The routine may be called by the names g01fbf or nagf_stat_inv_cdf_students_t.
The deviate, associated with the lower tail probability, , of the Student's -distribution with degrees of freedom is defined as the solution to
For or the integral equation is easily solved for .
For other values of a transformation to the beta distribution is used and the result obtained from g01fef.
For an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's -distribution Comm. ACM13(10) 617–619
1: – Character(1)Input
On entry: indicates which tail the supplied probability represents.
The upper tail probability, i.e., .
The lower tail probability, i.e., .
The two tail (significance level) probability, i.e., .
The two tail (confidence interval) probability, i.e., .
, , or .
2: – Real (Kind=nag_wp)Input
On entry: , the probability from the required Student's -distribution as defined by tail.
3: – Real (Kind=nag_wp)Input
On entry: , the degrees of freedom of the Student's -distribution.
4: – 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 since useful values can be provided in some output arguments even when on exit. 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:
Note: in some cases g01fbf may return useful information.
if , or on exit, then g01fbf returns zero.
On entry, .
Constraint: , , or .
On entry, .
On entry, .
On entry, .
The solution has failed to converge. However, the result should be a reasonable approximation.
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 results should be accurate to five significant digits, for most argument values. The error behaviour for various argument values is discussed in Hill (1970).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01fbf is not threaded in any implementation.
The value may be calculated by using the transformation described in Section 3 and using g01fef. This routine allows you to set the required accuracy.
Internal changes have been made to this routine as follows:
At Mark 27: The algorithm underlying this routine was altered to improve the accuracy in cases where .
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.
This example reads the probability, the tail that probability represents and the degrees of freedom for a number of Student's -distributions and computes the corresponding deviates.