nag_prob_lin_chi_sq (g01jdc) calculates the lower tail probability for a linear combination of (central) variables.
Let
be independent Normal variables with mean zero and unit variance, so that
have independent
-distributions with unit degrees of freedom. nag_prob_lin_chi_sq (g01jdc) evaluates the probability that
If
this is equivalent to the probability that
Alternatively let
then nag_prob_lin_chi_sq (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
nag_1d_quad_gen_1 (d01sjc)
is used.
Pan's procedure can only be used if the
are sufficiently distinct; nag_prob_lin_chi_sq (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:
method – Nag_LCCMethodInput
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:
n – IntegerInput
On entry: , the number of independent standard Normal variates, (central variates).
Constraint:
.
- 3:
rlam[n] – const doubleInput
On entry: the weights,
, for , of the central variables.
Constraint:
for at least one
. If
, then the
must be at least
distinct; see
Section 9, for
.
- 4:
d – doubleInput
On entry: , the multiplier of the central variables.
Constraint:
.
- 5:
c – doubleInput
On entry: , the value of the constant.
- 6:
prob – double *Output
On exit: the lower tail probability for the linear combination of central variables.
- 7:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
On successful exit at least four decimal places of accuracy should be achieved.
nag_prob_lin_chi_sq (g01jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
For the situation when all the
are positive
nag_prob_lin_non_central_chi_sq (g01jcc) may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by
nag_prob_durbin_watson (g01epc).