The function may be called by the names: g01jdc, nag_stat_prob_chisq_lincomb or nag_prob_lin_chi_sq.
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
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
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 Biometrika48 419–426
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan7 328–337
1: – 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.
, or .
2: – IntegerInput
On entry: , the number of independent standard Normal variates, (central variates).
3: – const doubleInput
On entry: the weights,
, for , of the central variables.
for at least one . If , the must be at least distinct; see Section 9, for .
4: – doubleInput
On entry: , the multiplier of the central variables.
5: – doubleInput
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).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
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.
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.
On entry, .
for all values of , for .
On entry, but two successive values of were not percent distinct.
On successful exit at least four decimal places of accuracy should be achieved.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01jdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.
Pan's procedure can only work if the are sufficiently distinct. g01jdc uses the check , where the are the ordered nonzero values of .
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.
For , the choice of method, values of and and the are input and the probabilities computed and printed.