NAG Library Manual, Mark 27.3
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NAG CL Interface
g01edc (prob_f)
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NAG Library Manual, Mark 27.3
Interfaces:
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CL
CPP
AD
NAG CL Interface Introduction
G01 (Stat) Chapter Contents
G01 (Stat) Chapter Introduction
g01ed:
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CL
CPP
AD
▸
▿
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
© The Numerical Algorithms Group Ltd. 2021
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1
Purpose
g01edc
returns the probability for the lower or upper tail of the
F
or variance-ratio distribution with real degrees of freedom.
2
Specification
copy
#include <nag.h>
double
g01edc
(
Nag_TailProbability
tail
,
double
f
,
double
df1
,
double
df2
,
NagError *
fail
)
The function may be called by the names:
g01edc
,
nag_stat_prob_f
or
nag_prob_f_dist
.
3
Description
The lower tail probability for the
F
, or variance-ratio distribution, with
ν
1
and
ν
2
degrees of freedom,
P
(
F
≤
f
:
ν
1
,
ν
2
)
, is defined by:
P
(
F
≤
f
:
ν
1
,
ν
2
)
=
ν
1
ν
1
/
2
ν
2
ν
2
/
2
Γ
(
(
ν
1
+
ν
2
)
/
2
)
Γ
(
ν
1
/
2
)
Γ
(
ν
2
/
2
)
∫
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
≥
0
.
The probability is computed by means of a transformation to a beta distribution,
P
β
(
B
≤
β
:
a
,
b
)
:
P
(
F
≤
f
:
ν
1
,
ν
2
)
=
P
β
(
B
≤
ν
1
f
ν
1
f
+
ν
2
:
ν
1
/
2
,
ν
2
/
2
)
and using a call to
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
or
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_TailProbability
Input
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
≤
f
:
ν
1
,
ν
2
)
.
tail
=
Nag_UpperTail
The upper tail probability is returned, i.e.,
P
(
F
≥
f
:
ν
1
,
ν
2
)
.
Constraint
:
tail
=
Nag_LowerTail
or
Nag_UpperTail
.
2:
f
–
double
Input
On entry
:
f
, the value of the
F
variate.
Constraint
:
f
≥
0.0
.
3:
df1
–
double
Input
On entry
: the degrees of freedom of the numerator variance,
ν
1
.
Constraint
:
df1
>
0.0
.
4:
df2
–
double
Input
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 7
in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
Note:
on any of the error conditions listed below except
fail
.
code
=
NE_PROBAB_CLOSE_TO_TAIL
g01edc
returns
0.0
.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See
Section 3.1.2
in the Introduction to the NAG Library CL Interface 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 7.5
in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See
Section 8
in the Introduction to the NAG Library CL Interface 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
≥
0.0
.
7
Accuracy
The result should be accurate to five significant digits.
8
Parallelism and Performance
g01edc
is not threaded in any implementation.
9
Further Comments
For higher accuracy
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.
10.1
Program Text
Program Text (g01edce.c)
10.2
Program Data
Program Data (g01edce.d)
10.3
Program Results
Program Results (g01edce.r)
NAG Library Manual, Mark 27.3
Interfaces:
FL
CL
CPP
AD
NAG CL Interface Introduction
G01 (Stat) Chapter Contents
G01 (Stat) Chapter Introduction
g01ed:
FL
CL
CPP
AD
© The Numerical Algorithms Group Ltd. 2021