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.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
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)