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Dataplot Vol 2 Vol 1

SEMCDF

Name:
    SEMCDF (LET)
Type:
    Library Function
Purpose:
    Compute the semi-circular cumulative distribution function.
Description:
    The semi-circular distribution is the distribution onto one axis of the points uniformly distributed within the unit circle. As such, it is useful for testing 2-dimensional uniformity.

    The formula for the cumulative distribution function is:

      F(x;mu,r) = 0.5 + x*SQRT(r**2 - (x-mu)**2)/(PI*r**2) + ARCSIN((x-mu)/r)/PI -r <= x <= r

    with mu and r denoting the location and scale parameters, respectively.

    The case where mu = 0 and r = 1 is referred to as the standard semi-circular distribution.

Syntax:
    LET <y> = SEMCDF(<x>,<mu>,<r>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable, number, or parameter;
                <mu> is a variable, number, or parameter that specifies the location parameter;
                <r> is a variable, number, or parameter that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed semi-circular cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    If <mu> and <r> are omitted, they default to 0 and 1, respectively.

Examples:
    LET A = SEMCDF(0.5)
    LET A = SEMCDF(0.5,0,5)
    LET X2 = SEMCDF(X1)
Default:
    None
Synonyms:
    None
Related Commands:
    SEMPDF = Compute the semi-circular probability density function.
    SEMPPF = Compute the semi-circular percent point function.
    UNIPDF = Compute the uniform probability density function.
    UNICDF = Compute the uniform cumulative distribution function.
    UNIPPF = Compute the uniform percent point function.
    NORCDF = Compute the normal cumulative distributoin function.
    NORPDF = Compute the normal probability density function.
    NORPPF = Compute the normal percent point function.
Reference:
    Johnson and Kotz (1970), Continuous Univariate Distributions - 2, Houghton Mifflin, (chapter 25).

    Filliben (1969), Simple and Robust Linear Estimation of the Location Parameter of a Symmetric Distribution, unpublished Ph.d dissertation, Princeton University, (pp. 21-44, 229-231).

Applications:
    Distributional Modeling
Implementation Date:
    1994/4: Implemented for the standard case
    2006/10: Implemented for the general case
Program: 
     
YLIMITS 0 1
MAJOR YTIC NUMBER 6
MINOR YTIC NUMBER 1
YTIC DECIMAL 1
XLIMITS -1 1
XTIC OFFSET 0.1 0.1
TITLE AUTOMATIC
PLOT SEMCDF(X) FOR X = -1 0.01 1
    plot generated by sample program

Date created: 1/8/2008
Last updated: 1/8/2008
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