NAG Library Routine Document

g01fbf (inv_cdf_students_t)


g01fbf returns the deviate associated with the given tail probability of Student's t-distribution with real degrees of freedom.


Fortran Interface
Function g01fbf ( tail, p, df, ifail)
Real (Kind=nag_wp):: g01fbf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: p, df
Character (1), Intent (In):: tail
C Header Interface
#include <nagmk26.h>
double  g01fbf_ (const char *tail, const double *p, const double *df, Integer *ifail, const Charlen length_tail)


The deviate, tp associated with the lower tail probability, p, of the Student's t-distribution with ν degrees of freedom is defined as the solution to
PT<tp:ν=p=Γν+1/2 νπΓν/2 -tp 1+T2ν -ν+1/2dT,  ν1; ​-<tp<.  
For ν=1 or 2 the integral equation is easily solved for tp.
For other values of ν<3 a transformation to the beta distribution is used and the result obtained from g01fef.
For ν3 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 t-distribution Comm. ACM 13(10) 617–619


1:     tail – Character(1)Input
On entry: indicates which tail the supplied probability represents.
The upper tail probability, i.e., PTtp:ν.
The lower tail probability, i.e., PTtp:ν.
The two tail (significance level) probability, i.e., PTtp:ν+PT-tp:ν.
The two tail (confidence interval) probability, i.e., PTtp:ν-PT-tp:ν.
Constraint: tail='U', 'L', 'S' or 'C'.
2:     p – Real (Kind=nag_wp)Input
On entry: p, the probability from the required Student's t-distribution as defined by tail.
Constraint: 0.0<p<1.0.
3:     df – Real (Kind=nag_wp)Input
On entry: ν, the degrees of freedom of the Student's t-distribution.
Constraint: df1.0.
4:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ifail0 on exit, the recommended value is -1. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01fbf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If ifail=1, 2 or 3 on exit, then g01fbf returns zero.
On entry, tail=value.
Constraint: tail='L', 'U', 'S' or 'C'.
On entry, p=value.
Constraint: p<1.0.
On entry, p=value.
Constraint: p>0.0.
On entry, df=value.
Constraint: df1.0.
The solution has failed to converge. However, the result should be a reasonable approximation.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 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 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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).

Parallelism and Performance

g01fbf is not threaded in any implementation.

Further Comments

The value tp may be calculated by using the transformation described in Section 3 and using g01fef. This routine allows you to set the required accuracy.


This example reads the probability, the tail that probability represents and the degrees of freedom for a number of Student's t-distributions and computes the corresponding deviates.

Program Text

Program Text (g01fbfe.f90)

Program Data

Program Data (g01fbfe.d)

Program Results

Program Results (g01fbfe.r)