NAG Library Routine Document

g01tbf  (inv_cdf_students_t_vector)

 Contents

    1  Purpose
    7  Accuracy

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 nagmk26.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)

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 – IntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tailltail – Character(1) arrayInput
On entry: indicates which tail the supplied probabilities represent. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,ldf:
tailj='L'
The lower tail probability, i.e., pi = P Ti tpi :νi .
tailj='U'
The upper tail probability, i.e., pi = P Ti tpi :νi .
tailj='C'
The two tail (confidence interval) probability,
i.e., pi = P Ti tpi :νi - P Ti - tpi :νi .
tailj='S'
The two tail (significance level) probability,
i.e., pi = P Ti tpi :νi + P Ti - tpi :νi .
Constraint: tailj='L', 'U', 'C' or 'S', for j=1,2,,ltail.
3:     lp – IntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     plp – Real (Kind=nag_wp) arrayInput
On entry: pi, the probability of the required Student's t-distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
Constraint: 0.0<pj<1.0, for j=1,2,,lp.
5:     ldf – IntegerInput
On entry: the length of the array df.
Constraint: ldf>0.
6:     dfldf – Real (Kind=nag_wp) arrayInput
On entry: νi, the degrees of freedom of the Student's t-distribution with νi=dfj, j=i-1 mod ldf+1.
Constraint: dfj1.0, for j=1,2,,ldf.
7:     t* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array t must be at least maxltail,lp,ldf.
On exit: tpi, the deviates for the Student's t-distribution.
8:     ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxltail,lp,ldf.
On exit: ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry,invalid value supplied in tail when calculating tpi.
ivalidi=2
On entry,pi0.0,
orpi1.0.
ivalidi=3
On entry,νi<1.0.
ivalidi=4
The solution has failed to converge. The result returned should represent an approximation to the solution.
9:     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, if you are not familiar with this argument, the recommended value is 0. 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 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017