Integer, Intent (In)	::	ltail, lt, ldf
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(*)
Real (Kind=nag_wp), Intent (In)	::	t(lt), df(ldf)
Real (Kind=nag_wp), Intent (Out)	::	p(*)
Character (1), Intent (In)	::	tail(ltail)

C Header Interface

#include <nag.h>

void	g01sbf_ (const Integer ltail, const char tail[], const Integer lt, const double t[], const Integer ldf, const double df[], double p[], Integer ivalid[], Integer ifail, const Charlen length_tail)

The routine may be called by the names g01sbf or nagf_stat_prob_students_t_vector.

3 Description

The lower tail probability for the Student's

t

-distribution with

ν_{i}

degrees of freedom,

P (T_{i} \leq t_{i} : ν_{i})

is defined by:

P (T_{i} \leq t_{i} : ν_{i}) = \frac{Γ ((ν_{i} + 1) / 2)}{\sqrt{π ν_{i}} Γ (ν_{i} / 2)} \int_{- \infty}^{t_{i}} {[1 + \frac{{T_{i}}^{2}}{ν_{i}}]}^{- (ν_{i} + 1) / 2} d T_{i}, ν_{i} \geq 1 .

Computationally, there are two situations:

(i)when $ν_{i} < 20$ , a transformation of the beta distribution, $P_{β_{i}} (B_{i} \leq β_{i} : a_{i}, b_{i})$ is used

$P (T_{i} \leq t_{i} : ν_{i}) = \frac{1}{2} P_{β_{i}} (B_{i} \leq \frac{ν_{i}}{ν_{i} + {t_{i}}^{2}} : ν_{i} / 2, \frac{1}{2}) when t_{i} < 0.0$

or

$P (T_{i} \leq t_{i} : ν_{i}) = \frac{1}{2} + \frac{1}{2} P_{β_{i}} (B_{i} \geq \frac{ν_{i}}{ν_{i} + {t_{i}}^{2}} : ν_{i} / 2, \frac{1}{2}) when t_{i} > 0.0;$
(ii)when $ν_{i} \geq 20$ , an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

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: $ltail$ – Integer Input

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) array Input

On entry: indicates which tail the returned probabilities should represent. For

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, lt, ldf)

$tail (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (T_{i} \leq t_{i} : ν_{i})$ .
$tail (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (T_{i} \geq t_{i} : ν_{i})$ .
$tail (j) ='C'$: The two tail (confidence interval) probability is returned, i.e., $p_{i} = P (T_{i} \leq | t_{i} | : ν_{i}) - P (T_{i} \leq - | t_{i} | : ν_{i})$ .
$tail (j) ='S'$: The two tail (significance level) probability is returned, i.e., $p_{i} = P (T_{i} \geq | t_{i} | : ν_{i}) + P (T_{i} \leq - | t_{i} | : ν_{i})$ .

Constraint:

tail (j) ='L'

'U'

'C'

'S'

, for

j = 1, 2, \dots, ltail

3: $lt$ – Integer Input

On entry: the length of the array t.

Constraint:

lt > 0

4: $t (lt)$ – Real (Kind=nag_wp) array Input

On entry:

t_{i}

, the values of the Student's

t

variates with

t_{i} = t (j)

j = ((i - 1) mod lt) + 1

5: $ldf$ – Integer Input

On entry: the length of the array df.

Constraint:

ldf > 0

6: $df (ldf)$ – Real (Kind=nag_wp) array Input

On entry:

ν_{i}

, the degrees of freedom of the Student's

t

-distribution with

ν_{i} = df (j)

j = ((i - 1) mod ldf) + 1

Constraint:

df (j) \geq 1.0

, for

j = 1, 2, \dots, ldf

7: $p (*)$ – Real (Kind=nag_wp) array Output

Note: the dimension of the array p must be at least

\max (ltail, lt, ldf)

On exit:

p_{i}

, the probabilities for the Student's

t

distribution.

8: $ivalid (*)$ – Integer array Output

Note: the dimension of the array ivalid must be at least

\max (ltail, lt, ldf)

On exit:

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: On entry, invalid value supplied in tail when calculating $p_{i}$ .
$ivalid (i) = 2$: On entry, $ν_{i} < 1.0$ .

9: $ifail$ – 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:

$ifail = 1$: On entry, at least one value of tail or df was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $array size = ⟨ value ⟩$ .
Constraint: $ltail > 0$ .

$ifail = 3$: On entry, $array size = ⟨ value ⟩$ .
Constraint: $lt > 0$ .

$ifail = 4$: On entry, $array size = ⟨ value ⟩$ .
Constraint: $ldf > 0$ .

$ifail = - 99$: 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

Background information to multithreading can be found in the Multithreading documentation.

g01sbf 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 g01sef. 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.

g01sb: FL CL CPP AD PY MB

NAG FL Interfaceg01sbf (prob_​students_​t_​vector)

▸▿ 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
g01sbf (prob_students_t_vector)