# NAG CL Interfaceg01tbc (inv_​cdf_​students_​t_​vector)

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## 1Purpose

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

## 2Specification

 #include
 void g01tbc (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double t[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01tbc, nag_stat_inv_cdf_students_t_vector or nag_deviates_students_t_vector.

## 3Description

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 ${\nu }_{i}=1$ 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 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.

## 4References

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: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbability Input
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., ${p}_{i}=P\left({T}_{i}\le |{t}_{{p}_{i}}|:{\nu }_{i}\right)-P\left({T}_{i}\le -|{t}_{{p}_{i}}|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability,
i.e., ${p}_{i}=P\left({T}_{i}\ge |{t}_{{p}_{i}}|:{\nu }_{i}\right)+P\left({T}_{i}\le -|{t}_{{p}_{i}}|:{\nu }_{i}\right)$.
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: $\mathbf{lp}$Integer Input
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4: $\mathbf{p}\left[{\mathbf{lp}}\right]$const double Input
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: $\mathbf{ldf}$Integer Input
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6: $\mathbf{df}\left[{\mathbf{ldf}}\right]$const double Input
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: $\mathbf{t}\left[\mathit{dim}\right]$double Output
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: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: ${\mathbf{lp}}>0$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.

## 7Accuracy

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).

## 8Parallelism and Performance

g01tbc is not threaded in any implementation.

The value ${t}_{{p}_{i}}$ may be calculated by using a transformation to the beta distribution and calling g01tec. This function allows you to set the required accuracy.

## 10Example

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.

### 10.1Program Text

Program Text (g01tbce.c)

### 10.2Program Data

Program Data (g01tbce.d)

### 10.3Program Results

Program Results (g01tbce.r)