void	nag_deviates_students_t_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double t[], Integer ivalid[], NagError *fail)

3 Description

The deviate,

t_{p_{i}}

associated with the lower tail probability,

p_{i}

, of the Student's

t

-distribution with

ν_{i}

degrees of freedom is defined as the solution to

P (T_{i} < t_{p_{i}} : ν_{i}) = p_{i} = \frac{Γ ((ν_{i} + 1) / 2)}{\sqrt{ν_{i} π} Γ (ν_{i} / 2)} \int_{- \infty}^{t_{p_{i}}} {(1 + \frac{{T_{i}}^{2}}{ν_{i}})}^{- (ν_{i} + 1) / 2} d T_{i}, ν_{i} \geq 1; ​ - \infty < t_{p_{i}} < \infty .

For

ν_{i} = 1 ​ or ​ 2

the integral equation is easily solved for

t_{p_{i}}

For other values of

ν_{i} < 3

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

For

ν_{i} \geq 3

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: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail [ltail]$ – const Nag_TailProbabilityInput

On entry: indicates which tail the supplied probabilities represent. For

j = (i - 1) mod ltail

, for

i = 1, 2, \dots, \max (ltail, lp, ldf)

$tail [j] = Nag_LowerTail$: The lower tail probability, i.e., $p_{i} = P (T_{i} \leq t_{p_{i}} : ν_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability, i.e., $p_{i} = P (T_{i} \geq t_{p_{i}} : ν_{i})$ .
$tail [j] = Nag_TwoTailConfid$: The two tail (confidence interval) probability,
i.e., $p_{i} = P (T_{i} \leq |t_{p_{i}}| : ν_{i}) - P (T_{i} \leq - |t_{p_{i}}| : ν_{i})$ .
$tail [j] = Nag_TwoTailSignif$: The two tail (significance level) probability,
i.e., $p_{i} = P (T_{i} \geq |t_{p_{i}}| : ν_{i}) + P (T_{i} \leq - |t_{p_{i}}| : ν_{i})$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailConfid

Nag_TwoTailSignif

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const doubleInput

On entry:

p_{i}

, the probability of the required Student's

t

-distribution as defined by tail with

p_{i} = p [j]

j = (i - 1) mod lp

Constraint:

0.0 < p [j - 1] < 1.0

, for

j = 1, 2, \dots, lp

5: $ldf$ – IntegerInput

On entry: the length of the array df.

Constraint:

ldf > 0

6: $df [ldf]$ – const doubleInput

On entry:

ν_{i}

, the degrees of freedom of the Student's

t

-distribution with

ν_{i} = df [j]

j = (i - 1) mod ldf

Constraint:

df [j - 1] \geq 1.0

, for

j = 1, 2, \dots, ldf

7: $t [\dim]$ – doubleOutput

Note: the dimension, dim, of the array t must be at least

\max (ltail, lp, ldf)

On exit:

t_{p_{i}}

, the deviates for the Student's

t

-distribution.

8: $ivalid [\dim]$ – IntegerOutput

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

t_{p_{i}}

$ivalid [i - 1] = 2$

On entry,	$p_{i} \leq 0.0$ ,
or	$p_{i} \geq 1.0$ .

$ivalid [i - 1] = 3$

On entry,

ν_{i} < 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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_deviates_students_t_vector (g01tbc) is not threaded in any implementation.

9 Further Comments

The value

t_{p_{i}}

may be calculated by using a transformation to the beta distribution and calling nag_deviates_beta_vector (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.

NAG Library Function Documentnag_deviates_students_t_vector (g01tbc)

▸▿ 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_vector (g01tbc)