NAG CL Interface
g01jdc (prob_​chisq_​lincomb)

Settings help

CL Name Style:

1 Purpose

g01jdc calculates the lower tail probability for a linear combination of (central) χ2 variables.

2 Specification

#include <nag.h>
void  g01jdc (Nag_LCCMethod method, Integer n, const double rlam[], double d, double c, double *prob, NagError *fail)
The function may be called by the names: g01jdc, nag_stat_prob_chisq_lincomb or nag_prob_lin_chi_sq.

3 Description

Let u1,u2,,un be independent Normal variables with mean zero and unit variance, so that u12,u22,,un2 have independent χ2-distributions with unit degrees of freedom. g01jdc evaluates the probability that
If c=0.0 this is equivalent to the probability that
λ1u12+λ2u22++λnun2 u12+u22++un2 <d.  
Alternatively let
λi*=λi-d,   ​ i= 1,2,,n,  
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 n6 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 λi* are sufficiently distinct; g01jdc requires the λi* to be at least 1% distinct; see Section 9. If the λi* are at least 1% distinct and n60, then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.

4 References

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

5 Arguments

1: method 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 λi*, for i=1,2,,n are at least 1% distinct and n60; otherwise Imhof's method is used.
Constraint: method=Nag_LCCPan, Nag_LCCImhof or Nag_LCCDefault.
2: n Integer Input
On entry: n, the number of independent standard Normal variates, (central χ2 variates).
Constraint: n1.
3: rlam[n] const double Input
On entry: the weights, λi, for i=1,2,,n, of the central χ2 variables.
Constraint: rlam[i-1]d for at least one i. If method=Nag_LCCPan, the λi* must be at least 1% distinct; see Section 9, for i=1,2,,n.
4: d double Input
On entry: d, the multiplier of the central χ2 variables.
Constraint: d0.0.
5: c double Input
On entry: c, the value of the constant.
6: prob double * Output
On exit: the lower tail probability for the linear combination of central χ2 variables.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
On entry, n=value.
Constraint: n1.
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, d=value.
Constraint: d0.0.
On entry, rlam[i-1]=d for all values of i, for i=1,2,,n.
On entry, method=Nag_LCCPan but two successive values of λ* were not 1 percent distinct.

7 Accuracy

On successful exit at least four decimal places of accuracy should be achieved.

8 Parallelism 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.

9 Further Comments

Pan's procedure can only work if the λi* are sufficiently distinct. g01jdc uses the check |wj-wj-1|0.01×max(|wj|,|wj-1|), where the wj are the ordered nonzero values of λi*.
For the situation when all the λi 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.

10 Example

For n=10, the choice of method, values of c and d and the λi are input and the probabilities computed and printed.

10.1 Program Text

Program Text (g01jdce.c)

10.2 Program Data

Program Data (g01jdce.d)

10.3 Program Results

Program Results (g01jdce.r)