# NAG CL Interfaceg01tcc (inv_​cdf_​chisq_​vector)

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## 1Purpose

g01tcc returns a number of deviates associated with the given probabilities of the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

 #include
 void g01tcc (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double x[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01tcc, nag_stat_inv_cdf_chisq_vector or nag_deviates_chi_sq_vector.

## 3Description

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
 $P( Xi ≤ xpi :νi) = pi = 1 2 νi/2 Γ (νi/2) ∫ 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi , 0 ≤ xpi < ∞ ; ​ νi > 0 .$
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 Section 2.6 in the G01 Chapter Introduction for further information.

## 4References

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

## 5Arguments

1: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbability Input
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left({X}_{i}\le {x}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left({X}_{i}\ge {x}_{{p}_{i}}:{\nu }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lp}$Integer Input
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4: $\mathbf{p}\left[{\mathbf{lp}}\right]$const double Input
On entry: ${p}_{i}$, the probability of the required ${\chi }^{2}$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraints:
• if ${\mathbf{tail}}\left[k\right]=\mathrm{Nag_LowerTail}$, $0.0\le {\mathbf{p}}\left[\mathit{j}\right]<1.0$;
• otherwise $0.0<{\mathbf{p}}\left[\mathit{j}\right]\le 1.0$.
Where and .
5: $\mathbf{ldf}$Integer Input
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6: $\mathbf{df}\left[{\mathbf{ldf}}\right]$const double Input
On entry: ${\nu }_{i}$, the degrees of freedom of the ${\chi }^{2}$-distribution with ${\nu }_{i}={\mathbf{df}}\left[j\right]$, .
Constraint: ${\mathbf{df}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.
7: $\mathbf{x}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
On exit: ${x}_{{p}_{i}}$, the deviates for the ${\chi }^{2}$-distribution.
8: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
On entry, invalid value supplied in tail when calculating ${x}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=3$
On entry, ${\nu }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
${p}_{i}$ is too close to $0.0$ or $1.0$ for the result to be calculated.
${\mathbf{ivalid}}\left[i-1\right]=5$
The solution has failed to converge. The result should be a reasonable approximation.
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldf}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lp}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.

## 7Accuracy

The results should be accurate to five significant digits for most argument values. Some accuracy is lost for ${p}_{i}$ close to $0.0$ or $1.0$.

## 8Parallelism and Performance

g01tcc is not threaded in any implementation.

For higher accuracy the relationship described in Section 3 may be used and a direct call to g01tfc made.

## 10Example

This example reads lower tail probabilities for several ${\chi }^{2}$-distributions, and calculates and prints the corresponding deviates.

### 10.1Program Text

Program Text (g01tcce.c)

### 10.2Program Data

Program Data (g01tcce.d)

### 10.3Program Results

Program Results (g01tcce.r)