NAG FL Interface
g01fbf (inv_​cdf_​students_​t)

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1 Purpose

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

2 Specification

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 <nag.h>
double  g01fbf_ (const char *tail, const double *p, const double *df, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01fbf or nagf_stat_inv_cdf_students_t.

3 Description

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
P(T<tp:ν)=p=Γ((ν+1)/2) νπΓ(ν/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).

4 References

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

5 Arguments

1: tail Character(1) Input
On entry: indicates which tail the supplied probability represents.
The upper tail probability, i.e., P(Ttp:ν).
The lower tail probability, i.e., P(Ttp:ν).
The two tail (significance level) probability, i.e., P(T|tp|:ν)+P(T-|tp|:ν).
The two tail (confidence interval) probability, i.e., P(T|tp|:ν)-P(T-|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 Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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).

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).
Errors or warnings detected by the routine:
Note: in some cases g01fbf may return useful information.
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 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.

7 Accuracy

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).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01fbf is not threaded in any implementation.

9 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.

9.1 Internal Changes

Internal changes have been made to this routine as follows:
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

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.

10.1 Program Text

Program Text (g01fbfe.f90)

10.2 Program Data

Program Data (g01fbfe.d)

10.3 Program Results

Program Results (g01fbfe.r)