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.
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(3,10,10,100)
LET X2 = DNTPDF(95,10,100,1)
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.
"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).
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)
Date created: 7/7/2004