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NAG Toolbox

NAG Toolbox: nag_stat_prob_chisq_noncentral (g01gc)

Purpose

nag_stat_prob_chisq_noncentral (g01gc) returns the probability associated with the lower tail of the noncentral χ2χ2-distribution via the function name.

Syntax

[result, ifail] = g01gc(x, df, rlamda, 'tol', tol, 'maxit', maxit)
[result, ifail] = nag_stat_prob_chisq_noncentral(x, df, rlamda, 'tol', tol, 'maxit', maxit)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 23: tol now optional (default 0)
.

Description

The lower tail probability of the noncentral χ2χ2-distribution with νν degrees of freedom and noncentrality parameter λλ, P(Xx : ν;λ)P(Xx:ν;λ), is defined by
P(Xx : ν;λ) = eλ / 2((λ / 2)j)/(j ! )P(Xx : ν + 2j;0),
j = 0
P(Xx:ν;λ)=j=0e-λ/2(λ/2)jj! P(Xx:ν+2j;0),
(1)
where P(Xx : ν + 2j;0)P(Xx:ν+2j;0) is a central χ2χ2-distribution with ν + 2jν+2j degrees of freedom.
The value of jj at which the Poisson weight, eλ / 2((λ / 2)j)/(j ! ) e-λ/2 (λ/2)jj! , is greatest is determined and the summation (1) is made forward and backward from that value of jj.
The recursive relationship:
P(Xx : a + 2;0) = P(Xx : a;0)((xa / 2)ex / 2)/(Γ(a + 1))
P(Xx:a+2;0)=P(Xx:a;0)-(xa/2)e-x/2 Γ(a+1)
(2)
is used during the summation in (1).

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Parameters

Compulsory Input Parameters

1:     x – double scalar
The deviate from the noncentral χ2χ2-distribution with νν degrees of freedom and noncentrality parameter λλ.
Constraint: x0.0x0.0.
2:     df – double scalar
νν, the degrees of freedom of the noncentral χ2χ2-distribution.
Constraint: df0.0df0.0.
3:     rlamda – double scalar
λλ, the noncentrality parameter of the noncentral χ2χ2-distribution.
Constraint: rlamda0.0rlamda0.0 if df > 0.0df>0.0 or rlamda > 0.0rlamda>0.0 if df = 0.0df=0.0.

Optional Input Parameters

1:     tol – double scalar
The required accuracy of the solution. If nag_stat_prob_chisq_noncentral (g01gc) is entered with tol greater than or equal to 1.01.0 or less than 10 × machine precision10×machine precision (see nag_machine_precision (x02aj)), then the value of 10 × machine precision10×machine precision is used instead.
Default: 0.00.0
2:     maxit – int64int32nag_int scalar
The maximum number of iterations to be performed.
Default: 100100. See Section [Further Comments] for further discussion.
Constraint: maxit1maxit1.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_stat_prob_chisq_noncentral (g01gc) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If on exit ifail = 1ifail=1, 22, 44 or 55, then nag_stat_prob_chisq_noncentral (g01gc) returns 0.00.0.
  ifail = 1ifail=1
On entry,df < 0.0df<0.0,
orrlamda < 0.0rlamda<0.0,
ordf = 0.0df=0.0 and rlamda = 0.0rlamda=0.0,
orx < 0.0x<0.0,
ormaxit < 1maxit<1.
  ifail = 2ifail=2
The initial value of the Poisson weight used in the summation (1) was too small to be calculated. The value of P(xx : ν;λ)P(xx:ν;λ) is likely to be zero.
  ifail = 3ifail=3
The solution has failed to converge in maxit iterations.
  ifail = 4ifail=4
The value of a term required in (2) is too large to be evaluated accurately. The most likely cause of this error is both x and rlamda being very large.
  ifail = 5ifail=5
The calculations for the central χ2χ2 probability has failed to converge. This is an unlikely error exit. A larger value of tol should be used.

Accuracy

The summations described in Section [Description] are made until an upper bound on the truncation error relative to the current summation value is less than tol.

Further Comments

The number of terms in (1) required for a given accuracy will depend on the following factors:
(i) The rate at which the Poisson weights tend to zero. This will be slower for larger values of λλ.
(ii) The rate at which the central χ2χ2 probabilities tend to zero. This will be slower for larger values of νν and xx.

Example

function nag_stat_prob_chisq_noncentral_example
x = 8.26;
df = 20;
rlamda = 3.5;
[result, ifail] = nag_stat_prob_chisq_noncentral(x, df, rlamda)
 

result =

    0.0032


ifail =

                    0


function g01gc_example
x = 8.26;
df = 20;
rlamda = 3.5;
[result, ifail] = g01gc(x, df, rlamda)
 

result =

    0.0032


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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