void	g01tbc (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double t[], Integer ivalid[], NagError *fail)

The function may be called by the names: g01tbc, nag_stat_inv_cdf_students_t_vector or nag_deviates_students_t_vector.

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

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 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: $ltail$ – Integer Input

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail [ltail]$ – const Nag_TailProbability Input

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: $lp$ – Integer Input

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const double Input

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: $ldf$ – Integer Input

On entry: the length of the array df.

Constraint:

ldf > 0

6: $df [ldf]$ – const double Input

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: $t [\dim]$ – double Output

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: $ivalid [\dim]$ – Integer Output

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: 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.
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

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 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.

g01tb: FL CL CPP AD

NAG CL Interfaceg01tbc (inv_​cdf_​students_​t_​vector)

▸▿ 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 CL Interface
g01tbc (inv_cdf_students_t_vector)