FNONCT Function

Returns the value of the noncentrality parameter of an F distribution.

Categories:	Mathematical
	CAS

Table of Contents

Syntax

FNONCT(x, ndf, ddf, probability)

Required Arguments

x

is a numeric random variable.

Range

x ≥ 0

ndf

is a numeric numerator degree of freedom parameter.

Range

ndf > 0

ddf

is a numeric denominator degree of freedom parameter.

Range

ddf > 0

probability

is a probability.

Range

0 < probability < 1

Details

The FNONCT function returns the nonnegative noncentrality parameter from a noncentral F distribution whose parameters are x, ndf, ddf, and nc. If probability is greater than the probability from the central F distribution whose parameters are x, ndf, and ddf, a root to this problem does not exist. In this case a missing value is returned. A Newton-type algorithm is used to find a nonnegative root nc of the equation

multiline equation line 1 upper P Subscript f Baseline left-parenthesis x vertical-bar n d f comma d d f comma n c right-parenthesis minus p r o b equals 0

The following relationship applies to the preceding equation:

multiline equation line 1 upper P Subscript f Baseline left-parenthesis x vertical-bar n d f comma d d f comma n c right-parenthesis equals epsilon Superscript StartFraction minus n c Over 2 EndFraction Baseline ModifyingAbove normal upper Sigma With infinity Underscript j equals 0 Endscripts StartFraction left-parenthesis StartFraction n c Over 2 EndFraction right-parenthesis Superscript j Baseline Over j factorial EndFraction upper I Subscript StartFraction left-parenthesis n d f right-parenthesis x Over d d f plus left-parenthesis n d f right-parenthesis x EndFraction Baseline left-parenthesis StartFraction d d f Over 2 EndFraction plus j comma StartFraction d d f Over 2 EndFraction right-parenthesis

In the equation, I (. . .) is the probability from the beta distribution that is given by the following equation:

multiline equation line 1 upper I Subscript x Baseline left-parenthesis a comma b right-parenthesis equals StartStartFraction OverOver StartFraction upper Gamma left-parenthesis a right-parenthesis upper Gamma left-parenthesis b right-parenthesis Over upper Gamma left-parenthesis a plus b right-parenthesis EndFraction EndEndFraction integral Subscript 0 Superscript x Baseline t Superscript a minus 1 Baseline left-parenthesis 1 minus t right-parenthesis Superscript b minus 1 Baseline d t

If the algorithm fails to converge to a fixed point, a missing value is returned.

Example

data work;
   x=2;
   df=4;
   ddf=5;
   do nc=1 to 3 by .5;
      prob=probf(x, df, ddf, nc);
      ncc=fnonct(x, df, ddf, prob);
      output;
   end;
run;
proc print;
run;

Output from the FNONCT Function

Last updated: March 16, 2017