# NAG CL Interfaceg01scc (prob_​chisq_​vector)

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## 1Purpose

g01scc returns a number of lower or upper tail probabilities for the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

 #include
 void g01scc (Integer ltail, const Nag_TailProbability tail[], Integer lx, const double x[], Integer ldf, const double df[], double p[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01scc, nag_stat_prob_chisq_vector or nag_prob_chi_sq_vector.

## 3Description

The lower tail probability for the ${\chi }^{2}$-distribution with ${\nu }_{i}$ degrees of freedom, $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ is defined by:
 $P = (Xi≤xi:νi) = 1 2 νi/2 Γ (νi/2) ∫ 0.0 xi Xi νi/2-1 e -Xi/2 dXi , xi ≥ 0 , νi > 0 .$
To calculate $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ 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 Section 2.6 in the G01 Chapter Introduction for further information.

## 4References

NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 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 whether the lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lx}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({X}_{i}\ge {x}_{i}:{\nu }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lx}$Integer Input
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4: $\mathbf{x}\left[{\mathbf{lx}}\right]$const double Input
On entry: ${x}_{i}$, the values of the ${\chi }^{2}$ variates with ${\nu }_{i}$ degrees of freedom with ${x}_{i}={\mathbf{x}}\left[j\right]$, .
Constraint: ${\mathbf{x}}\left[\mathit{j}-1\right]\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lx}}$.
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: $\mathbf{p}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{ldf}},{\mathbf{lx}}\right)$.
On exit: ${p}_{i}$, the probabilities 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{ldf}},{\mathbf{lx}}\right)$.
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 ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
On entry, ${x}_{i}<0.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
On entry, ${\nu }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.
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: ${\mathbf{ltail}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lx}}>0$.
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 x, df or tail was invalid, or the solution failed to converge.

## 7Accuracy

A relative accuracy of five significant figures is obtained in most cases.

## 8Parallelism and Performance

g01scc is not threaded in any implementation.

For higher accuracy the transformation described in Section 3 may be used with a direct call to s14bac.

## 10Example

Values from various ${\chi }^{2}$-distributions are read, the lower tail probabilities calculated, and all these values printed out.

### 10.1Program Text

Program Text (g01scce.c)

### 10.2Program Data

Program Data (g01scce.d)

### 10.3Program Results

Program Results (g01scce.r)