naginterfaces.library.stat.prob_​chisq_​vector

naginterfaces.library.stat.prob_chisq_vector(tail, x, df)

prob_chisq_vector returns a number of lower or upper tail probabilities for the ${\chi }^2$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01sc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01scf.html

Parameters

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

Indicates whether the lower or upper tail probabilities are required. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lx},\text{ldf})$:

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

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

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

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

xfloat, array-like, shape $\left(\text{lx}\right)$

$x_i$, the values of the ${\chi }^2$ variates with ${\nu }_i$ degrees of freedom.

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

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

Returns

pfloat, ndarray, shape $(max(\text{ltail},\text{ldf}))$

$p_i$, the probabilities for the ${\chi }^2$ distribution.

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

$ivalid[i-1]$ 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 $p_i$.

$ivalid[i-1]=2$

On entry, $x_i<0.0$.

$ivalid[i-1]=3$

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

$ivalid[i-1]=4$

The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.

Raises

NagValueError

(errno $2$)

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

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

(errno $3$)

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

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

(errno $4$)

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

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

Warns

NagAlgorithmicWarning

(errno $1$)

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

Check $ivalid$ for more information.

Notes

The lower tail probability for the ${\chi }^2$-distribution with ${\nu }_i$ degrees of freedom, $P=(X_i\le x_i:{\nu }_i)$ is defined by:

$$P=(X_i\le x_i:{\nu }_i)=\frac{1}{2^{{\nu }_i/2}\Gamma \left({\nu }_i/2\right)}{\int }_{0.0}^{x_i}{X}_i^{{\nu }_i/2-1}{e}^{-X_i/2}d{X}_i\text{,}{x}_i\ge 0,{\nu }_i>0\text{.}$$

To calculate $P=(X_i\le x_i:{\nu }_i)$ a transformation of a gamma distribution is employed, 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 ${\nu }_i/2$.

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

NIST Digital Library of Mathematical Functions

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