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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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_prob_students_t (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 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 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