(i)when $ν < 20$ , a transformation of the beta distribution, $P_{β} (B \leq β : a, b)$ is used

$P (T \leq t : ν) = \frac{1}{2} P_{β} (B \leq \frac{ν}{ν + t^{2}} : ν / 2, \frac{1}{2}) when t < 0.0$

or

$P (T \leq t : ν) = \frac{1}{2} + \frac{1}{2} P_{β} (B \geq \frac{ν}{ν + t^{2}} : ν / 2, \frac{1}{2}) when t > 0.0;$
(ii)when $ν \geq 20$ , 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 Arguments

1: $tail$ – Character(1) Input

On entry: indicates which tail the returned probability should represent.

$tail ='U'$: The upper tail probability is returned, i.e., $P (T \geq t : ν)$ .
$tail ='S'$: The two tail (significance level) probability is returned, i.e., $P (T \geq | t | : ν) + P (T \leq - | t | : ν)$ .
$tail ='C'$: The two tail (confidence interval) probability is returned, i.e., $P (T \leq | t | : ν) - P (T \leq - | t | : ν)$ .
$tail ='L'$: The lower tail probability is returned, i.e., $P (T \leq t : ν)$ .

Constraint:

tail ='U'

'S'

'C'

'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:

ν

, the degrees of freedom of the Student's

t

-distribution.

Constraint:

df \geq 1.0

4: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

ifail \neq 0

, then g01ebf returns

0.0

$ifail = 1$: On entry, $tail = ⟨ value ⟩$ .
Constraint: $tail ='L'$ , $'U'$ , $'S'$ or $'C'$ .

$ifail = 2$: On entry, $df = ⟨ value ⟩$ .
Constraint: $df \geq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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).

8 Parallelism and Performance

g01ebf is not threaded in any implementation.

9 Further Comments

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.

10 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.

g01eb: FL CL CPP AD

NAG FL Interfaceg01ebf (prob_​students_​t)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g01ebf (prob_students_t)