NAG CL Interface
g01tbc (inv_​cdf_​students_​t_​vector)

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1 Purpose

g01tbc returns a number of deviates associated with given probabilities of Student's t-distribution with real degrees of freedom.

2 Specification

#include <nag.h>
void  g01tbc (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double t[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01tbc, nag_stat_inv_cdf_students_t_vector or nag_deviates_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 g01fec.
For νi3 an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).
The input arrays to this function 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] const Nag_TailProbability Input
On entry: indicates which tail the supplied probabilities represent. For j= (i-1) mod ltail , for i=1,2,,max(ltail,lp,ldf):
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi = P( Ti tpi :νi) .
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi = P( Ti tpi :νi) .
tail[j]=Nag_TwoTailConfid
The two tail (confidence interval) probability,
i.e., pi = P( Ti |tpi| :νi) - P( Ti - |tpi| :νi) .
tail[j]=Nag_TwoTailSignif
The two tail (significance level) probability,
i.e., pi = P( Ti |tpi| :νi) + P( Ti - |tpi| :νi) .
Constraint: tail[j-1]=Nag_LowerTail, Nag_UpperTail, Nag_TwoTailConfid or Nag_TwoTailSignif, for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p[lp] const double 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.
Constraint: 0.0<p[j-1]<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] const double Input
On entry: νi, the degrees of freedom of the Student's t-distribution with νi=df[j], j=(i-1) mod ldf.
Constraint: df[j-1]1.0, for j=1,2,,ldf.
7: t[dim] double Output
Note: the dimension, dim, 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[dim] Integer Output
Note: the dimension, dim, of the array ivalid must be at least max(ltail,lp,ldf).
On exit: ivalid[i-1] indicates any errors with the input arguments, with
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry, invalid value supplied in tail when calculating tpi.
ivalid[i-1]=2
On entry, pi0.0, or, pi1.0.
ivalid[i-1]=3
On entry, νi<1.0.
ivalid[i-1]=4
The solution has failed to converge. The result returned should represent an approximation to the solution.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, array size=value.
Constraint: ldf>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ltail>0.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more 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.
g01tbc 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 g01tec. This function 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 (g01tbce.c)

10.2 Program Data

Program Data (g01tbce.d)

10.3 Program Results

Program Results (g01tbce.r)