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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$t$-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$t$-distribution with νi${\nu }_{i}$ degrees of freedom, P( Ti ti : νi) $P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$ is defined by:
 ti P( Ti ≤ ti : νi) = ( Γ ((νi + 1) / 2) )/( sqrt(πνi) Γ(νi / 2) ) ∫ [1 + (Ti2)/(νi)] − (νi + 1) / 2 dTi,  νi ≥ 1. − ∞
$P( Ti ≤ ti :νi) = Γ ( (νi+1) / 2 ) πνi Γ(νi/2) ∫ -∞ ti [ 1+ Ti2νi ] -(νi+1) / 2 dTi , νi≥1 .$
Computationally, there are two situations:
(i) when νi < 20${\nu }_{i}<20$, a transformation of the beta distribution, Pβi( Bi βi : ai,bi) ${P}_{{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is used
 P( Ti ≤ ti : νi) = (1/2) Pβi( Bi ≤ (νi)/(νi + ti2) : νi / 2,(1/2))   when ​ ti < 0.0 $P( Ti ≤ ti :νi) = 12 Pβi( Bi≤ νi νi+ti2 :νi/2,12) when ​ ti<0.0$
or
 P( Ti ≤ ti : νi) = (1/2) + (1/2) Pβi( Bi ≥ (νi)/( νi + ti2 ) : νi / 2,(1/2))   when ​ ti > 0.0 ; $P( Ti ≤ ti :νi) = 12 + 12 Pβi( Bi ≥ νi νi + ti2 :νi/2,12) when ​ ti>0.0 ;$
(ii) when νi20${\nu }_{i}\ge 20$, 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 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
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 returned probabilities should represent. For j = ((i1)  mod  ltail) + 1 , for i = 1,2,,max (ltail,lt,ldf)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., pi = P( Ti ti : νi) ${p}_{i}=P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., pi = P( Ti ti : νi) ${p}_{i}=P\left({T}_{i}\ge {t}_{i}:{\nu }_{i}\right)$.
tail(j) = 'C'${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability is returned,
i.e., pi = P( Ti |ti| : νi) P( Ti |ti| : νi) ${p}_{i}=P\left({T}_{i}\le |{t}_{i}|:{\nu }_{i}\right)-P\left({T}_{i}\le -|{t}_{i}|:{\nu }_{i}\right)$.
tail(j) = 'S'${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned,
i.e., pi = P( Ti |ti| : νi) + P( Ti |ti| : νi) ${p}_{i}=P\left({T}_{i}\ge |{t}_{i}|:{\nu }_{i}\right)+P\left({T}_{i}\le -|{t}_{i}|:{\nu }_{i}\right)$.
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:     t(lt) – double array
lt, the dimension of the array, must satisfy the constraint lt > 0${\mathbf{lt}}>0$.
ti${t}_{i}$, the values of the Student's t$t$ variates with ti = t(j)${t}_{i}={\mathbf{t}}\left(j\right)$, j = ((i1)  mod  lt) + 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 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:     lt – int64int32nag_int scalar
Default: The dimension of the array t.
The length of the array t.
Constraint: lt > 0${\mathbf{lt}}>0$.
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,lt,ldf)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
pi${p}_{i}$, the probabilities 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,lt,ldf)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
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 pi${p}_{i}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, νi < 1.0${\nu }_{i}<1.0$.
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 or df was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ltail > 0${\mathbf{ltail}}>0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: lt > 0${\mathbf{lt}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: ldf > 0${\mathbf{ldf}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
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 1010${10}^{-10}$), 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

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

fprintf('\n    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   %5d\n', t(mod(i,lt)+1), df(mod(i,ldf)+1), ...
prob(i+1), tail{mod(i,ltail)+1}, ivalid(i+1));
end
```
```

T      DF     PROB  TAIL
0.850  20.000  0.7973  L       0
0.850  20.000  0.4054  S       0
0.850  20.000  0.5946  C       0
0.850  20.000  0.2027  U       0

```
```function g01sb_example
t = [0.85];
df = [20];
tail = {'L'; 'S'; 'C'; 'U'};
% calculate probability
[prob, ivalid, ifail] = g01sb(tail, t, df);

fprintf('\n    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   %5d\n', t(mod(i,lt)+1), df(mod(i,ldf)+1), ...
prob(i+1), tail{mod(i,ltail)+1}, ivalid(i+1));
end
```
```

T      DF     PROB  TAIL
0.850  20.000  0.7973  L       0
0.850  20.000  0.4054  S       0
0.850  20.000  0.5946  C       0
0.850  20.000  0.2027  U       0

```