NAG CL Interface
g01fbc (inv_​cdf_​students_​t)

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1 Purpose

g01fbc returns the deviate associated with the given tail probability of Student's t-distribution with real degrees of freedom.

2 Specification

#include <nag.h>
double  g01fbc (Nag_TailProbability tail, double p, double df, NagError *fail)
The function may be called by the names: g01fbc, nag_stat_inv_cdf_students_t or nag_deviates_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 g01fec.
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 Nag_TailProbability 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=Nag_UpperTail, Nag_LowerTail, Nag_TwoTailSignif or Nag_TwoTailConfid.
2: p double 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 double Input
On entry: ν, the degrees of freedom of the Student's t-distribution.
Constraint: df1.0.
4: 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

Note: on any of the error conditions listed below except fail.code= NE_SOL_NOT_CONV g01fbc returns 0.0.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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.
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.
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.

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

g01fbc 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 g01fec. This function allows you to set the required accuracy.

9.1 Internal Changes

Internal changes have been made to this function 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 (g01fbce.c)

10.2 Program Data

Program Data (g01fbce.d)

10.3 Program Results

Program Results (g01fbce.r)