(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: $tail$ – Nag_TailProbability Input

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

$tail = Nag_UpperTail$: The upper tail probability is returned, i.e., $P (T \geq t : ν)$ .
$tail = Nag_TwoTailSignif$: The two tail (significance level) probability is returned, i.e., $P (T \geq | t | : ν) + P (T \leq - | t | : ν)$ .
$tail = Nag_TwoTailConfid$: The two tail (confidence interval) probability is returned, i.e., $P (T \leq | t | : ν) - P (T \leq - | t | : ν)$ .
$tail = Nag_LowerTail$: The lower tail probability is returned, i.e., $P (T \leq t : ν)$ .

Constraint:

tail = Nag_UpperTail

Nag_TwoTailSignif

Nag_TwoTailConfid

Nag_LowerTail

2: $t$ – double Input

On entry:

t

, the value of the Student's

t

variate.

3: $df$ – double Input

On entry:

ν

, the degrees of freedom of the Student's

t

-distribution.

Constraint:

df \geq 1.0

4: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

fail . code =

NE_NOERROR, then g01ebc returns

0.0

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_BAD_PARAM: On entry, argument $⟨ value ⟩$ 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.
NE_REAL_ARG_LT: On entry, $df = ⟨ value ⟩$ .
Constraint: $df \geq 1.0$ .

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).

8 Parallelism and Performance

g01ebc 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 g01eec. This function 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 CL Interfaceg01ebc (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 CL Interface
g01ebc (prob_students_t)