NAG FL Interface
g01tbf (inv_​cdf_​students_​t_​vector)

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1 Purpose

g01tbf returns a number of deviates associated with given probabilities of Student's t-distribution with real degrees of freedom.

2 Specification

Fortran Interface
Subroutine g01tbf ( ltail, tail, lp, p, ldf, df, t, ivalid, ifail)
Integer, Intent (In) :: ltail, lp, ldf
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(*)
Real (Kind=nag_wp), Intent (In) :: p(lp), df(ldf)
Real (Kind=nag_wp), Intent (Out) :: t(*)
Character (1), Intent (In) :: tail(ltail)
C Header Interface
#include <nag.h>
void  g01tbf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf, const double df[], double t[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01tbf or nagf_stat_inv_cdf_students_t_vector.

3 Description

The deviate, tpi associated with the lower tail probability, pi, of the Student's t-distribution with νi degrees of freedom is defined as the solution to
P( Ti < tpi :νi) = pi = Γ ((νi+1)/2) νiπ Γ (νi/2) - tpi (1+ Ti2 νi ) - (νi+1) / 2 d Ti ,   νi 1 ; ​ - < tpi < .  
For νi=1 or 2 the integral equation is easily solved for tpi.
For other values of νi<3 a transformation to the beta distribution is used and the result obtained from g01fef.
For νi3 an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

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: ltail Integer Input
On entry: the length of the array tail.
Constraint: ltail>0.
2: tail(ltail) Character(1) array Input
On entry: indicates which tail the supplied probabilities represent. For j= ((i-1) mod ltail) +1 , for i=1,2,,max(ltail,lp,ldf):
tail(j)='L'
The lower tail probability, i.e., pi = P( Ti tpi :νi) .
tail(j)='U'
The upper tail probability, i.e., pi = P( Ti tpi :νi) .
tail(j)='C'
The two tail (confidence interval) probability,
i.e., pi = P( Ti |tpi| :νi) - P( Ti - |tpi| :νi) .
tail(j)='S'
The two tail (significance level) probability,
i.e., pi = P( Ti |tpi| :νi) + P( Ti - |tpi| :νi) .
Constraint: tail(j)='L', 'U', 'C' or 'S', for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p(lp) Real (Kind=nag_wp) array Input
On entry: pi, the probability of the required Student's t-distribution as defined by tail with pi=p(j), j=((i-1) mod lp)+1.
Constraint: 0.0<p(j)<1.0, for j=1,2,,lp.
5: ldf Integer Input
On entry: the length of the array df.
Constraint: ldf>0.
6: df(ldf) Real (Kind=nag_wp) array Input
On entry: νi, the degrees of freedom of the Student's t-distribution with νi=df(j), j=((i-1) mod ldf)+1.
Constraint: df(j)1.0, for j=1,2,,ldf.
7: t(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array t must be at least max(ltail,lp,ldf).
On exit: tpi, the deviates for the Student's t-distribution.
8: ivalid(*) Integer array Output
Note: the dimension of the array ivalid must be at least max(ltail,lp,ldf).
On exit: ivalid(i) indicates any errors with the input arguments, with
ivalid(i)=0
No error.
ivalid(i)=1
On entry, invalid value supplied in tail when calculating tpi.
ivalid(i)=2
On entry, pi0.0, or, pi1.0.
ivalid(i)=3
On entry, νi<1.0.
ivalid(i)=4
The solution has failed to converge. The result returned should represent an approximation to the solution.
9: 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 0 is recommended. 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:
ifail=1
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.
ifail=2
On entry, array size=value.
Constraint: ltail>0.
ifail=3
On entry, array size=value.
Constraint: lp>0.
ifail=4
On entry, array size=value.
Constraint: ldf>0.
ifail=-99
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.
ifail=-399
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.
ifail=-999
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

g01tbf is not threaded in any implementation.

9 Further Comments

The value tpi may be calculated by using a transformation to the beta distribution and calling g01tef. This routine allows you to set the required accuracy.

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 (g01tbfe.f90)

10.2 Program Data

Program Data (g01tbfe.d)

10.3 Program Results

Program Results (g01tbfe.r)