The routine may be called by the names g01gbf or nagf_stat_prob_students_t_noncentral.
The lower tail probability of the noncentral Student's -distribution with degrees of freedom and noncentrality parameter , , is defined by
The probability is computed in one of two ways.
(i)When , 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 -distributions Biometrika51 451–458
1: – Real (Kind=nag_wp)Input
On entry: , the deviate from the Student's -distribution with degrees of freedom.
2: – Real (Kind=nag_wp)Input
On entry: , the degrees of freedom of the Student's -distribution.
3: – Real (Kind=nag_wp)Input
On entry: , the noncentrality parameter of the Students -distribution.
4: – 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 or less than (see x02ajf), the value of is used instead.
5: – IntegerInput
On entry: the maximum number of terms that are used in each of the summations.
. See Section 9 for further comments.
6: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , 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 , then g01gbf returns .
On entry, .
On entry, .
One of the series has failed to converge with and . Reconsider the requested tolerance and/or the maximum number of iterations.
The probability is too close to or . 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.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01gbf is not threaded in any implementation.
The rate of convergence of the series depends, in part, on the quantity . The smaller this quantity the faster the convergence. Thus for large and small the convergence may be slow. If 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 -distribution to the -distribution can be used:
Note that g01gbf only allows degrees of freedom greater than or equal to although values between and are theoretically possible.
This example reads values from, and degrees of freedom for, and noncentrality parameters of the noncentral Student's -distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.