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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_students_t_vector (g01sb)

## Purpose

nag_stat_prob_students_t_vector (g01sb) returns a number of one or two tail probabilities for the Student's $t$-distribution with real degrees of freedom.

## Syntax

[p, ivalid, ifail] = g01sb(tail, t, df, 'ltail', ltail, 'lt', lt, 'ldf', ldf)
[p, ivalid, ifail] = nag_stat_prob_students_t_vector(tail, t, df, 'ltail', ltail, 'lt', lt, 'ldf', ldf)

## Description

The lower tail probability for the Student's $t$-distribution with ${\nu }_{i}$ degrees of freedom, $P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$ is defined by:
 $P Ti ≤ ti :νi = Γ νi+1 / 2 πνi Γνi/2 ∫ -∞ ti 1+ Ti2νi -νi+1 / 2 dTi , νi≥1 .$
Computationally, there are two situations:
(i) when ${\nu }_{i}<20$, a transformation of the beta distribution, ${P}_{{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is used
 $P Ti ≤ ti :νi = 12 Pβi Bi≤ νi νi+ti2 :νi/2,12 when ​ ti<0.0$
or
 $P Ti ≤ ti :νi = 12 + 12 Pβi Bi ≥ νi νi + ti2 :νi/2,12 when ​ ti>0.0 ;$
(ii) when ${\nu }_{i}\ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see 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

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
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 returned probabilities should represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\ge {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability is returned,
i.e., ${p}_{i}=P\left({T}_{i}\le \left|{t}_{i}\right|:{\nu }_{i}\right)-P\left({T}_{i}\le -\left|{t}_{i}\right|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned,
i.e., ${p}_{i}=P\left({T}_{i}\ge \left|{t}_{i}\right|:{\nu }_{i}\right)+P\left({T}_{i}\le -\left|{t}_{i}\right|:{\nu }_{i}\right)$.
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:     $\mathrm{t}\left({\mathbf{lt}}\right)$ – double array
${t}_{i}$, the values of the Student's $t$ variates with ${t}_{i}={\mathbf{t}}\left(j\right)$, .
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:     $\mathrm{lt}$int64int32nag_int scalar
Default: the dimension of the array t.
The length of the array t.
Constraint: ${\mathbf{lt}}>0$.
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:     $\mathrm{p}\left(:\right)$ – double array
The dimension of the array p will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$
${p}_{i}$, the probabilities for the Student's $t$ distribution.
2:     $\mathrm{ivalid}\left(:\right)$int64int32nag_int array
The dimension of the array ivalid will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$
${\mathbf{ivalid}}\left(i\right)$ 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 ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${\nu }_{i}<1.0$.
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 or df was invalid.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lt}}>0$.
${\mathbf{ifail}}=4$
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 computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than ${10}^{-10}$), see Hastings and Peacock (1975).

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_stat_prob_beta_vector (g01se). This function allows you to set the required accuracy.

## Example

This example reads values from, and degrees of freedom for Student's $t$-distributions along with the required tail. The probabilities are calculated and printed.
```function g01sb_example

fprintf('g01sb example results\n\n');

t    = [0.85];
df   = [20];
tail = {'L'; 'S'; 'C'; 'U'};
% calculate probability
[prob, ivalid, ifail] = g01sb(...
tail, t, df);

fprintf('    t      df     prob  tail\n');
lt    = numel(t);
ldf   = numel(df);
ltail = numel(tail);
len   = max ([lt, ldf, ltail]);
for i=0:len-1
fprintf('%7.3f%8.3f%8.4f  %c\n', t(mod(i,lt)+1), df(mod(i,ldf)+1), ...
prob(i+1), tail{mod(i,ltail)+1});
end

```
```g01sb example results

t      df     prob  tail
0.850  20.000  0.7973  L
0.850  20.000  0.4054  S
0.850  20.000  0.5946  C
0.850  20.000  0.2027  U
```