DNTPDF

DNTPDF (LET)

Library Function

Compute the doubly non-central t probability density function with degrees of freedom parameter and with non-centrality parameters and .

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 degrees of freedom, then

follows a t distribution with degrees of freedom.

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

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

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

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.

LET A = DNTPDF(0.7,1,1,1)
LET A = DNTPDF(3,10,10,100)
LET X2 = DNTPDF(95,10,100,1)

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.

The degrees of freedom parameter and the non-centrality parameters can be non-negative real numbers. The non-centrality parameters is restricted to value under 100 and the 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.

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.

None

None

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.

"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).

Distributional Modeling

2004/5

 
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)