CNONCT Function

Returns the noncentrality parameter from a chi-square distribution.

Category:

Mathematical

Syntax

CNONCT(x,df,prob)

Required Arguments

x

is a numeric random variable.

Range

x ≥ 0

df

is a numeric degrees of freedom parameter.

Range

df > 0

prob

is a probability.

Range

0 < prob < 1

Details

The CNONCT function returns the nonnegative noncentrality parameter from a noncentral chi-square distribution whose parameters are x, df, and nc. If prob 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

\begin{matrix} P_{c} (x | df, n c) - p r o b = 0 \end{matrix}

The following relationship applies to the preceding equation:

\begin{matrix} P_{c} (x | df, nc) = ε^{\frac{- n c}{2}} Σ_{j = 0}^{\infty} \frac{{(\frac{n c}{2})}^{j}}{j!} P_{g} (\frac{x}{2} | \frac{d f}{2} + j) \end{matrix}

The following relationship applies to the preceding equation:

P_{g} (x | a)

is the probability from the gamma distribution given by

\begin{matrix} P_{g} (x | a) = \frac{1}{Γ (a)} \int_{0}^{x} t^{a - 1} ε^{- t} d t \end{matrix}

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;
      prob=probchi(x,df,nc);
      ncc=cnonct(x,df,prob);
      output;
   end;
run;
proc print;
run;

Computations of the Noncentrality Parameters from the Chi-squared Distribution