# NAG FL Interfaceg01fbf (inv_​cdf_​students_​t)

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## 1Purpose

g01fbf returns the deviate associated with the given tail probability of Student's $t$-distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Function g01fbf ( tail, p, df,
 Real (Kind=nag_wp) :: g01fbf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p, df Character (1), Intent (In) :: tail
#include <nag.h>
 double g01fbf_ (const char *tail, const double *p, const double *df, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01fbf or nagf_stat_inv_cdf_students_t.

## 3Description

The deviate, ${t}_{p}$ associated with the lower tail probability, $p$, of the Student's $t$-distribution with $\nu$ degrees of freedom is defined as the solution to
 $P(T
For $\nu =1$ or $2$ the integral equation is easily solved for ${t}_{p}$.
For other values of $\nu <3$ a transformation to the beta distribution is used and the result obtained from g01fef.
For $\nu \ge 3$ an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).

## 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: $\mathbf{tail}$Character(1) Input
On entry: indicates which tail the supplied probability represents.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability, i.e., $P\left(T\ge {t}_{p}:\nu \right)$.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability, i.e., $P\left(T\le {t}_{p}:\nu \right)$.
${\mathbf{tail}}=\text{'S'}$
The two tail (significance level) probability, i.e., $P\left(T\ge |{t}_{p}|:\nu \right)+P\left(T\le -|{t}_{p}|:\nu \right)$.
${\mathbf{tail}}=\text{'C'}$
The two tail (confidence interval) probability, i.e., $P\left(T\le |{t}_{p}|:\nu \right)-P\left(T\le -|{t}_{p}|:\nu \right)$.
Constraint: ${\mathbf{tail}}=\text{'U'}$, $\text{'L'}$, $\text{'S'}$ or $\text{'C'}$.
2: $\mathbf{p}$Real (Kind=nag_wp) Input
On entry: $p$, the probability from the required Student's $t$-distribution as defined by tail.
Constraint: $0.0<{\mathbf{p}}<1.0$.
3: $\mathbf{df}$Real (Kind=nag_wp) Input
On entry: $\nu$, the degrees of freedom of the Student's $t$-distribution.
Constraint: ${\mathbf{df}}\ge 1.0$.
4: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g01fbf may return useful information.
if ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$ on exit, then g01fbf returns zero.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}<1.0$.
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}>0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df}}\ge 1.0$.
${\mathbf{ifail}}=5$
The solution has failed to converge. However, the result should be a reasonable approximation.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 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

g01fbf is not threaded in any implementation.

The value ${t}_{p}$ may be calculated by using the transformation described in Section 3 and using g01fef. This routine allows you to set the required accuracy.

### 9.1Internal Changes

Internal changes have been made to this routine as follows:
• At Mark 27: The algorithm underlying this routine was altered to improve the accuracy in cases where ${\mathbf{df}}<3$.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

## 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 (g01fbfe.f90)

### 10.2Program Data

Program Data (g01fbfe.d)

### 10.3Program Results

Program Results (g01fbfe.r)