NAG Library Routine Document

G01SBF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01SBF returns a number of one or two tail probabilities for the Student's

t

-distribution with real degrees of freedom.

2 Specification

SUBROUTINE G01SBF ( LTAIL, TAIL, LT, T, LDF, DF, P, IVALID, IFAIL)

INTEGER LTAIL, LT, LDF, IVALID(*), IFAIL

REAL (KIND=nag_wp) T(LT), DF(LDF), P(*)

CHARACTER(1) TAIL(LTAIL)

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

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 parameters 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 Parameters

1: LTAIL – INTEGERInput

On entry: the length of the array TAIL.

Constraint:

LTAIL > 0

2: TAIL(LTAIL) – CHARACTER(1) arrayInput

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 – INTEGERInput

On entry: the length of the array T.

Constraint:

LT > 0

4: T(LT) – REAL (KIND=nag_wp) arrayInput

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 – INTEGERInput

On entry: the length of the array DF.

Constraint:

LDF > 0

6: DF(LDF) – REAL (KIND=nag_wp) arrayInput

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) arrayOutput

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 arrayOutput

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 – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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: $LDF > 0$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

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 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.

9 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.

NAG Library Routine DocumentG01SBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01SBF