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 3.6 in the Essential Introduction).

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.
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 $1.0$ . 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

Not applicable.

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)