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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 3.7 in How to Use the NAG Library and its Documentation).

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.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
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

nag_prob_f_dist (g01edc) is not threaded in any implementation.

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 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

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