nag_prob_lin_non_central_chi_sq (g01jcc) returns the lower tail probability of a distribution of a positive linear combination of random variables.
For a linear combination of noncentral
random variables with integer degrees of freedom the lower tail probability is
where
and
are positive constants and where
represents an independent
random variable with
degrees of freedom and noncentrality argument
. The linear combination may arise from considering a quadratic form in Normal variables.
Ruben's method as described in
Farebrother (1984) is used. Ruben has shown that
(1) may be expanded as an infinite series of the form
where
, i.e., the probability that a central
is less than
.
- 1:
a[n] – const doubleInput
On entry: the weights, .
Constraint:
, for .
- 2:
mult[n] – const IntegerInput
On entry: the degrees of freedom, .
Constraint:
, for .
- 3:
rlamda[n] – const doubleInput
On entry: the noncentrality parameters, .
Constraint:
, for .
- 4:
n – IntegerInput
On entry:
, the number of
random variables in the combination, i.e., the number of terms in equation
(1).
Constraint:
.
- 5:
c – doubleInput
On entry: , the point for which the lower tail probability is to be evaluated.
Constraint:
.
- 6:
p – double *Output
On exit: the lower tail probability associated with the linear combination of random variables with
degrees of freedom, and noncentrality arguments , for .
- 7:
pdf – double *Output
On exit: the value of the probability density function of the linear combination of variables.
- 8:
tol – doubleInput
On entry: the relative accuracy required by you in the results. If nag_prob_lin_non_central_chi_sq (g01jcc) is entered with
tol greater than or equal to
or less than
(see
nag_machine_precision (X02AJC)), then the value of
is used instead.
- 9:
maxit – IntegerInput
On entry: the maximum number of terms that should be used during the summation.
Suggested value:
.
Constraint:
.
- 10:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
The series
(2) is summed until a bound on the truncation error is less than
tol. See
Farebrother (1984) for further discussion.
Not applicable.
None.