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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_stat_prob_chisq_noncentral_lincomb (g01jc)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_stat_prob_chisq_noncentral_lincomb (g01jc) returns the lower tail probability of a distribution of a positive linear combination of

χ^{2}

random variables.

Syntax

[p, pdf, ifail] = g01jc(a, mult, rlamda, c, 'n', n, 'tol', tol, 'maxit', maxit)

[p, pdf, ifail] = nag_stat_prob_chisq_noncentral_lincomb(a, mult, rlamda, c, 'n', n, 'tol', tol, 'maxit', maxit)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:

tol was made optional (default 0)

Description

For a linear combination of noncentral

χ^{2}

random variables with integer degrees of freedom the lower tail probability is

P (\sum_{j = 1}^{n} a_{j} χ^{2} (m_{j}, λ_{j}) \leq c),

(1)

where

a_{j}

and

c

are positive constants and where

χ^{2} (m_{j}, λ_{j})

represents an independent

χ^{2}

random variable with

m_{j}

degrees of freedom and noncentrality argument

λ_{j}

. 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

\sum_{k = 0}^{\infty} d_{k} F (m + 2 k, c / β),

(2)

where

F (m + 2 k, c / β) = P (χ^{2} (m + 2 k) < c / β)

, i.e., the probability that a central

χ^{2}

is less than

c / β

The value of

β

is set at

β = β_{B} = \frac{2}{(1 / a_{\min} + 1 / a_{\max})}

unless

β_{B} > 1.8 a_{\min}

, in which case

β = β_{A} = a_{\min}

is used, where

a_{\min} = \min \{a_{j}\}

and

a_{\max} = \max \{a_{j}\}

, for

j = 1, 2, \dots, n

References

Farebrother R W (1984) The distribution of a positive linear combination of

χ^{2}

random variables Appl. Statist. 33(3)

Parameters

Compulsory Input Parameters

1: $a (n)$ – double array: The weights, $a_{1}, a_{2}, \dots, a_{n}$ .

Constraint: $a (i) > 0.0$ , for $i = 1, 2, \dots, n$ .
2: $mult (n)$ – int64int32nag_int array: The degrees of freedom, $m_{1}, m_{2}, \dots, m_{n}$ .

Constraint: $mult (i) \geq 1$ , for $i = 1, 2, \dots, n$ .
3: $rlamda (n)$ – double array: The noncentrality parameters, $λ_{1}, λ_{2}, \dots, λ_{n}$ .

Constraint: $rlamda (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ .
4: $c$ – double scalar: $c$ , the point for which the lower tail probability is to be evaluated.

Constraint: $c \geq 0.0$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays a, mult, rlamda. (An error is raised if these dimensions are not equal.)
$n$ , the number of $χ^{2}$ random variables in the combination, i.e., the number of terms in equation (1).

Constraint: $n \geq 1$ .
2: $tol$ – double scalar: Default: $0$
The relative accuracy required by you in the results. If nag_stat_prob_chisq_noncentral_lincomb (g01jc) is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see nag_machine_precision (x02aj)), then the value of $10 \times machine precision$ is used instead.
3: $maxit$ – int64int32nag_int scalar: Default: $500$ .
The maximum number of terms that should be used during the summation.

Constraint: $maxit \geq 1$ .

Output Parameters

1: $p$ – double scalar: The lower tail probability associated with the linear combination of $n$ $χ^{2}$ random variables with $m_{j}$ degrees of freedom, and noncentrality arguments $λ_{j}$ , for $j = 1, 2, \dots, n$ .
2: $pdf$ – double scalar: The value of the probability density function of the linear combination of $χ^{2}$ variables.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_stat_prob_chisq_noncentral_lincomb (g01jc) 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 = 1

2

, then nag_stat_prob_chisq_noncentral_lincomb (g01jc) returns

0.0

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$n < 1$ ,
or	$maxit < 1$ ,
or	$c < 0.0$ .

$ifail = 2$

On entry,	a has an element $\leq 0.0$ ,
or	mult has an element $< 1$ ,
or	rlamda has an element $< 0.0$ .

$ifail = 3$: The central $χ^{2}$ calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.

W $ifail = 4$: The solution has failed to converge within maxit iterations. A larger value of maxit or tol should be used. The returned value should be a reasonable approximation to the correct value.

W $ifail = 5$: The solution appears to be too close to $0$ or $1$ for accurate calculation. The value returned is $0$ or $1$ as appropriate.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The series (2) is summed until a bound on the truncation error is less than tol. See Farebrother (1984) for further discussion.

Further Comments

None.

Example

The number of

χ^{2}

variables is read along with their coefficients, degrees of freedom and noncentrality arguments. The lower tail probability is then computed and printed.

Open in the MATLAB editor: g01jc_example

function g01jc_example


fprintf('g01jc example results\n\n');

fprintf('        a     mult  rlamda\n');

% First set
a      = [        6   3  1];
mult   = [int64(1)  1  1];
rlamda = [        0   0  0];
c      = 20;

[p1, pdf, ifail] = g01jc( ...
			  a, mult, rlamda, c);

fprintf('%10.2f%6d%9.2f\n',[a; mult; rlamda]);
fprintf('c = %6.2f    prob = %6.4f\n\n', c, p1);

% Second set
a      = [        7   3];
mult   = [int64(1)  1];
rlamda = [        6   2];
c      = 10;

[p2, pdf, ifail] = g01jc( ...
			  a, mult, rlamda, c);

fprintf('%10.2f%6d%9.2f\n',[a; mult; rlamda]);
fprintf('c = %6.2f    prob = %6.4f\n', c, p2);

g01jc example results

        a     mult  rlamda
      6.00     1     0.00
      3.00     1     0.00
      1.00     1     0.00
c =  20.00    prob = 0.8760

      7.00     1     6.00
      3.00     1     2.00
c =  10.00    prob = 0.0451

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Chapter Contents

Chapter Introduction

NAG Toolbox