nag_prob_lin_chi_sq (g01jdc) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_prob_lin_chi_sq (g01jdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_prob_lin_chi_sq (Nag_LCCMethod method, Integer n, const double rlam[], double d, double c, double *prob, NagError *fail)

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. nag_prob_lin_chi_sq (g01jdc) evaluates the probability that
λ1u12+λ2u22++λnun2<du12+u22++un2+c.
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 nag_prob_lin_chi_sq (g01jdc) returns the probability that
λ1*u12+λ2*u22++λn*un2<c.
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 nag_1d_quad_gen_1 (d01sjc) is used.
Pan's procedure can only be used if the λi* are sufficiently distinct; nag_prob_lin_chi_sq (g01jdc) requires the λi* to be at least 1% distinct; see Section 8. 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:     methodNag_LCCMethodInput
On entry: indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.
method=Nag_LCCPan
Pan's method is used.
method=Nag_LCCImhof
Imhof's method is used.
method=Nag_LCCDefault
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:     nIntegerInput
On entry: n, the number of independent standard Normal variates, (central χ2 variates).
Constraint: n1.
3:     rlam[n]const doubleInput
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, then the λi* must be at least 1% distinct; see Section 8, for i=1,2,,n.
4:     ddoubleInput
On entry: d, the multiplier of the central χ2 variables.
Constraint: d0.0.
5:     cdoubleInput
On entry: c, the value of the constant.
6:     probdouble *Output
On exit: the lower tail probability for the linear combination of central χ2 variables.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n1.
NE_INTERNAL_ERROR
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.
NE_REAL
On entry, d=value.
Constraint: d0.0.
NE_REAL_ARRAY
On entry, all values of rlam=d.
NE_REAL_ARRAY_ENUM
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  Further Comments

Pan's procedure can only work if the λi* are sufficiently distinct. nag_prob_lin_chi_sq (g01jdc) uses the check wj-wj-10.01×maxwj,wj-1, where the wj are the ordered nonzero values of λi*.
For the situation when all the λi 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).

9  Example

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

9.1  Program Text

Program Text (g01jdce.c)

9.2  Program Data

Program Data (g01jdce.d)

9.3  Program Results

Program Results (g01jdce.r)


nag_prob_lin_chi_sq (g01jdc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012