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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_inv_cdf_chisq_vector (g01tc)

## Purpose

nag_stat_inv_cdf_chisq_vector (g01tc) returns a number of deviates associated with the given probabilities of the χ2${\chi }^{2}$-distribution with real degrees of freedom.

## Syntax

[x, ivalid, ifail] = g01tc(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)
[x, ivalid, ifail] = nag_stat_inv_cdf_chisq_vector(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)

## Description

The deviate, xpi${x}_{{p}_{i}}$, associated with the lower tail probability pi${p}_{i}$ of the χ2${\chi }^{2}$-distribution with νi${\nu }_{i}$ degrees of freedom is defined as the solution to
 xpi P( Xi ≤ xpi : νi) = pi = 1/( 2νi / 2 Γ (νi / 2) ) ∫ e − Xi / 2Xi vi / 2 − 1 dXi,  0 ≤ xpi < ∞; ​νi > 0. 0
$P( Xi ≤ xpi :νi) = pi = 1 2 νi/2 Γ (νi/2) ∫ 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi , 0 ≤ xpi < ∞ ; ​ νi > 0 .$
The required xpi${x}_{{p}_{i}}$ is found by using the relationship between a χ2${\chi }^{2}$-distribution and a gamma distribution, 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${\nu }_{i}/2$.
For very large values of νi${\nu }_{i}$, greater than 105${10}^{5}$, Wilson and Hilferty's Normal approximation to the χ2${\chi }^{2}$ is used; see Kendall and Stuart (1969).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector 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

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2${\chi }^{2}$ 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

## 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 which tail the supplied probabilities represent. For j = ((i1)  mod  ltail) + 1 , for i = 1,2,,max (ltail,lp,ldf)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., pi = P( Xi xpi : νi) ${p}_{i}=P\left({X}_{i}\le {x}_{{p}_{i}}:{\nu }_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., pi = P( Xi xpi : νi) ${p}_{i}=P\left({X}_{i}\ge {x}_{{p}_{i}}:{\nu }_{i}\right)$.
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:     p(lp) – double array
lp, the dimension of the array, must satisfy the constraint lp > 0${\mathbf{lp}}>0$.
pi${p}_{i}$, the probability of the required χ2${\chi }^{2}$-distribution as defined by tail with pi = p(j)${p}_{i}={\mathbf{p}}\left(j\right)$, j = ((i1)  mod  lp) + 1.
Constraints:
• if tail(k) = 'L'${\mathbf{tail}}\left(k\right)=\text{'L'}$, 0.0p(j) < 1.0$0.0\le {\mathbf{p}}\left(\mathit{j}\right)<1.0$;
• otherwise 0.0 < p(j)1.0$0.0<{\mathbf{p}}\left(\mathit{j}\right)\le 1.0$.
Where k = (i1)  mod  ltail + 1 and j = (i1)  mod  lp + 1.
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:     lp – int64int32nag_int scalar
Default: The dimension of the array p.
The length of the array p.
Constraint: lp > 0${\mathbf{lp}}>0$.
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:     x( : $:$) – double array
Note: the dimension of the array x must be at least max (ltail,lp,ldf)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
xpi${x}_{{p}_{i}}$, the deviates for the χ2${\chi }^{2}$-distribution.
2:     ivalid( : $:$) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lp,ldf)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
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 xpi${x}_{{p}_{i}}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, invalid value for pi${p}_{i}$.
ivalid(i) = 3${\mathbf{ivalid}}\left(i\right)=3$
 On entry, νi ≤ 0.0${\nu }_{i}\le 0.0$.
ivalid(i) = 4${\mathbf{ivalid}}\left(i\right)=4$
pi${p}_{i}$ is too close to 0.0$0.0$ or 1.0$1.0$ for the result to be calculated.
ivalid(i) = 5${\mathbf{ivalid}}\left(i\right)=5$
The solution has failed to converge. The result should be a reasonable approximation.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and 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 tail, p or df 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: lp > 0${\mathbf{lp}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: ldf > 0${\mathbf{ldf}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

For higher accuracy the relationship described in Section [Description] may be used and a direct call to nag_stat_inv_cdf_gamma_vector (g01tf) made.

## Example

```function nag_stat_inv_cdf_chisq_vector_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
df = [20; 7.5; 45];
[x, ivalid, ifail] = nag_stat_inv_cdf_chisq_vector(tail, p, df);

fprintf('\n  TAIL     P       DF      X     IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf = numel(df);
len = max ([ltail, lp, ldf]);
for i=0:len-1
fprintf('%5c%8.3f%8.3f%8.3f%8d\n',  cell2mat(tail(mod(i, ltail)+1)), ...
p(mod(i,lp)+1), df(mod(i,ldf)+1), x(i+1), ivalid(i+1));
end
```
```

TAIL     P       DF      X     IVALID
L   0.010  20.000   8.260       0
L   0.428   7.500   6.201       0
L   0.869  45.000  55.738       0

```
```function g01tc_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
df = [20; 7.5; 45];
[x, ivalid, ifail] = g01tc(tail, p, df);

fprintf('\n  TAIL     P       DF      X     IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf = numel(df);
len = max ([ltail, lp, ldf]);
for i=0:len-1
fprintf('%5c%8.3f%8.3f%8.3f%8d\n',  cell2mat(tail(mod(i, ltail)+1)), ...
p(mod(i,lp)+1), df(mod(i,ldf)+1), x(i+1), ivalid(i+1));
end
```
```

TAIL     P       DF      X     IVALID
L   0.010  20.000   8.260       0
L   0.428   7.500   6.201       0
L   0.869  45.000  55.738       0

```