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Dataplot Vol 2 Vol 1

DNTPDF

Name:
    DNTPDF (LET)
Type:
    Library Function
Purpose:
    Compute the doubly non-central t probability density function with degrees of freedom parameter nu and with non-centrality parameters lambda1 and lambda2.
Description:
    If Z and X are mutually independent random variables with Z following a standard normal distribution and X following a chi-square random variable with nu degrees of freedom, then

      Y = Z(0,1)/SQRT(X(nu)/nu)

    follows a t distribution with nu degrees of freedom.

    If Z is replaced with a normal distribution with location parameter lambda1 and X is replaced with a non-central chi-square distribution with non-centrality parameter lambda2, then

      Y = Z(lambda1,1)/SQRT(X(nu,lambda2)/nu)

    follows a doubly non-central t distribution with non-centrality parameters lambda1 and lambda2.

    The doubly non-central t probability density function can be generalized with location and scale parameters in the usual way.

Syntax:
    LET <y> = DNTPDF(<x>,<nu>,<lambda1>,<lambda2>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, variable or a parameter;
                <nu> is a non-negative number, parameter or variable that specifies the degrees of freedom parameter;
                <lambda1> is a non-negative number, parameter or variable that specifies the first non-centrality parameter;
                <lambda2> is a non-negative number, parameter or variable that specifies the second non-centrality parameter;
                <loc> is a number or parameter that specifies the location parameter;
                <scale> is a number or parameter that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The location and scale parameters are optional.

Examples:
    LET A = DNTPDF(0.7,1,1,1)
    LET A = DNTPDF(3,10,10,100)
    LET X2 = DNTPDF(95,10,100,1)
Note:
    Dataplot computes the doubly non-central t probability density function by finding the numerical derivative of the doubly non-central t cumulative distribution function. It uses the DIFF routine from the SLATEC library to compute the numerical derivative.

    The doubly non-central t cumulative distribution function is computed using an algorithm written by Charles Reeves while he was a member of the Statistical Engineering Division at NIST. The algorithm is described in the paper listed in the Reference section below and is based on a series representation given by Krishnan (see the Reference below) of the exact form of the doubly non-central t distribution.

Note:
    The degrees of freedom parameter and the non-centrality parameters can be non-negative real numbers. The lambda1 non-centrality parameters is restricted to value under 100 and the lambda2 non-centrality parameter is restricted to values under 10,000. The compute time increases as the value of the non-centrality parameters increases. The degrees of freedom parameter need not be an integer.
Note:
    DATAPLOT also supports the central t and the singly non-central t distributions (see the documentation for TPDF and NCTPDF). The DNTPDF function can be used for these cases as well by setting one or both non-centrality parameters to zero. However, it uses a different algorithm.
Default:
    None
Synonyms:
    None
Related Commands:
    DNTCDF = Compute the doubly non-central t cumulative distribution function.
    DNTPPF = Compute the doubly non-central t percent point function.
    NCTPDF = Compute the singly non-central t probability density function.
    TPDF = Compute the t probability density function.
    DNFPDF = Compute the doubly non-central F probability density function.
    CHSPDF = Compute the chi-square probability density function.
    NORPDF = Compute the normal probability density function.
    FPDF = Compute the F probability density function.
Reference:
    "An Algorithm for Computing the Doubly Non-Central t C.D.F. to a Specified Accuracy", Charles Reeve, SED Note 86-5, December, 1986.

    "Series Representation of the Doubly Non-Central t-Distribution", Marakatha Krishnan, Journal of the the American Statistical Association, Vol. 63, No. 323, 1968 (pp. 1004-1012).

Applications:
    Distributional Modeling
Implementation Date:
    2004/5
Program:
     
    LABEL CASE ASIS
    Y1LABEL Probability
    X1LABEL X
    Y1LABEL DISPLACEMENT 12
    X1LABEL DISPLACEMENT 12
    TITLE DISPLACEMENT 2
    YLIMITS 0 0.5
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    MULTIPLOT SCALE FACTOR 2
    MULTIPLOT 2 2
    TITLE LAMBDA1 = 0.5, LAMBDA2 = 0.5
    PLOT DNTPDF(X,5,0.5,0.5) FOR X = -5 0.1 5
    TITLE LAMBDA1 = 0.5, LAMBDA2 = 2
    PLOT DNTPDF(X,5,0.5,2) FOR X = -5 0.1 5
    TITLE LAMBDA1 = 2, LAMBDA2 = 0.5
    PLOT DNTPDF(X,5,2,0.5) FOR X = -5 0.1 5
    TITLE LAMBDA1 = 2, LAMBDA2 = 2
    PLOT DNTPDF(X,5,2,2) FOR X = -5 0.1 5
    END OF MULTIPLOT
    CASE ASIS
    JUSTIFICATION CENTER
    MOVE 50 97
    TEXT Doubly Non-Central t Distribution PDF (NU = 5)
        
    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.