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NAG Library Function Document

nag_prob_non_central_f_dist (g01gdc)

▸▿ 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_non_central_f_dist (g01gdc) returns the probability associated with the lower tail of the noncentral

F

or variance-ratio distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_non_central_f_dist (double f, double df1, double df2, double lambda, double tol, Integer max_iter, NagError *fail)

3 Description

The lower tail probability of the noncentral

F

-distribution with

ν_{1}

and

ν_{2}

degrees of freedom and noncentrality parameter

λ

P (F \leq f : ν_{1}, ν_{2}; λ)

, is defined by

P (F \leq f : ν_{1}, ν_{2}; λ) = \int_{0}^{x} p (F : ν_{1}, ν_{2}; λ) d F,

where

P (F : ν_{1}, ν_{2}; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!} \times \frac{{(ν_{1} + 2 j)}^{(ν_{1} + 2 j) / 2} ν_{2}^{ν_{2} / 2}}{B ((ν_{1} + 2 j) / 2, ν_{2} / 2)}

\times u^{(ν_{1} + 2 j - 2) / 2} {[ν_{2} + (ν_{1} + 2 j) u]}^{- (ν_{1} + 2 j + ν_{2}) / 2}

and

B (\cdot, \cdot)

is the beta function.

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

P (F \leq f : ν_{1}, ν_{2}; λ) = P_{β} (X \leq x : a, b; λ),

where

x = \frac{ν_{1} f}{ν_{1} f + ν_{2}}

and

P_{β} (X \leq x : a, b; λ)

is the lower tail probability integral of the noncentral beta distribution with parameters

a

b

, and

λ

ν_{2}

is very large, greater than

10^{6}

, then a

χ^{2}

approximation is used.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5 Arguments

1: $f$ – doubleInput: On entry: $f$ , the deviate from the noncentral $F$ -distribution.

Constraint: $f > 0.0$ .
2: $df1$ – doubleInput: On entry: the degrees of freedom of the numerator variance, $ν_{1}$ .

Constraint: $0.0 < df1 \leq 10^{6}$ .
3: $df2$ – doubleInput: On entry: the degrees of freedom of the denominator variance, $ν_{2}$ .

Constraint: $df2 > 0.0$ .
4: $lambda$ – doubleInput: On entry: $λ$ , the noncentrality parameter.

Constraint: $0.0 \leq lambda \leq - 2.0 \log (U)$ where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
5: $tol$ – doubleInput: On entry: the relative accuracy required by you in the results. If nag_prob_non_central_f_dist (g01gdc) is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see nag_machine_precision (X02AJC)), then the value of $10 \times machine precision$ is used instead.
6: $max_iter$ – IntegerInput: On entry: the maximum number of iterations to be used.

Suggested value: $500$ . See nag_prob_non_central_chi_sq (g01gcc) and nag_prob_non_central_beta_dist (g01gec) for further details.

Constraint: $max_iter \geq 1$ .
7: $fail$ – 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

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_CONV: The solution has failed to converge in $〈value〉$ iterations. Consider increasing max_iter or tol.
NE_INT_ARG_LT: On entry, $max_iter = 〈value〉$ .
Constraint: $max_iter \geq 1$ .
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_PROB_F: The required probability cannot be computed accurately. This may happen if the result would be very close to zero or one. Alternatively the values of df1 and f may be too large. In the latter case you could try using a normal approximation, see Abramowitz and Stegun (1972).
NE_PROB_F_INIT: The required accuracy was not achieved when calculating the initial value of the central $F$ or $χ^{2}$ probability. You should try a larger value of tol. If the $χ^{2}$ approximation is being used then nag_prob_non_central_f_dist (g01gdc) returns zero otherwise the value returned should be an approximation to the correct value.
NE_REAL_ARG_CONS: On entry, $df1 = 〈value〉$ .
Constraint: $0.0 < df1 \leq 10^{6}$ .

On entry, $df1 = 〈value〉$ .
Constraint: $df1 > 0.0$ .

On entry, $lambda = 〈value〉$ .
Constraint: $0.0 \leq lambda \leq - 2.0 \times \log (U)$ , where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
NE_REAL_ARG_LE: On entry, $df2 = 〈value〉$ .
Constraint: $df2 > 0.0$ .

On entry, $f = 〈value〉$ .
Constraint: $f > 0.0$ .

7 Accuracy

The relative accuracy should be as specified by tol. For further details see nag_prob_non_central_chi_sq (g01gcc) and nag_prob_non_central_beta_dist (g01gec).

8 Parallelism and Performance

nag_prob_non_central_f_dist (g01gdc) is not threaded in any implementation.

9 Further Comments

When both

ν_{1}

and

ν_{2}

are large a Normal approximation may be used and when only

ν_{1}

is large a

χ^{2}

approximation may be used. In both cases

λ

is required to be of the same order as

ν_{1}

. See Abramowitz and Stegun (1972) for further details.

10 Example

This example reads values from, and degrees of freedom for,

F

-distributions, computes the lower tail probabilities and prints all these values until the end of data is reached.

NAG Library Function Documentnag_prob_non_central_f_dist (g01gdc)