DNFPDF

DNFPDF (LET)

Library Function

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

If U and V are mutually independent chi-square random variables with degrees of freedom parameter and , respectively, then

follows a F distribution. If U and V are replaced with non-central chi-square distributions with non-centrality parameters and , respectively, then the above ratio follows a doubly non-central F distribution with non-centrality parameters and .

The probability density function of the doubly non-central F distribution is computed by finding the numerical derivative of the doubly non-central F cumulative distribution function.

The doubly non-central F distribution can be generalized with location and scale parameters in the usual way.

LET <y> = DNFPDF(<x>,<v1>,<v2>,<lambda>,<loc>,<scale>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, variable or a parameter containing non-negative values;
            <v1> is a non-negative number, parameter or variable that specifies the first degrees of freedom parameter;
            <v2> is a non-negative number, parameter or variable that specifies the second 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, parameter or variable that specifies the location parameter;
            <scale> is a number, parameter or variable that specifies the scale parameter;
            <y> is a variable or a parameter (depending on what <y1> is) where the computed pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Note that the location and scale parameters are optional.

LET A = DNFPDF(2,3,3,5)
LET A = DNFPDF(2,10,10,5)
LET X2 = DNFPDF(1.1,14,15,10000)

Dataplot computes the doubly non-central F probability density function by finding the numerical derivative of the doubly non-central F cumulative distribution function. It uses the DIFF routine from the SLATEC library to compute the numerical derivative.

The doubly non-central F 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 Bulgren (see the Reference below) of the exact form of the doubly non-central F distribution.

Both the degrees of freedom parameters and the non-centrality parameters can be non-negative real numbers. The non-centrality parameters are restricted to values under 10,000. The compute time increases as the value of the non-centrality parameters increases. The degrees of freedom parameters need not be integers.

DATAPLOT also supports the central F and the non-central F distributions (see the documentation for FPDF and NCFPDF). The DNFPDF routine can be used to compute the central F distribution and the non-central F distribution (set the appropriate non-centrality parameters to zero). For example, this can be used for the non-integer degrees of freedom case for the standard F distribution.

To generate doubly non-central F random numbers, enter the commands
LET NU1 = <value>
LET NU2 = <value>
LET LAMBDA1 = <value>
LET LAMBDA2 = <value>
LET Y = DOUBLY NON-CENTRAL F RANDOM NUMBERS ...

FOR I = 1 1 N


To generate a non-central F probability plot or an non-central F Kolmogorov-Smirnov or chi-square goodness of fit test, enter the following commands

LET NU1 = <value>
LET NU2 = <value>
LET LAMBDA1 = <value>
LET LAMBDA2 = <value>
DOUBLY NON-CENTRAL F PROBABILITY PLOT Y
DOUBLY NON-CENTRAL F KOLMOGOROV SMIRNOV ...

GOODNESS OF FIT Y

DOUBLY NON-CENTRAL F CHI-SQUARE GOODNESS OF FIT Y

None

None

DNFCDF	= Compute the doubly non-central F cumulative distribution function.
DNFPPF	= Compute the doubly non-central F percent point function.
NCFPDF	= Compute the singly non-central F probability density function.
FPDF	= Compute the F probability density function.
NCBPDF	= Compute the non-central beta probability density function.
NCCPDF	= Compute the non-central chi-square probability density function.
DNTPDF	= Compute the doubly non-central t probability density function.
NCTPDF	= Compute the non-central t probability density function.
CHSPDF	= Compute the chi-square probability density function.
NORPDF	= Compute the normal probability density function.

"An Algorithm for Computing the Doubly Non-Central F C.D.F. to a Specified Accuracy", Charles Reeve, SED Note 86-4, November, 1986.

"On Representations of the Doubly Non-Central F Distribution", W. G. Bulgren, Journal of the the American Statistical Association, Vol. 66, No. 333, 1971 (pp. 184-186).

"Continuous Univariate Distributions: Volume 2", Johnson, Kotz, and Balakrishnan, Wiley and Sons, 1994, chapter 30.

"Statistical Distributions", Third Edition, Evans, Hastings, and Peacock, 2000 pp. 95-97.

Distributional Modeling

2004/5

 
LABEL CASE ASIS
Y1LABEL Probability
X1LABEL X
Y1LABEL DISPLACEMENT 12
X1LABEL DISPLACEMENT 12
TITLE DISPLACEMENT 2
YLIMITS 0 0.9
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 2
TITLE LAMBDA1 = 0.5, LAMBDA2 = 0.5
PLOT DNFPDF(X,10,5,0.5,0.5) FOR X = 0.01 0.01 5
TITLE LAMBDA1 = 0.5, LAMBDA2 = 2
PLOT DNFPDF(X,10,5,0.5,2) FOR X = 0.01 0.01 5
TITLE LAMBDA1 = 2, LAMBDA2 = 0.5
PLOT DNFPDF(X,10,5,2,0.5) FOR X = 0.01 0.01 5
TITLE LAMBDA1 = 2, LAMBDA2 = 2
PLOT DNFPDF(X,10,5,2,2) FOR X = 0.01 0.01 5
END OF MULTIPLOT
CASE ASIS
JUSTIFICATION CENTER
MOVE 50 97
TEXT Doubly Non-Central F Distribution PDF (NU1 = 10, NU2 = 5)