CNONCT Function

Returns the noncentrality parameter from a chi-square distribution.

Categories:	Mathematical
	CAS

Table of Contents

Syntax

CNONCT(x, df, probability)

Required Arguments

x

is a numeric random variable.

Range

x ≥ 0

df

is a numeric degrees of freedom parameter.

Range

df > 0

probability

is a probability.

Range

0 < probability < 1

Details

The CNONCT function returns the nonnegative noncentrality parameter from a noncentral chi-square distribution whose parameters are x, df, and nc. If probability is greater than the probability from the central chi-square distribution with the parameters x and df, 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 c Baseline left-parenthesis x vertical-bar 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 c Baseline left-parenthesis x vertical-bar 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 P Subscript g Baseline left-parenthesis StartFraction x Over 2 EndFraction vertical-bar StartFraction d f Over 2 EndFraction plus j right-parenthesis

The following relationship applies to the preceding equation:

upper P Subscript g Baseline left-parenthesis x vertical-bar a right-parenthesis

is the probability from the gamma distribution given by

multiline equation line 1 upper P Subscript g Baseline left-parenthesis x vertical-bar a right-parenthesis equals StartFraction 1 Over upper Gamma left-parenthesis a right-parenthesis EndFraction integral Subscript 0 Superscript x Baseline t Superscript a minus 1 Baseline epsilon Superscript negative t 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;
   do nc=1 to 3 by .5;
      probability=probchi(x, df, nc);
      ncc=cnonct(x, df, probability);
      output;
   end;
run;
proc print;
run;

Computations of the Noncentrality Parameters from the Chi-squared Distribution

Last updated: March 16, 2017