NAG FL Interfaceg01ebf (prob_​students_​t)

1Purpose

g01ebf returns the lower tail, upper tail or two tail probability for the Student's $t$-distribution with real degrees of freedom.

2Specification

Fortran Interface
 Function g01ebf ( tail, t, df,
 Real (Kind=nag_wp) :: g01ebf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: t, df Character (1), Intent (In) :: tail
#include <nag.h>
 double g01ebf_ (const char *tail, const double *t, const double *df, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01ebf or nagf_stat_prob_students_t.

3Description

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:
1. (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
 $P T≤t:ν = 12 + 12 Pβ B≥ ν ν+t2 : ν/2, 12 when ​ t>0.0 ;$
2. (ii)when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

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: $\mathbf{tail}$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: $\mathbf{t}$Real (Kind=nag_wp) Input
On entry: $t$, the value of the Student's $t$ variate.
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 $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value 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:
If ${\mathbf{ifail}}\ne {\mathbf{0}}$, then g01ebf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1.0$.
${\mathbf{ifail}}=-99$
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 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

g01ebf 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 g01eef. This routine 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 until the end of data is reached.

10.1Program Text

Program Text (g01ebfe.f90)

10.2Program Data

Program Data (g01ebfe.d)

10.3Program Results

Program Results (g01ebfe.r)