Let
be independent Normal variables with mean zero and unit variance, so that
have independent
-distributions with unit degrees of freedom.
g01jdc evaluates the probability that
If
this is equivalent to the probability that
Alternatively let
then
g01jdc returns the probability that
Two methods are available. One due to
Pan (1964) (see
Farebrother (1980)) makes use of series approximations. The other method due to
Imhof (1961) reduces the problem to a one-dimensional integral. If
then a non-adaptive method
is used to compute the value of the integral otherwise
d01sjc
is used.
Pan's procedure can only be used if the
are sufficiently distinct;
g01jdc requires the
to be at least
distinct; see
Section 9. If the
are at least
distinct and
, then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.
Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227
Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337
-
1:
– Nag_LCCMethod
Input
-
On entry: indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.
- Pan's method is used.
- Imhof's method is used.
- Pan's method is used if
, for are at least distinct and ; otherwise Imhof's method is used.
Constraint:
, or .
-
2:
– Integer
Input
-
On entry: , the number of independent standard Normal variates, (central variates).
Constraint:
.
-
3:
– const double
Input
-
On entry: the weights,
, for , of the central variables.
Constraint:
for at least one
. If
, the
must be at least
distinct; see
Section 9, for
.
-
4:
– double
Input
-
On entry: , the multiplier of the central variables.
Constraint:
.
-
5:
– double
Input
-
On entry: , the value of the constant.
-
6:
– double *
Output
-
On exit: the lower tail probability for the linear combination of central variables.
-
7:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
On successful exit at least four decimal places of accuracy should be achieved.
Background information to multithreading can be found in the
Multithreading documentation.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
For the situation when all the
are positive
g01jcc may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by
g01epc.