NAG C Library Function Document

nag_prob_non_central_students_t (g01gbc)


nag_prob_non_central_students_t (g01gbc) returns the lower tail probability for the noncentral Student's t-distribution.


#include <nag.h>
#include <nagg01.h>
double  nag_prob_non_central_students_t (double t, double df, double delta, double tol, Integer max_iter, NagError *fail)


The lower tail probability of the noncentral Student's t-distribution with ν degrees of freedom and noncentrality parameter δ, PTt:ν;δ, is defined by
PTt:ν;δ=Cν0 12π- αu-δe-x2/2dx uν-1e-u2/2du,  ν>0.0  
Cν=1Γ 12ν 2ν- 2/2 ,   α=tν.  
The probability is computed in one of two ways.
(i) When t=0.0, the relationship to the normal is used:
(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.


Amos D E (1964) Representations of the central and non-central t-distributions Biometrika 51 451–458


1:     t doubleInput
On entry: t, the deviate from the Student's t-distribution with ν degrees of freedom.
2:     df doubleInput
On entry: ν, the degrees of freedom of the Student's t-distribution.
Constraint: df1.0.
3:     delta doubleInput
On entry: δ, the noncentrality parameter of the Students t-distribution.
4:     tol doubleInput
On entry: the absolute accuracy required by you in the results. If nag_prob_non_central_students_t (g01gbc) is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see nag_machine_precision (X02AJC)), the value of 10×machine precision is used instead.
5:     max_iter IntegerInput
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_iter1.
6:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, max_iter=value.
Constraint: max_iter1.
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.
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.
The probability is too close to 0 or 1. The returned value should be a reasonable estimate of the true value.
Unable to calculate the probability as it is too close to zero or one.
On entry, df=value.
Constraint: df1.0.
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.


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.

Parallelism and Performance

nag_prob_non_central_students_t (g01gbc) is not threaded in any implementation.

Further Comments

The rate of convergence of the series depends, in part, on the quantity t2/t2+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=T2,λ=δ2,ν1=1  and  ν2=ν,  
and a call made to nag_prob_non_central_f_dist (g01gdc).
Note that nag_prob_non_central_students_t (g01gbc) only allows degrees of freedom greater than or equal to 1 although values between 0 and 1 are theoretically possible.


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.

Program Text

Program Text (g01gbce.c)

Program Data

Program Data (g01gbce.d)

Program Results

Program Results (g01gbce.r)