(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 Arguments

1: $t$ – double Input: On entry: $t$ , the deviate from the Student's $t$ -distribution with $ν$ degrees of freedom.
2: $df$ – double Input: On entry: $ν$ , the degrees of freedom of the Student's $t$ -distribution.

Constraint: $df \geq 1.0$ .
3: $delta$ – double Input: On entry: $δ$ , the noncentrality parameter of the Students $t$ -distribution.
4: $tol$ – double Input: On entry: the absolute accuracy required by you in the results. If g01gbc is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see X02AJC), the value of $10 \times machine precision$ is used instead.
5: $max_iter$ – Integer Input: On entry: the maximum number of terms that are used in each of the summations.

Suggested value: $100$ . See Section 9 for further comments.

Constraint: $max_iter \geq 1$ .
6: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit

fail . code =

NE_NOERROR, then g01gbc returns

0.0

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INT_ARG_LT: On entry, $max_iter = ⟨ value ⟩$ .
Constraint: $max_iter \geq 1$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PROB_LIMIT: The probability is too close to $0$ or $1$ . The returned value should be a reasonable estimate of the true value.
NE_PROBABILITY: Unable to calculate the probability as it is too close to zero or one.
NE_REAL_ARG_LT: On entry, $df = ⟨ value ⟩$ .
Constraint: $df \geq 1.0$ .
NE_SERIES: One of the series has failed to converge with $max_iter = ⟨ value ⟩$ and $tol = ⟨ value ⟩$ . Reconsider the requested tolerance and/or the maximum number of iterations.

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 Parallelism and Performance

g01gbc is not threaded in any implementation.

9 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 g01gdc.

Note that g01gbc only allows degrees of freedom greater than or equal to

1

although values between

0

and

1

are theoretically possible.

10 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.

g01gb: FL CL CPP AD

NAG CL Interfaceg01gbc (prob_​students_​t_​noncentral)

▸▿ 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 CL Interface
g01gbc (prob_students_t_noncentral)