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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_inv_cdf_students_t_vector (g01tb)

## Purpose

nag_stat_inv_cdf_students_t_vector (g01tb) returns a number of deviates associated with given probabilities of Student's $t$-distribution with real degrees of freedom.

## Syntax

[t, ivalid, ifail] = g01tb(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)
[t, ivalid, ifail] = nag_stat_inv_cdf_students_t_vector(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)

## Description

The deviate, ${t}_{{p}_{i}}$ associated with the lower tail probability, ${p}_{i}$, of the Student's $t$-distribution with ${\nu }_{i}$ degrees of freedom is defined as the solution to
 $P Ti < tpi :νi = pi = Γ νi+1 / 2 νiπ Γ νi/2 ∫ -∞ tpi 1 + Ti2 νi - νi+1 / 2 d Ti , νi ≥ 1 ; ​ -∞ < tpi < ∞ .$
For ${\nu }_{i}=1\text{​ or ​}2$ the integral equation is easily solved for ${t}_{{p}_{i}}$.
For other values of ${\nu }_{i}<3$ a transformation to the beta distribution is used and the result obtained from nag_stat_inv_cdf_beta (g01fe).
For ${\nu }_{i}\ge 3$ an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).
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 Vectorized Routines in the G01 Chapter Introduction for further information.

## References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's $t$-distribution Comm. ACM 13(10) 617–619

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{tail}\left({\mathbf{ltail}}\right)$ – cell array of strings
Indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left({T}_{i}\le {t}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left({T}_{i}\ge {t}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability,
i.e., ${p}_{i}=P\left({T}_{i}\le \left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)-P\left({T}_{i}\le -\left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability,
i.e., ${p}_{i}=P\left({T}_{i}\ge \left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)+P\left({T}_{i}\le -\left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$, $\text{'U'}$, $\text{'C'}$ or $\text{'S'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     $\mathrm{p}\left({\mathbf{lp}}\right)$ – double array
${p}_{i}$, the probability of the required Student's $t$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{p}}\left(\mathit{j}\right)<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
3:     $\mathrm{df}\left({\mathbf{ldf}}\right)$ – double array
${\nu }_{i}$, the degrees of freedom of the Student's $t$-distribution with ${\nu }_{i}={\mathbf{df}}\left(j\right)$, .
Constraint: ${\mathbf{df}}\left(\mathit{j}\right)\ge 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{t}\left(:\right)$ – double array
The dimension of the array t will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$
${t}_{{p}_{i}}$, the deviates for the Student's $t$-distribution.
2:     $\mathrm{ivalid}\left(:\right)$int64int32nag_int array
The dimension of the array ivalid will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$
${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating ${t}_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${p}_{i}\le 0.0$, or ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${\nu }_{i}<1.0$.
${\mathbf{ivalid}}\left(i\right)=4$
The solution has failed to converge. The result returned should represent an approximation to the solution.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ 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  ${\mathbf{ifail}}=1$
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{ldf}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The results should be accurate to five significant digits, for most argument values. The error behaviour for various argument values is discussed in Hill (1970).

The value ${t}_{{p}_{i}}$ may be calculated by using a transformation to the beta distribution and calling nag_stat_inv_cdf_beta_vector (g01te). This function allows you to set the required accuracy.

## Example

This example reads the probability, the tail that probability represents and the degrees of freedom for a number of Student's $t$-distributions and computes the corresponding deviates.
```function g01tb_example

fprintf('g01tb example results\n\n');

p    = [0.01; 0.01; 0.99];
df   = [20;   7.5;  45];
tail = {'S';  'L';  'C'};
[x, ivalid, ifail] = g01tb( ...
tail, p, df);

fprintf('     p      df    tail      x\n');
lp    = numel(p);
ldf   = numel(df);
ltail = numel(tail);
len   = max ([lp, ldf, ltail]);
for i=0:len-1
fprintf('%8.3f%8.3f   %c   %8.3f\n', p(mod(i,lp)+1), df(mod(i,ldf)+1), ...
tail{mod(i,ltail)+1}, x(i+1));
end

```
```g01tb example results

p      df    tail      x
0.010  20.000   S      2.845
0.010   7.500   L     -2.943
0.990  45.000   C      2.690
```