NAG Library Function Document

nag_prob_f_dist (g01edc)

+− 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

nag_prob_f_dist (g01edc) returns the probability for the lower or upper tail of the

F

or variance-ratio distribution with real degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_f_dist (Nag_TailProbability tail, double f, double df1, double df2, NagError *fail)

3 Description

The lower tail probability for the

F

, or variance-ratio distribution, with

ν_{1}

and

ν_{2}

degrees of freedom,

P (F \leq f : ν_{1}, ν_{2})

, is defined by:

P (F \leq f : ν_{1}, ν_{2}) = \frac{ν_{1}^{ν_{1} / 2} ν_{2}^{ν_{2} / 2} Γ ((ν_{1} + ν_{2}) / 2)}{Γ (ν_{1} / 2) Γ (ν_{2} / 2)} \int_{0}^{f} F^{(ν_{1} - 2) / 2} {(ν_{1} F + ν_{2})}^{- (ν_{1} + ν_{2}) / 2} d F,

for

ν_{1}

ν_{2} > 0

f \geq 0

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

P_{β} (B \leq β : a, b)

P (F \leq f : ν_{1}, ν_{2}) = P_{β} (B \leq \frac{ν_{1} f}{ν_{1} f + ν_{2}} : ν_{1} / 2, ν_{2} / 2)

and using a call to nag_prob_beta_dist (g01eec).

For very large values of both

ν_{1}

and

ν_{2}

, greater than

10^{5}

, a normal approximation is used. If only one of

ν_{1}

ν_{2}

is greater than

10^{5}

then a

χ^{2}

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

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: tail – Nag_TailProbabilityInput

On entry: indicates whether an upper or lower tail probability is required.

$tail = Nag_LowerTail$: The lower tail probability is returned, i.e., $P (F \leq f : ν_{1}, ν_{2})$ .
$tail = Nag_UpperTail$: The upper tail probability is returned, i.e., $P (F \geq f : ν_{1}, ν_{2})$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

2: f – doubleInput

On entry:

f

, the value of the

F

variate.

Constraint:

f \geq 0.0

3: df1 – doubleInput

On entry: the degrees of freedom of the numerator variance,

ν_{1}

Constraint:

df1 > 0.0

4: df2 – doubleInput

On entry: the degrees of freedom of the denominator variance,

ν_{2}

Constraint:

df2 > 0.0

5: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

On any of the error conditions listed below except NE_PROBAB_CLOSE_TO_TAIL nag_prob_f_dist (g01edc) returns 0.0.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_PROBAB_CLOSE_TO_TAIL: The probability is too close to $0.0$ or $1.0$ . f is too far out into the tails for the probability to be evaluated exactly. The result tends to approach $1.0$ if $f$ is large, or $0.0$ if $f$ is small. The result returned is a good approximation to the required solution.
NE_REAL_ARG_LE: On entry, $df1 = ⟨value⟩$ and $df2 = ⟨value⟩$ .
Constraint: $df1 > 0.0$ and $df2 > 0.0$ .
NE_REAL_ARG_LT: On entry, $f = ⟨value⟩$ .
Constraint: $f \geq 0.0$ .

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_dist (g01eec) 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_dist (g01edc)

+− 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_dist (g01edc)