naginterfaces.library.stat.inv_​cdf_​chisq

naginterfaces.library.stat.inv_cdf_chisq(p, df)

inv_cdf_chisq returns the deviate associated with the given lower tail probability of the ${\chi }^2$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01fc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01fcf.html

Parameters

pfloat

$p$, the lower tail probability from the required ${\chi }^2$-distribution.

dffloat

$\nu$, the degrees of freedom of the ${\chi }^2$-distribution.

Returns

xfloat

The deviate associated with the given lower tail probability of the ${\chi }^2$-distribution with real degrees of freedom.

Raises

NagValueError

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p<1.0$.

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\ge 0.0$.

(errno $2$)

On entry, $df=⟨value⟩$.

Constraint: $df>0.0$.

(errno $3$)

The probability is too close to $0.0$ or $1.0$.

(errno $5$)

The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.

Warns

NagAlgorithmicWarning

(errno $4$)

The algorithm has failed to converge in $⟨value⟩$ iterations. The result should be a reasonable approximation.

Notes

The deviate, $x_p$, associated with the lower tail probability $p$ of the ${\chi }^2$-distribution with $\nu$ degrees of freedom is defined as the solution to

$$P(X\le x_p:\nu )=p=\frac{1}{2^{\nu /2}\Gamma \left(\nu /2\right)}{\int }_0^{x_p}{e}^{-X/2}{X}^{v/2-1}dX\text{,}0\le x_p<\infty \text{;}\nu >0\text{.}$$

The required $x_p$ is found by using the relationship between a ${\chi }^2$-distribution and a gamma distribution, i.e., a ${\chi }^2$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.

For very large values of $\nu$, greater than ${10}^5$, Wilson and Hilferty’s normal approximation to the ${\chi }^2$ is used; see Kendall and Stuart (1969).

References

Best, D J and Roberts, D E, 1975, Algorithm AS 91. The percentage points of the ${\chi }^2$ distribution, Appl. Statist. (24), 385–388

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth

Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin