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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$t$-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, tpi${t}_{{p}_{i}}$ associated with the lower tail probability, pi${p}_{i}$, of the Student's t$t$-distribution with νi${\nu }_{i}$ degrees of freedom is defined as the solution to
 tpi P( Ti < tpi : νi) = pi = ( Γ ((νi + 1) / 2) )/( sqrt(νiπ) Γ (νi / 2) ) ∫ (1 + (Ti2)/(νi)) − (νi + 1) / 2 dTi,  νi ≥ 1; ​ − ∞ < tpi < ∞. − ∞
$P( Ti < tpi :νi) = pi = Γ ( (νi+1) / 2 ) νiπ Γ (νi/2) ∫ -∞ tpi ( 1 + Ti2 νi ) - (νi+1) / 2 d Ti , νi ≥ 1 ; ​ -∞ < tpi < ∞ .$
For νi = 1​ or ​2${\nu }_{i}=1\text{​ or ​}2$ the integral equation is easily solved for tpi${t}_{{p}_{i}}$.
For other values of νi < 3${\nu }_{i}<3$ a transformation to the beta distribution is used and the result obtained from nag_stat_inv_cdf_beta (g01fe).
For νi3${\nu }_{i}\ge 3$ 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 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

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

## 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( Ti tpi : νi) ${p}_{i}=P\left({T}_{i}\le {t}_{{p}_{i}}:{\nu }_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., pi = P( Ti tpi : νi) ${p}_{i}=P\left({T}_{i}\ge {t}_{{p}_{i}}:{\nu }_{i}\right)$.
tail(j) = 'C'${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability,
i.e., pi = P( Ti |tpi| : νi) P( Ti |tpi| : νi) ${p}_{i}=P\left({T}_{i}\le |{t}_{{p}_{i}}|:{\nu }_{i}\right)-P\left({T}_{i}\le -|{t}_{{p}_{i}}|:{\nu }_{i}\right)$.
tail(j) = 'S'${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability,
i.e., pi = P( Ti |tpi| : νi) + P( Ti |tpi| : νi) ${p}_{i}=P\left({T}_{i}\ge |{t}_{{p}_{i}}|:{\nu }_{i}\right)+P\left({T}_{i}\le -|{t}_{{p}_{i}}|:{\nu }_{i}\right)$.
Constraint: tail(j) = 'L'${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$, 'U'$\text{'U'}$, 'C'$\text{'C'}$ or 'S'$\text{'S'}$, 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 Student's t$t$-distribution as defined by tail with pi = p(j)${p}_{i}={\mathbf{p}}\left(j\right)$, j = ((i1)  mod  lp) + 1.
Constraint: 0.0 < p(j) < 1.0$0.0<{\mathbf{p}}\left(\mathit{j}\right)<1.0$, for j = 1,2,,lp$\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
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 Student's t$t$-distribution with νi = df(j)${\nu }_{i}={\mathbf{df}}\left(j\right)$, j = ((i1)  mod  ldf) + 1.
Constraint: df(j)1.0${\mathbf{df}}\left(\mathit{j}\right)\ge 1.0$, 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:     t( : $:$) – double array
Note: the dimension of the array t must be at least max (ltail,lp,ldf)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
tpi${t}_{{p}_{i}}$, the deviates for the Student's t$t$-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 tpi${t}_{{p}_{i}}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, pi ≤ 0.0${p}_{i}\le 0.0$, or pi ≥ 1.0${p}_{i}\ge 1.0$.
ivalid(i) = 3${\mathbf{ivalid}}\left(i\right)=3$
 On entry, νi < 1.0${\nu }_{i}<1.0$.
ivalid(i) = 4${\mathbf{ivalid}}\left(i\right)=4$
The solution has failed to converge. 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

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. The error behaviour for various parameter values is discussed in Hill (1970).

The value tpi${t}_{{p}_{i}}$ 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

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

fprintf('\n     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%8d\n', p(mod(i,lp)+1), df(mod(i,ldf)+1), ...
tail{mod(i,ltail)+1}, x(i+1), ivalid(i+1));
end
```
```

P      DF    TAIL      X
0.010  20.000   S      2.845       0
0.010   7.500   L     -2.943       0
0.990  45.000   C      2.690       0

```
```function g01tb_example
p = [0.01; 0.01; 0.99];
df = [20; 7.5; 45];
tail = {'S'; 'L'; 'C'};
[x, ivalid, ifail] = g01tb(tail, p, df);

fprintf('\n     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%8d\n', p(mod(i,lp)+1), df(mod(i,ldf)+1), ...
tail{mod(i,ltail)+1}, x(i+1), ivalid(i+1));
end
```
```

P      DF    TAIL      X
0.010  20.000   S      2.845       0
0.010   7.500   L     -2.943       0
0.990  45.000   C      2.690       0

```