naginterfaces.library.stat.inv_​cdf_​chisq_​vector

naginterfaces.library.stat.inv_cdf_chisq_vector(tail, p, df)

inv_cdf_chisq_vector returns a number of deviates associated with the given probabilities of the ${\chi }^2$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01tc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01tcf.html

Parameters

tailstr, length 1, array-like, shape $\left(\text{ltail}\right)$

Indicates which tail the supplied probabilities represent. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lp},\text{ldf})$:

$tail\left[j\right]=\text{'L'}$

The lower tail probability, i.e., $p_i=(X_i\le x_{p_i}:{\nu }_i)$.

$tail\left[j\right]=\text{'U'}$

The upper tail probability, i.e., $p_i=(X_i\ge x_{p_i}:{\nu }_i)$.

pfloat, array-like, shape $\left(\text{lp}\right)$

$p_i$, the probability of the required ${\chi }^2$-distribution as defined by $tail$.

dffloat, array-like, shape $\left(\text{ldf}\right)$

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

Returns

xfloat, ndarray, shape $(max(\text{ltail},\text{lp}))$

$x_{p_i}$, the deviates for the ${\chi }^2$-distribution.

ivalidint, ndarray, shape $(max(\text{ltail},\text{lp}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, invalid value supplied in $tail$ when calculating $x_{p_i}$.

$ivalid[i-1]=2$

On entry, invalid value for $p_i$.

$ivalid[i-1]=3$

On entry, ${\nu }_i\le 0.0$.

$ivalid[i-1]=4$

$p_i$ is too close to $0.0$ or $1.0$ for the result to be calculated.

$ivalid[i-1]=5$

The solution has failed to converge. The result should be a reasonable approximation.

Raises

NagValueError

(errno $2$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ltail}>0$.

(errno $3$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lp}>0$.

(errno $4$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ldf}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $tail$, $p$ or $df$ was invalid, or the solution failed to converge.

Check $ivalid$ for more information.

Notes

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

$$(X_i\le x_{p_i}:{\nu }_i)=p_i=\frac{1}{2^{{\nu }_i/2}\Gamma \left({\nu }_i/2\right)}{\int }_0^{x_{p_i}}{e}^{-X_i/2}{X}_i^{v_i/2-1}d{X}_i\text{,}0\le x_{p_i}<\infty \text{;}{\nu }_i>0\text{.}$$

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

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

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

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