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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_chisq_vector (g01sc)

## Purpose

nag_stat_prob_chisq_vector (g01sc) returns a number of lower or upper tail probabilities for the χ2${\chi }^{2}$-distribution with real degrees of freedom.

## Syntax

[p, ivalid, ifail] = g01sc(tail, x, df, 'ltail', ltail, 'lx', lx, 'ldf', ldf)
[p, ivalid, ifail] = nag_stat_prob_chisq_vector(tail, x, df, 'ltail', ltail, 'lx', lx, 'ldf', ldf)

## Description

The lower tail probability for the χ2${\chi }^{2}$-distribution with νi${\nu }_{i}$ degrees of freedom, P = ( Xi xi : νi) $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ is defined by:
 xi P = (Xi ≤ xi : νi) = 1/( 2νi / 2 Γ (νi / 2) ) ∫ Xiνi / 2 − 1e − Xi / 2dXi,  xi ≥ 0 , νi > 0. 0.0
$P = (Xi≤xi:νi) = 1 2 νi/2 Γ (νi/2) ∫ 0.0 xi Xi νi/2-1 e -Xi/2 dXi , xi ≥ 0 , νi > 0 .$
To calculate P = ( Xi xi : νi) $P=\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$ a transformation of a gamma distribution is employed, i.e., a χ2${\chi }^{2}$-distribution with νi${\nu }_{i}$ degrees of freedom is equal to a gamma distribution with scale parameter 2$2$ and shape parameter νi / 2${\nu }_{i}/2$.
The input arrays to this function are designed to allow maximum flexibility in the supply of vector parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section [Vectorized s] in the G01 Chapter Introduction for further information.

## 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

## Parameters

### Compulsory Input Parameters

1:     tail(ltail) – cell array of strings
ltail, the dimension of the array, must satisfy the constraint ltail > 0${\mathbf{ltail}}>0$.
Indicates whether the lower or upper tail probabilities are required. For j = ((i1)  mod  ltail) + 1 , for i = 1,2,,max (ltail,lx,ldf)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lx}},{\mathbf{ldf}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., pi = P( Xi xi : νi) ${p}_{i}=P\left({X}_{i}\le {x}_{i}:{\nu }_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., pi = P( Xi xi : νi) ${p}_{i}=P\left({X}_{i}\ge {x}_{i}:{\nu }_{i}\right)$.
Constraint: tail(j) = 'L'${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or 'U'$\text{'U'}$, for j = 1,2,,ltail$\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     x(lx) – double array
lx, the dimension of the array, must satisfy the constraint lx > 0${\mathbf{lx}}>0$.
xi${x}_{i}$, the values of the χ2${\chi }^{2}$ variates with νi${\nu }_{i}$ degrees of freedom with xi = x(j)${x}_{i}={\mathbf{x}}\left(j\right)$, j = ((i1)  mod  lx) + 1.
Constraint: x(j)0.0${\mathbf{x}}\left(\mathit{j}\right)\ge 0.0$, for j = 1,2,,lx$\mathit{j}=1,2,\dots ,{\mathbf{lx}}$.
3:     df(ldf) – double array
ldf, the dimension of the array, must satisfy the constraint ldf > 0${\mathbf{ldf}}>0$.
νi${\nu }_{i}$, the degrees of freedom of the χ2${\chi }^{2}$-distribution with νi = df(j)${\nu }_{i}={\mathbf{df}}\left(j\right)$, j = ((i1)  mod  ldf) + 1.
Constraint: df(j) > 0.0${\mathbf{df}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,ldf$\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.

### Optional Input Parameters

1:     ltail – int64int32nag_int scalar
Default: The dimension of the array tail.
The length of the array tail.
Constraint: ltail > 0${\mathbf{ltail}}>0$.
2:     lx – int64int32nag_int scalar
Default: The dimension of the array x.
The length of the array x.
Constraint: lx > 0${\mathbf{lx}}>0$.
3:     ldf – int64int32nag_int scalar
Default: The dimension of the array df.
The length of the array df.
Constraint: ldf > 0${\mathbf{ldf}}>0$.

None.

### Output Parameters

1:     p( : $:$) – double array
Note: the dimension of the array p must be at least max (ltail,ldf,lx)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{ldf}},{\mathbf{lx}}\right)$.
pi${p}_{i}$, the probabilities for the χ2${\chi }^{2}$ distribution.
2:     ivalid( : $:$) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,ldf,lx)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{ldf}},{\mathbf{lx}}\right)$.
ivalid(i)${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating pi${p}_{i}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, xi < 0.0${x}_{i}<0.0$.
ivalid(i) = 3${\mathbf{ivalid}}\left(i\right)=3$
 On entry, νi ≤ 0.0${\nu }_{i}\le 0.0$.
ivalid(i) = 4${\mathbf{ivalid}}\left(i\right)=4$
The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Note: nag_stat_prob_chisq_vector (g01sc) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of x, df or tail was invalid, or the solution failed to converge.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ltail > 0${\mathbf{ltail}}>0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: lx > 0${\mathbf{lx}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: ldf > 0${\mathbf{ldf}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

For higher accuracy the transformation described in Section [Description] may be used with a direct call to nag_specfun_gamma_incomplete (s14ba).

## Example

```function nag_stat_prob_chisq_vector_example
x = [8.26; 6.2; 55.76];
df = [20; 7.5; 45];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = nag_stat_prob_chisq_vector(tail, x, df);

fprintf('\n    X      DF     PROB\n');
lx = numel(x);
ldf = numel(df);
ltail = numel(tail);
len = max ([lx, ldf, ltail]);
for i=0:len-1
fprintf('%7.3f%8.3f%8.3f  %5d\n', x(mod(i,lx)+1), df(mod(i,ldf)+1), ...
prob(i+1), ivalid(i+1));
end
```
```

X      DF     PROB
8.260  20.000   0.010      0
6.200   7.500   0.428      0
55.760  45.000   0.869      0

```
```function g01sc_example
x = [8.26; 6.2; 55.76];
df = [20; 7.5; 45];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = g01sc(tail, x, df);

fprintf('\n    X      DF     PROB\n');
lx = numel(x);
ldf = numel(df);
ltail = numel(tail);
len = max ([lx, ldf, ltail]);
for i=0:len-1
fprintf('%7.3f%8.3f%8.3f  %5d\n', x(mod(i,lx)+1), df(mod(i,ldf)+1), ...
prob(i+1), ivalid(i+1));
end
```
```

X      DF     PROB
8.260  20.000   0.010      0
6.200   7.500   0.428      0
55.760  45.000   0.869      0

```