nag_deviates_students_t (g01fbc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_deviates_students_t (g01fbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_deviates_students_t (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>
#include <nagg01.h>
double  nag_deviates_students_t (Nag_TailProbability tail, double p, double df, NagError *fail)

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
PT<tp:ν=p=Γν+1/2 νπΓν/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 nag_deviates_beta (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:     tailNag_TailProbabilityInput
On entry: indicates which tail the supplied probability represents.
tail=Nag_UpperTail
The upper tail probability, i.e., PTtp:ν.
tail=Nag_LowerTail
The lower tail probability, i.e., PTtp:ν.
tail=Nag_TwoTailSignif
The two tail (significance level) probability, i.e., PTtp:ν+PT-tp:ν.
tail=Nag_TwoTailConfid
The two tail (confidence interval) probability, i.e., PTtp:ν-PT-tp:ν.
Constraint: tail=Nag_UpperTail, Nag_LowerTail, Nag_TwoTailSignif or Nag_TwoTailConfid.
2:     pdoubleInput
On entry: p, the probability from the required Student's t-distribution as defined by tail.
Constraint: 0.0<p<1.0.
3:     dfdoubleInput
On entry: ν, the degrees of freedom of the Student's t-distribution.
Constraint: df1.0.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On any of the error conditions listed below except NE_SOL_NOT_CONV nag_deviates_students_t (g01fbc) returns 0.0 .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NE_REAL_ARG_GE
On entry, p=value.
Constraint: p<1.0.
NE_REAL_ARG_LE
On entry, p=value.
Constraint: p>0.0.
NE_REAL_ARG_LT
On entry, df=value.
Constraint: df1.0.
NE_SOL_NOT_CONV
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  Further Comments

The value tp may be calculated by using the transformation described in Section 3 and using nag_deviates_beta (g01fec). This function allows you to set the required accuracy.

9  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.

9.1  Program Text

Program Text (g01fbce.c)

9.2  Program Data

Program Data (g01fbce.d)

9.3  Program Results

Program Results (g01fbce.r)


nag_deviates_students_t (g01fbc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012