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NAG Library Function Document

nag_deviates_f_dist (g01fdc)

▸▿ 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_deviates_f_dist (g01fdc) returns the deviate associated with the given lower tail probability of the

F

or variance-ratio distribution with real degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_f_dist (double p, double df1, double df2, NagError *fail)

3 Description

The deviate,

f_{p}

, associated with the lower tail probability,

p

, of the

F

-distribution with degrees of freedom

ν_{1}

and

ν_{2}

is defined as the solution to

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

where

ν_{1}, ν_{2} > 0

;

0 \leq f_{p} < \infty

The value of

f_{p}

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_deviates_beta (g01fec).

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: $p$ – doubleInput: On entry: $p$ , the lower tail probability from the required $F$ -distribution.

Constraint: $0.0 \leq p < 1.0$ .
2: $df1$ – doubleInput: On entry: the degrees of freedom of the numerator variance, $ν_{1}$ .

Constraint: $df1 > 0.0$ .
3: $df2$ – doubleInput: On entry: the degrees of freedom of the denominator variance, $ν_{2}$ .

Constraint: $df2 > 0.0$ .
4: $fail$ – 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

On any of the error conditions listed below except $fail . code =$ NE_SOL_NOT_CONV nag_deviates_f_dist (g01fdc) 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_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.
NE_PROBAB_CLOSE_TO_TAIL: The probability is too close to $0.0$ or $1.0$ . The value of $f_{p}$ cannot be computed. This will only occur when the large sample approximations are used.
NE_REAL_ARG_GE: On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .
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, $p = 〈value〉$ .
Constraint: $p \geq 0.0$ .
NE_SOL_NOT_CONV: The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively, nag_deviates_beta (g01fec) can be used with a suitable setting of the argument tol.

7 Accuracy

The result should be accurate to five significant digits.

8 Parallelism and Performance

nag_deviates_f_dist (g01fdc) is not threaded in any implementation.

9 Further Comments

For higher accuracy nag_deviates_beta (g01fec) can be used along with the transformations given in Section 3.

10 Example

This example reads the lower tail probabilities for several

F

-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

NAG Library Function Documentnag_deviates_f_dist (g01fdc)