g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_deviates_students_t_vector (g01tbc)

## 1  Purpose

nag_deviates_students_t_vector (g01tbc) returns a number of deviates associated with given probabilities of Student's $t$-distribution with real degrees of freedom.

## 2  Specification

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

## 3  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_deviates_beta (g01fec).
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 Section 2.6 in the g01 Chapter Introduction for further information.

## 4  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

## 5  Arguments

1:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     tail[${\mathbf{ltail}}$]const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left({T}_{i}\le {t}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left({T}_{i}\ge {t}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability,
i.e., ${p}_{i}=P\left({T}_{i}\le \left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)-P\left({T}_{i}\le -\left|{t}_{{p}_{i}}\right|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
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: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_TwoTailSignif}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:     lpIntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:     p[lp]const doubleInput
On entry: ${p}_{i}$, the probability of the required Student's $t$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{p}}\left[\mathit{j}-1\right]<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
5:     ldfIntegerInput
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6:     df[ldf]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:     t[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
On exit: ${t}_{{p}_{i}}$, the deviates for the Student's $t$-distribution.
8:     ivalid[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
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 ${t}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${p}_{i}\le 0.0$, or ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${\nu }_{i}<1.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
The solution has failed to converge. The result returned should represent an approximation to the solution.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
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.
NW_IVALID
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.

## 7  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_deviates_beta_vector (g01tec). This function allows you to set the required accuracy.

## 9  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.

### 9.1  Program Text

Program Text (g01tbce.c)

### 9.2  Program Data

Program Data (g01tbce.d)

### 9.3  Program Results

Program Results (g01tbce.r)