUBLE EXPONENTIAL DEFINITION MU

    To reset the kappa parameterization, enter the command

      SET ASYMMETRIC DOUBLE EXPONENTIAL DEFINITION K
Default:
    None
Synonyms:
    None
Related Commands:
    ADEPDF = Compute the asymmetric double exponential probability density function.
    ADEPPF = Compute the asymmetric double exponential percent point function.
    SDECDF = Compute the skew double exponential probability density function.
    DEXCDF = Compute the double exponential probability density function.
    EXPCDF = Compute the exponential probability density function.
    SNCDF = Compute the skew normal probability density function.
    STCDF = Compute the skew t probability density function.
Reference:
    "The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance", Birkhauser, 2001, pp. 133-174.

    "A Class of Distributions Which Includes the Normal Ones", Azzalini, Scandinavian Journal of Statistics, 12, 171-178.

Applications:
    Distributional Modeling
Implementation Date:
    6/2004
Program: 
     
Y1LABEL Probability
X1LABEL X
LABEL CASE ASIS
TITLE CASE ASIS
CASE ASIS
Y1LABEL DISPLACEMENT 12
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
TITLE Kappa = 0.5
PLOT ADECDF(X,0.5) FOR X = -5 0.1 5
TITLE Kappa = 1
PLOT ADECDF(X,1) FOR X = -5 0.1 5
TITLE Kappa = 2
PLOT ADECDF(X,2) FOR X = -5 0.1 5
TITLE Kappa = 5
PLOT ADECDF(X,5) FOR X = -5 0.1 5
END OF MULTIPLOT
MOVE 50 97
JUSTIFICATIONC CENTER
TEXT Asymmetric Double Exponential Distribution
    plot generated by sample program

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