NAG Library Function Document

nag_prob_f_vector (g01sdc)

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

Not applicable.

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 Documentnag_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

NAG Library Function Document

nag_prob_f_vector (g01sdc)