NAG Library Routine Document
G01JDF calculates the lower tail probability for a linear combination of (central) variables.
||RLAM(N), D, C, PROB, WORK(N+1)
be independent Normal variables with mean zero and unit variance, so that
-distributions with unit degrees of freedom. G01JDF evaluates the probability that
this is equivalent to the probability that
then G01JDF 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
described in D01BDF
is used to compute the value of the integral otherwise
Pan's procedure can only be used if the
are sufficiently distinct; G01JDF requires the
to be at least
distinct; see Section 8
. If the
are at least
, 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: METHOD – CHARACTER(1)Input
: 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.
, or .
- 2: N – INTEGERInput
On entry: , the number of independent standard Normal variates, (central variates).
- 3: RLAM(N) – REAL (KIND=nag_wp) arrayInput
On entry: the weights,
, for , of the central variables.
for at least one
, then the
must be at least
distinct; see Section 8
- 4: D – REAL (KIND=nag_wp)Input
On entry: , the multiplier of the central variables.
- 5: C – REAL (KIND=nag_wp)Input
On entry: , the value of the constant.
- 6: PROB – REAL (KIND=nag_wp)Output
On exit: the lower tail probability for the linear combination of central variables.
- 7: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 8: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|or||, or .|
On entry, for all values of , for .
On entry, yet two successive values of the ordered , for , were not at least distinct.
On successful exit at least four decimal places of accuracy should be achieved.
Pan's procedure can only work if the are sufficiently distinct. G01JDF uses the check , where the are the ordered nonzero values of .
For the situation when all the
are positive G01JCF
may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by G01EPF
For , the choice of method, values of and and the are input and the probabilities computed and printed.
9.1 Program Text
Program Text (g01jdfe.f90)
9.2 Program Data
Program Data (g01jdfe.d)
9.3 Program Results
Program Results (g01jdfe.r)