NAG Library Function Document

nag_prob_students_t (g01ebc)

+− 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

1 Purpose

nag_prob_students_t (g01ebc) returns the lower tail, upper tail or two tail probability for the Student's

t

-distribution with real degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_students_t (Nag_TailProbability tail, double t, double df, NagError *fail)

3 Description

The lower tail probability for the Student's

t

-distribution with

ν

degrees of freedom,

P (T \leq t : ν)

is defined by:

P (T \leq t : ν) = \frac{Γ ((ν + 1) / 2)}{\sqrt{π ν} Γ (ν / 2)} \int_{- \infty}^{t} {[1 + \frac{T^{2}}{ν}]}^{- (ν + 1) / 2} d T, ν \geq 1 .

Computationally, there are two situations:

(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

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_TailProbabilityInput

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 – doubleInput

On entry:

t

, the value of the Student's

t

variate.

3: df – doubleInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

Not applicable.

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 nag_prob_beta_dist (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.

NAG Library Function Documentnag_prob_students_t (g01ebc)