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Dataplot Vol 2 Vol 1

ADEPDF

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

      f(x,kappa) = (SQRT(2)*kappa/(1+kappa**2))*EXP[-SQRT(2)*kappa*ABS(x)] x >= 0; = (SQRT(2)*kappa/(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> = ADEPDF(<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 pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET Y = ADEPDF(3,1)
    LET Y = ADEPDF(X1,K)
    PLOT ADEPDF(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
Note:
    To generate asymmetric double exponential random numbers, enter the commands

      LET K = <value> LET Y = ASYMMETRIC DOUBLE EXPONENTIAL RANDOM NUMBERS ...
        FOR I = 1 1 N

    To generate an asymmetric double exponential probability plot or an asymmetric double exponential Kolmogorov-Smirnov or chi-square goodness of fit test, enter the following commands

      LET K = <value> ASYMMETRIC DOUBLE EXPONENTIAL PROBABILITY PLOT Y
      ASYMMETRIC DOUBLE EXPONENTIAL KOLMOGOROV SMIRNOV ...
        GOODNESS OF FIT Y
      ASYMMETRIC DOUBLE EXPONENTIAL CHI-SQUARE GOODNESS OF FIT Y

    To generate a PPCC or Kolmogorov-Smirnov plot, enter the following commands

      LET K1 = <value>
      LET K2 = <value>
      ASYMMETRIC DOUBLE EXPONENTIAL PPCC PLOT Y
      ASYMMETRIC DOUBLE EXPONENTIAL KS PLOT Y

    The default values for K1 and K2 are 0.2 and 10.

    If you have requested the mu parameterization, then you you need to enter the command

      LET MU = <value>

    in place of

      LET K = <value>
    and

      LET MU1 = <value>
      LET MU2 = <value>

    in place of

      LET K1 = <value>
      LET K2 = <value>
Default:
    None
Synonyms:
    None
Related Commands:
    ADECDF = Compute the asymmetric double exponential cumulative distribution function.
    ADEPPF = Compute the asymmetric double exponential percent point function.
    SDEPDF = Compute the skew double exponential probability density function.
    DEXPDF = Compute the double exponential probability density function.
    EXPPDF = Compute the exponential probability density function.
    SNPDF = Compute the skew normal probability density function.
    STPDF = 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 ADEPDF(X,0.5) FOR X = -5 0.1 5
TITLE Kappa = 1
PLOT ADEPDF(X,1) FOR X = -5 0.1 5
TITLE Kappa = 2
PLOT ADEPDF(X,2) FOR X = -5 0.1 5
TITLE Kappa = 5
PLOT ADEPDF(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
Please email comments on this WWW page to alan.heckert@nist.gov.