NAG Library Function Document

nag_deviates_students_t (g01fbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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,

t_{p}

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 < t_{p} : ν) = p = \frac{Γ ((ν + 1) / 2)}{\sqrt{ν π} Γ (ν / 2)} \int_{- \infty}^{t_{p}} {(1 + \frac{T^{2}}{ν})}^{- (ν + 1) / 2} d T, ν \geq 1; ​ - \infty < t_{p} < \infty .

For

ν = 1 ​ or ​ 2

the integral equation is easily solved for

t_{p}

For other values of

ν < 3

a transformation to the beta distribution is used and the result obtained from nag_deviates_beta (g01fec).

For

ν \geq 3

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_TailProbabilityInput

On entry: indicates which tail the supplied probability represents.

$tail = Nag_UpperTail$: The upper tail probability, i.e., $P (T \geq t_{p} : ν)$ .
$tail = Nag_LowerTail$: The lower tail probability, i.e., $P (T \leq t_{p} : ν)$ .
$tail = Nag_TwoTailSignif$: The two tail (significance level) probability, i.e., $P (T \geq |t_{p}| : ν) + P (T \leq - |t_{p}| : ν)$ .
$tail = Nag_TwoTailConfid$: The two tail (confidence interval) probability, i.e., $P (T \leq |t_{p}| : ν) - P (T \leq - |t_{p}| : ν)$ .

Constraint:

tail = Nag_UpperTail

Nag_LowerTail

Nag_TwoTailSignif

Nag_TwoTailConfid

2: p – doubleInput

On entry:

p

, the probability from the required Student's

t

-distribution as defined by tail.

Constraint:

0.0 < p < 1.0

3: df – doubleInput

On entry:

ν

, the degrees of freedom of the Student's

t

-distribution.

Constraint:

df \geq 1.0

4: fail – NagError *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 $fail . code =$ 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: $df \geq 1.0$ .
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 Parallelism and Performance

Not applicable.

9 Further Comments

The value

t_{p}

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.

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.

NAG Library Function Documentnag_deviates_students_t (g01fbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_deviates_students_t (g01fbc)