The routine may be called by the names g01ebf or nagf_stat_prob_students_t.
The lower tail probability for the Student's -distribution with degrees of freedom, is defined by:
Computationally, there are two situations:
(i)when , a transformation of the beta distribution, is used
(ii)when , an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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 returned probability should represent.
The upper tail probability is returned, i.e., .
The two tail (significance level) probability is returned, i.e., .
The two tail (confidence interval) probability is returned, i.e., .
The lower tail probability is returned, i.e., .
, , or .
2: – Real (Kind=nag_wp)Input
On entry: , the value of the Student's variate.
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. 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 , then g01ebf returns .
On entry, .
Constraint: , , or .
On entry, .
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 computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than ), see Hastings and Peacock (1975).
8Parallelism and Performance
g01ebf is not threaded in any implementation.
The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using g01eef. This routine allows you to set the required accuracy.
This example reads values from, and degrees of freedom for Student's -distributions along with the required tail. The probabilities are calculated and printed until the end of data is reached.