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

3 Description

The lower tail probability for the Student's

t

-distribution with

ν_{i}

degrees of freedom,

P (T_{i} \leq t_{i} : ν_{i})

is defined by:

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

Computationally, there are two situations:

(i)

when

ν_{i} < 20

, a transformation of the beta distribution,

P_{β_{i}} (B_{i} \leq β_{i} : a_{i}, b_{i})

is used

P (T_{i} \leq t_{i} : ν_{i}) = \frac{1}{2} P_{β_{i}} (B_{i} \leq \frac{ν_{i}}{ν_{i} + {t_{i}}^{2}} : ν_{i} / 2, \frac{1}{2}) when ​ t_{i} < 0.0

P (T_{i} \leq t_{i} : ν_{i}) = \frac{1}{2} + \frac{1}{2} P_{β_{i}} (B_{i} \geq \frac{ν_{i}}{ν_{i} + {t_{i}}^{2}} : ν_{i} / 2, \frac{1}{2}) when ​ t_{i} > 0.0;

(ii)

when

ν_{i} \geq 20

, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see 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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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 returned probabilities should represent. For

j = (i - 1) mod ltail

, for

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

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

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailConfid

Nag_TwoTailSignif

, for

j = 1, 2, \dots, ltail

3: $lt$ – IntegerInput

On entry: the length of the array t.

Constraint:

lt > 0

4: $t [lt]$ – const doubleInput

On entry:

t_{i}

, the values of the Student's

t

variates with

t_{i} = t [j]

j = (i - 1) mod lt

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: $p [\dim]$ – doubleOutput

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

\max (ltail, lt, ldf)

On exit:

p_{i}

, the probabilities for the Student's

t

distribution.

8: $ivalid [\dim]$ – IntegerOutput

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

\max (ltail, lt, 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

p_{i}

$ivalid [i - 1] = 2$

On entry,

ν_{i} < 1.0

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: $lt > 0$ .

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 or df was invalid.
Check ivalid for more information.

7 Accuracy

The computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than

10^{- 10}

), see Hastings and Peacock (1975).

8 Parallelism and Performance

nag_prob_students_t_vector (g01sbc) is not threaded in any implementation.

9 Further Comments

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_prob_beta_vector (g01sec). This function allows you to set the required accuracy.

10 Example

This example reads values from, and degrees of freedom for Student's

t

-distributions along with the required tail. The probabilities are calculated and printed.

NAG Library Function Documentnag_prob_students_t_vector (g01sbc)

▸▿ 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_prob_students_t_vector (g01sbc)