NAG Library Routine Document

G01GBF

+− 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

G01GBF returns the lower tail probability for the noncentral Student's

t

-distribution, via the routine name.

2 Specification

FUNCTION G01GBF (

T, DF, DELTA, TOL, MAXIT, IFAIL)

REAL (KIND=nag_wp) G01GBF

INTEGER	MAXIT, IFAIL
REAL (KIND=nag_wp)	T, DF, DELTA, TOL

3 Description

The lower tail probability of the noncentral Student's

t

-distribution with

ν

degrees of freedom and noncentrality parameter

δ

P (T \leq t : ν; δ)

, is defined by

P (T \leq t : ν; δ) = C_{ν} \int_{0}^{\infty} (\frac{1}{\sqrt{2 π}} \int_{- \infty}^{α u - δ} e^{- x^{2} / 2} d x) u^{ν - 1} e^{- u^{2} / 2} d u, ν > 0.0

with

C_{ν} = \frac{1}{Γ (\frac{1}{2} ν) 2^{(ν - 2) / 2}}, α = \frac{t}{\sqrt{ν}} .

The probability is computed in one of two ways.

(i)

When

t = 0.0

, the relationship to the normal is used:

P (T \leq t : ν; δ) = \frac{1}{\sqrt{2 π}} \int_{δ}^{\infty} e^{- u^{2} / 2} d u .

(ii)

Otherwise the series expansion described in Equation 9 of Amos (1964) is used. This involves the sums of confluent hypergeometric functions, the terms of which are computed using recurrence relationships.

4 References

Amos D E (1964) Representations of the central and non-central

t

-distributions Biometrika 51 451–458

5 Parameters

1: T – REAL (KIND=nag_wp)Input: On entry: $t$ , the deviate from the Student's $t$ -distribution with $ν$ degrees of freedom.
2: DF – REAL (KIND=nag_wp)Input: On entry: $ν$ , the degrees of freedom of the Student's $t$ -distribution.
Constraint: $DF \geq 1.0$ .
3: DELTA – REAL (KIND=nag_wp)Input: On entry: $δ$ , the noncentrality parameter of the Students $t$ -distribution.
4: TOL – REAL (KIND=nag_wp)Input: On entry: the absolute accuracy required by you in the results. If G01GBF is entered with TOL greater than or equal to $1.0$ or less than $10 \times machine precision$ (see X02AJF), then the value of $10 \times machine precision$ is used instead.
5: MAXIT – INTEGERInput: On entry: the maximum number of terms that are used in each of the summations.

Suggested value: $100$ . See Section 8 for further comments.
Constraint: $MAXIT \geq 1$ .
6: 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:

If on exit

IFAIL \neq 0

, then G01GBF returns

0.0

$IFAIL = 1$

On entry,

DF < 1.0

$IFAIL = 2$

On entry,

MAXIT < 1

$IFAIL = 3$: One of the series has failed to converge. Reconsider the requested tolerance and/or maximum number of iterations.

$IFAIL = 4$: The probability is too small to calculate accurately.

7 Accuracy

The series described in Amos (1964) are summed until an estimated upper bound on the contribution of future terms to the probability is less than TOL. There may also be some loss of accuracy due to calculation of gamma functions.

8 Further Comments

The rate of convergence of the series depends, in part, on the quantity

t^{2} / (t^{2} + ν)

. The smaller this quantity the faster the convergence. Thus for large

t

and small

ν

the convergence may be slow. If

ν

is an integer then one of the series to be summed is of finite length.

If two tail probabilities are required then the relationship of the

t

-distribution to the

F

-distribution can be used:

F = T^{2}, λ = δ^{2}, ν_{1} = 1 and ν_{2} = ν,

and a call made to G01GDF.

Note that G01GBF only allows degrees of freedom greater than or equal to

1

although values between

0

and

1

are theoretically possible.

9 Example

This example reads values from, and degrees of freedom for, and noncentrality parameters of the noncentral Student's

t

-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

NAG Library Routine DocumentG01GBF