g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_chi_sq_vector (g01scc)

## 1  Purpose

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

## 2  Specification

 #include #include
 void nag_prob_chi_sq_vector (Integer ltail, const Nag_TailProbability tail[], Integer lx, const double x[], Integer ldf, const double df[], double p[], Integer ivalid[], NagError *fail)

## 3  Description

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.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5  Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
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:    $\mathbf{lx}$IntegerInput
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4:    $\mathbf{x}\left[{\mathbf{lx}}\right]$const doubleInput
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:    $\mathbf{ldf}$IntegerInput
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6:    $\mathbf{df}\left[{\mathbf{ldf}}\right]$const doubleInput
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:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
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:    $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
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 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of x, df or tail was invalid, or the solution failed to converge.

## 7  Accuracy

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

## 8  Parallelism and Performance

nag_prob_chi_sq_vector (g01scc) is not threaded in any implementation.

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

## 10  Example

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

### 10.1  Program Text

Program Text (g01scce.c)

### 10.2  Program Data

Program Data (g01scce.d)

### 10.3  Program Results

Program Results (g01scce.r)