Dataplot Vol 2 Vol 1

# NCFPDF

Name:
NCFPDF (LET)
Type:
Library Function
Purpose:
Compute the non-central F probability density function with degrees of freedom parameters and and with non-centrality parameter .
Description:
If U and V are mutually independent chi-square random variables with degrees of freedom parameters and , respectively, then

follows a F distribution. If U is replaced with a non-central chi-square distribution with non-centrality parameter , then the above ratio follows a non-central F distribution with non-centrality parameter .

The probability density function of the non-central F distribution is rather complicated and not given here. It is given on page 95 of Evans, Hastings, and Peacock (see the Reference section below). The input value and both degrees of freedom parameters should be positive and the non-centrality parameter should be non-negative ( = 0 reduces to the standard F distribution).

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

Syntax:
LET <y> = NCFPDF(<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;
<lambda> is a non-negative number, parameter or variable that specifies the 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 <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.

Examples:
LET A = NCFPDF(2,3,3,5)
LET A = NCFPDF(2,10,10,5)
LET X2 = NCFPDF(1.1,14,15,10000)
Note:
Dataplot computes the non-central F probability density by finding the numerical derivative of the non-central F cumulative distribution function.

DATAPLOT computes the non-central F distribution by converting it to an equivalent non-central beta distribution. It then uses algorithm AS 226 (see the REFERENCE section below) obtained from the statlib archive to compute the non-central beta cdf. It uses the DBETAI and DLNGAM routines from the SLATEC library rather than the corresponding algorithms from the Applied Statistics series to compute the log gamma and incomplete beta functions.

Note:
DATAPLOT also supports the central F and the doubly non-central F distributions (see the documentation for FPDF and DNFPDF). The NCFPDF routine can be used to compute the central F distribution (set the non-centrality parameter to zero). For example, this can be used for the non-integer degrees of freedom case.
Note:
To generate non-central F random numbers, enter the commands

LET NU1 = <value>
LET NU2 = <value>
LET LAMBDA = <value>
LET Y = 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 LAMBDA = <value>
NON-CENTRAL F PROBABILITY PLOT Y
NON-CENTRAL F KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
NON-CENTRAL F CHI-SQUARE GOODNESS OF FIT Y
Default:
None
Synonyms:
None
Related Commands:
 NCFCDF = Compute the singly non-central F cumulative distribution function. NCFPPF = Compute the singly non-central F percent point function. DNFCDF = Compute the doubly non-central F cumulative distribution function. DNFPDF = Compute the doubly 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. NCTPDF = Compute the non-central t probability density function. CHSPDF = Compute the chi-square probability density function. NORPDF = Compute the normal probability density function.
Reference:
"Computing Noncentral Beta Probabilities", Lenth, Applied Statistics, Vol. 39, No. 2, 1987, pp. 241-244.

"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.

Applications:
Distributional Modeling
Implementation Date:
2004/5
Program:
```
LABEL CASE ASIS
Y1LABEL Probability
X1LABEL X
Y1LABEL DISPLACEMENT 12
X1LABEL DISPLACEMENT 12
TITLE DISPLACEMENT 2
Y1LIMITS 0 0.7
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 2
TITLE LAMBDA = 0
PLOT NCFPDF(X,10,5,0) FOR X = 0.01 0.01 5
TITLE LAMBDA = 0.5
PLOT NCFPDF(X,10,5,0.5) FOR X = 0.01 0.01 5
TITLE LAMBDA = 1
PLOT NCFPDF(X,10,5,1) FOR X = 0.01 0.01 5
TITLE LAMBDA = 2
PLOT NCFPDF(X,10,5,2) FOR X = 0.01 0.01 5
END OF MULTIPLOT
CASE ASIS
JUSTIFICATION CENTER
MOVE 50 97
TEXT Non-Central F Distribution PDF (NU1 = 10, NU2 = 5)
```

Date created: 7/7/2004
Last updated: 7/7/2004