g01 Chapter Contents
g01 Chapter Introduction
NAG C 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:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     tail[${\mathbf{ltail}}$]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:     lxIntegerInput
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4:     x[lx]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:     ldfIntegerInput
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6:     df[ldf]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:     p[$\mathit{dim}$]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:     ivalid[$\mathit{dim}$]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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
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.

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

## 9  Example

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

### 9.1  Program Text

Program Text (g01scce.c)

### 9.2  Program Data

Program Data (g01scce.d)

### 9.3  Program Results

Program Results (g01scce.r)