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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

Not applicable.

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)