NON_CENTRAL_F_PDF Function (PV-WAVE Advantage)
Evaluates the noncentral F probability density function (PDF).
Usage
result = NON_CENTRAL_F_PDF (f, df_numerator, df_denominator, lambda)
Input Parameters
f—Scalar float value for which the noncentral F probability density function is to be evaluated. f must be non-negative.
df_numerator—Scalar float value indicating the numerator degrees of freedom of the noncentral F distribution. df_numerator must be positive.
df_denominator—Scalar float value indicating the denominator degrees of freedom of the noncentral F distribution. df_denominator must be positive.
lambda—Scalar float value indicating the noncentrality parameter. lambda must be non-negative.
Returned Value
result—The probability density associated with a noncentral F random variable with value f.
Input Keywords
Double—If present and nonzero, then double precision is used.
Discussion
If X is a noncentral chi-square random variable with noncentrality parameter λ and ν1 degrees of freedom, and Y is a chi-square random variable with ν2 degrees of freedom which is statistically independent of X, then:
F = (X/ν1)/(Y/ν2)
is a noncentral F-distributed random variable whose PDF is given by:
where:
and Γ(.) is the gamma function, ν1 = df_numerator, ν2 = df_denominator, λ = lambda, and f = f.
With a noncentrality parameter of zero, the noncentral F distribution is the same as the F distribution.
The efficiency of the calculation of the above series is enhanced by:
1. calculating each term Φk in the series recursively in terms of either the term Φk-1 preceding it or the term Φk+1 following it, and
2. initializing the sum with the largest series term and adding the subsequent terms in order of decreasing magnitude.
Special cases:
Example
This example traces out a portion of a noncentral F distribution with parameters df_numerator = 100, df_denominator = 10, and lambda = 10.
PRO t_non_central_f_pdf
f = [0., .4, .8, 3.2, 5.6, 8.8, 14., 18.]
df_numerator = 100.
df_denominator = 10.
lambda =10.
PRINT,"df_numerator:",$
STRING(df_numerator,Format="(f4.0)")
PRINT,"df_denominator:",$
STRING(df_denominator,Format="(f4.0)")
PRINT,"lambda:",$
STRING(lambda,Format="(f4.0)")
PRINT," f pdf(f)"
FOR i=0L, 7 DO BEGIN
pdfv = NON_CENTRAL_F_PDF( f(i),df_numerator, $
df_denominator, lambda)
PRINT,STRING(f(i),Format="(f5.1)")," ",$
STRING(pdfv,Format="(E12.4)")
ENDFOR
END
Output
df_numerator: 100
df_denominator: 10
lambda: 10
f pdf(f)
0.0 0.0000e+00
0.4 9.7488e-02
0.8 8.1312e-01
3.2 3.6948e-02
5.6 2.8302e-03
8.8 2.7661e-04
14.0 2.1963e-05
18.0 5.3483e-06
Version 2017.0
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