# NAG C Library Function Document

## 1Purpose

nag_prob_students_t_vector (g01sbc) returns a number of one or two tail probabilities for the Student's $t$-distribution with real degrees of freedom.

## 2Specification

 #include #include
 void nag_prob_students_t_vector (Integer ltail, const Nag_TailProbability tail[], Integer lt, const double t[], Integer ldf, const double df[], double p[], Integer ivalid[], NagError *fail)

## 3Description

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 Section 2.6 in the g01 Chapter Introduction for further information.

## 4References

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

## 5Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates which tail the returned probabilities should represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\ge {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\le \left|{t}_{i}\right|:{\nu }_{i}\right)-P\left({T}_{i}\le -\left|{t}_{i}\right|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
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: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_TwoTailSignif}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lt}$IntegerInput
On entry: the length of the array t.
Constraint: ${\mathbf{lt}}>0$.
4:    $\mathbf{t}\left[{\mathbf{lt}}\right]$const doubleInput
On entry: ${t}_{i}$, the values of the Student's $t$ variates with ${t}_{i}={\mathbf{t}}\left[j\right]$, .
5:    $\mathbf{ldf}$IntegerInput
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6:    $\mathbf{df}\left[{\mathbf{ldf}}\right]$const doubleInput
On entry: ${\nu }_{i}$, the degrees of freedom of the Student's $t$-distribution with ${\nu }_{i}={\mathbf{df}}\left[j\right]$, .
Constraint: ${\mathbf{df}}\left[\mathit{j}-1\right]\ge 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.
7:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
On exit: ${p}_{i}$, the probabilities for the Student's $t$ distribution.
8:    $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${\nu }_{i}<1.0$.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lt}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of tail or df was invalid.

## 7Accuracy

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).

## 8Parallelism and Performance

nag_prob_students_t_vector (g01sbc) is not threaded in any implementation.

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_prob_beta_vector (g01sec). This function allows you to set the required accuracy.

## 10Example

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.

### 10.1Program Text

Program Text (g01sbce.c)

### 10.2Program Data

Program Data (g01sbce.d)

### 10.3Program Results

Program Results (g01sbce.r)