G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01EBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01EBF returns the lower tail, upper tail or two tail probability for the Student's $t$-distribution with real degrees of freedom, via the routine name.

## 2  Specification

 FUNCTION G01EBF ( TAIL, T, DF, IFAIL)
 REAL (KIND=nag_wp) G01EBF
 INTEGER IFAIL REAL (KIND=nag_wp) T, DF CHARACTER(1) TAIL

## 3  Description

The lower tail probability for the Student's $t$-distribution with $\nu$ degrees of freedom, $P\left(T\le t:\nu \right)$ is defined by:
 $P T≤t:ν = Γ ν+1 / 2 πν Γν/2 ∫ -∞ t 1+ T2ν -ν+1 / 2 dT , ν≥1 .$
Computationally, there are two situations:
(i) when $\nu <20$, a transformation of the beta distribution, ${P}_{\beta }\left(B\le \beta :a,b\right)$ is used
 $P T≤t:ν = 12 Pβ B≤ ν ν+t2 : ν/2, 12 when ​ t<0.0$
or
 $P T≤t:ν = 12 + 12 Pβ B≥ ν ν+t2 : ν/2, 12 when ​ t>0.0 ;$
(ii) when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

## 4  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 $t$-distribution Comm. ACM 13(10) 617–619

## 5  Parameters

1:     TAIL – CHARACTER(1)Input
On entry: indicates which tail the returned probability should represent.
${\mathbf{TAIL}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(T\ge t:\nu \right)$.
${\mathbf{TAIL}}=\text{'S'}$
The two tail (significance level) probability is returned, i.e., $P\left(T\ge \left|t\right|:\nu \right)+P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{TAIL}}=\text{'C'}$
The two tail (confidence interval) probability is returned, i.e., $P\left(T\le \left|t\right|:\nu \right)-P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{TAIL}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(T\le t:\nu \right)$.
Constraint: ${\mathbf{TAIL}}=\text{'U'}$, $\text{'S'}$, $\text{'C'}$ or $\text{'L'}$.
2:     T – REAL (KIND=nag_wp)Input
On entry: $t$, the value of the Student's $t$ variate.
3:     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:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
If ${\mathbf{IFAIL}}\ne {\mathbf{0}}$, then G01EBF returns $0.0$.
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{TAIL}}\ne \text{'U'}$, $\text{'S'}$, $\text{'C'}$ or $\text{'L'}$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{DF}}<1.0$.

## 7  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 ${10}^{-10}$), 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 G01EEF. This routine allows you to set the required accuracy.

## 9  Example

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 until the end of data is reached.

### 9.1  Program Text

Program Text (g01ebfe.f90)

### 9.2  Program Data

Program Data (g01ebfe.d)

### 9.3  Program Results

Program Results (g01ebfe.r)