Cdf.InverseChi Method

Evaluates the inverse of the chi-squared cumulative probability distribution function.

public static double InverseChi(
double p,
double df
);

Parameters

p: A double specifying the probability for which the inverse chi-squared function is to be evaluated.
df: A double specifying the number of degrees of freedom. This must be at least 0.5.

Return Value

A double specifying the probability that a chi-squared random variable takes a value less than or equal to this value is p.

Remarks

Method InverseChi evaluates the inverse distribution function of a chi-squared random variable with df degrees of freedom, that is, with P = p and v = df, it determines x (equal to InverseChi(p, df)), such that

$P = \frac{1}{{2^{\nu /2} \Gamma \left( {\nu /2} \right)}}\int_0^x {e^{ - t/2} t^{\nu /2 - 1} } dt$

where $\Gamma(\cdot)$ is the gamma function. The probability that the random variable takes a value less than or equal to x is P.

For $v \lt 40$ , InverseChi uses bisection, if $v \ge 2$ or $P \gt 0.98$ , or regula falsi to find the point at which the chi-squared distribution function is equal to P. The distribution function is evaluated using Chi.

For $40 \le v \lt 100$ , a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.18) to the normal distribution is used, and InverseNormal is used to evaluate the inverse of the normal distribution function. For $v \ge 100$ , the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.17) is used.

Cdf.InverseChi Method

Parameters

Return Value

Remarks

See Also