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Dataplot Vol 2 Vol 1

STCDF

Name:
    STCDF (LET)
Type:
    Library Function
Purpose:
    Compute the skew-t cumulative distribution function.
Description:
    The skew-t distribution has the following probability density function:

      f(x,nu,lambda)=2*TCDF(lambda*x*SQRT((1+nu)/(x**2+nu)),nu+1)* TPDF(x,nu) -infinity < x, lambda < infinity

    with nu, lambda, TCDF, and TPDF denoting the degrees of freedom parameter, the skewness parameter, the cumulative distribution function of the t distribution, and the probability density function of the t distribution, respectively.

    For lambda = 0, the skew-t reduces to a t distribution. As lambda goes to infinity, the skew-t tends to the folded-t distribution.

    The cumulative distribution is computed by numerically integrating the skew-t probability density function.

    The standard skew-t distribution can be generalized with location and scale parameters.

Syntax:
    LET <y> = STCDF(<x>,<nu>,<lambda>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable or a parameter;
                <nu> is a number of parameter that specifies the value of the degrees of freedom shape parameter;
                <lambda> is a number of parameter that specifies the value of the skewness shape parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed skew-t cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = STCDF(3,5,1)
    LET A = STCDF(A1,DF,LAMBDA)
    LET X2 = STCDF(X1,NU,0.5)
Default:
    None
Synonyms:
    None
Related Commands:
    STPDF = Compute the skew-t probability density function.
    STPPF = Compute the skew-t percent point function.
    SNPDF = Compute the skew-normal probability density function.
    TPDF = Compute the t probability density function.
    FTPDF = Compute the folded t probability density function.
    NORPDF = Compute the normal density function.
    CHSPDF = Compute the chi-square probability density function.
Reference:
    "A Class of Distributions Which Includes the Normal Ones", Azzalini, Scandinavian Journal of Statistics, 12, 171-178.

    "Log-Skew-Normal and Log-Skew-t Distributions as Models for Familiy Income Data", Azzalini and Dal Cappello, unpublished paper downloaded from Azzallini web site.

Applications:
    Distributional Modeling
Implementation Date:
    1/2004
Program: 
     
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 100
TITLE SKEW-T (NU=3): LAMBDA = 0
PLOT STCDF(X,3,0) FOR X = -5 0.1 5
TITLE SKEW-T (NU=3): LAMBDA = 1
PLOT STCDF(X,3,1) FOR X = -5 0.1 5
TITLE SKEW-T (NU=3): LAMBDA = 5
PLOT STCDF(X,3,5) FOR X = -5 0.1 5
TITLE SKEW-T (NU=3): LAMBDA = 10
PLOT STCDF(X,3,10) FOR X = -5 0.1 5
END OF MULTIPLOT
    plot generated by sample program

Date created: 2/3/2004
Last updated: 2/3/2004
Please email comments on this WWW page to alan.heckert@nist.gov.