void	nag_prob_f_vector (Integer ltail, const Nag_TailProbability tail[], Integer lf, const double f[], Integer ldf1, const double df1[], Integer ldf2, const double df2[], double p[], Integer ivalid[], NagError *fail)

3

Description

The lower tail probability for the

F

, or variance-ratio, distribution with

u_{i}

and

v_{i}

degrees of freedom,

P (F_{i} \leq f_{i} : u_{i}, v_{i})

, is defined by:

P (F_{i} \leq f_{i} : u_{i}, v_{i}) = \frac{{u_{i}}^{u_{i} / 2} {v_{i}}^{v_{i} / 2} Γ ((u_{i} + v_{i}) / 2)}{Γ (u_{i} / 2) Γ (v_{i} / 2)} \int_{0}^{f_{i}} {F_{i}}^{(u_{i} - 2) / 2} {(u_{i} F_{i} + v_{i})}^{- (u_{i} + v_{i}) / 2} d F_{i},

for

u_{i}

v_{i} > 0

f_{i} \geq 0

The probability is computed by means of a transformation to a beta distribution,

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

P (F_{i} \leq f_{i} : u_{i}, v_{i}) = P_{β_{i}} (B_{i} \leq \frac{u_{i} f_{i}}{u_{i} f_{i} + v_{i}} : u_{i} / 2, v_{i} / 2)

and using a call to nag_prob_beta_dist (g01eec).

For very large values of both

u_{i}

and

v_{i}

, greater than

10^{5}

, a normal approximation is used. If only one of

u_{i}

v_{i}

is greater than

10^{5}

then a

χ^{2}

approximation is used, see Abramowitz and Stegun (1972).

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

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 whether the lower or upper tail probabilities are required. For

j = (i - 1) mod ltail

, for

i = 1, 2, \dots, \max (ltail, lf, ldf1, ldf2)

$tail [j] = Nag_LowerTail$: The lower tail probability is returned, i.e., $p_{i} = P (F_{i} \leq f_{i} : u_{i}, v_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability is returned, i.e., $p_{i} = P (F_{i} \geq f_{i} : u_{i}, v_{i})$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $lf$ – IntegerInput

On entry: the length of the array f.

Constraint:

lf > 0

4: $f [lf]$ – const doubleInput

On entry:

f_{i}

, the value of the

F

variate with

f_{i} = f [j]

j = (i - 1) mod lf

Constraint:

f [j - 1] \geq 0.0

, for

j = 1, 2, \dots, lf

5: $ldf1$ – IntegerInput

On entry: the length of the array df1.

Constraint:

ldf1 > 0

6: $df1 [ldf1]$ – const doubleInput

On entry:

u_{i}

, the degrees of freedom of the numerator variance with

u_{i} = df1 [j]

j = (i - 1) mod ldf1

Constraint:

df1 [j - 1] > 0.0

, for

j = 1, 2, \dots, ldf1

7: $ldf2$ – IntegerInput

On entry: the length of the array df2.

Constraint:

ldf2 > 0

8: $df2 [ldf2]$ – const doubleInput

On entry:

v_{i}

, the degrees of freedom of the denominator variance with

v_{i} = df2 [j]

j = (i - 1) mod ldf2

Constraint:

df2 [j - 1] > 0.0

, for

j = 1, 2, \dots, ldf2

9: $p [\dim]$ – doubleOutput

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

\max (ltail, lf, ldf1, ldf2)

On exit:

p_{i}

, the probabilities for the

F

-distribution.

10: $ivalid [\dim]$ – IntegerOutput

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

\max (ltail, lf, ldf1, ldf2)

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,

f_{i} < 0.0

$ivalid [i - 1] = 3$

On entry,	$u_{i} \leq 0.0$ ,
or	$v_{i} \leq 0.0$ .

$ivalid [i - 1] = 4$

The solution has failed to converge. The result returned should represent an approximation to the solution.

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 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: $ldf1 > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $ldf2 > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $lf > 0$ .

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 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 f, df1, df2 or tail was invalid, or the solution failed to converge.
Check ivalid for more information.

7

Accuracy

The result should be accurate to five significant digits.

8

Parallelism and Performance

nag_prob_f_vector (g01sdc) is not threaded in any implementation.

9

Further Comments

For higher accuracy nag_prob_beta_vector (g01sec) can be used along with the transformations given in Section 3.

10

Example

This example reads values from, and degrees of freedom for, a number of

F

-distributions and computes the associated lower tail probabilities.

NAG Library Function Document

nag_prob_f_vector (g01sdc)

▸▿ 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

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