NAG Library Function Document

nag_prob_non_central_chi_sq (g01gcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_prob_non_central_chi_sq (g01gcc) returns the probability associated with the lower tail of the noncentral

χ^{2}

-distribution .

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_non_central_chi_sq (double x, double df, double lambda, double tol, Integer max_iter, NagError *fail)

3 Description

The lower tail probability of the noncentral

χ^{2}

-distribution with

ν

degrees of freedom and noncentrality parameter

λ

P (X \leq x : ν; λ)

, is defined by

P (X \leq x : ν; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!} P (X \leq x : ν + 2 j; 0),

(1)

where

P (X \leq x : ν + 2 j; 0)

is a central

χ^{2}

-distribution with

ν + 2 j

degrees of freedom.

The value of

j

at which the Poisson weight,

e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!}

, is greatest is determined and the summation (1) is made forward and backward from that value of

j

The recursive relationship:

P (X \leq x : a + 2; 0) = P (X \leq x : a; 0) - \frac{(x^{a} / 2) e^{- x / 2}}{Γ (a + 1)}

(2)

is used during the summation in (1).

4 References

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

5 Arguments

1: x – doubleInput: On entry: the deviate from the noncentral $χ^{2}$ -distribution with $ν$ degrees of freedom and noncentrality parameter $λ$ .
Constraint: $x \geq 0.0$ .
2: df – doubleInput: On entry: $ν$ , the degrees of freedom of the noncentral $χ^{2}$ -distribution.
Constraint: $df \geq 0.0$ .
3: lambda – doubleInput: On entry: $λ$ , the noncentrality parameter of the noncentral $χ^{2}$ -distribution.
Constraint: $lambda \geq 0.0$ if $df > 0.0$ or $lambda > 0.0$ if $df = 0.0$ .
4: tol – doubleInput: On entry: the required accuracy of the solution. If nag_prob_non_central_chi_sq (g01gcc) is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see nag_machine_precision (X02AJC)), then the value of $10 \times machine precision$ is used instead.
5: max_iter – IntegerInput: On entry: the maximum number of iterations to be performed.

Suggested value: $100$ . See Section 9 for further discussion.
Constraint: $max_iter \geq 1$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_REAL_ARG_CONS: On entry, $df = 0.0$ and $lambda = 0.0$ .
Constraint: $lambda > 0.0$ if $df = 0.0$ .
NE_CHI_PROB: The calculations for the central chi-square probability has failed to converge. A larger value of tol should be used.
NE_CONV: The solution has failed to converge in $⟨value⟩$ iterations. Consider increasing max_iter or tol.
NE_INT_ARG_LT: On entry, $max_iter = ⟨value⟩$ .
Constraint: $max_iter \geq 1$ .
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_POISSON_WEIGHT: The initial value of the Poisson weight used in the summation of (1) (see Section 3) was too small to be calculated. The computed probability is likely to be zero.
NE_REAL_ARG_LT: On entry, $df = ⟨value⟩$ .
Constraint: $df \geq 0.0$ .
On entry, $lambda = ⟨value⟩$ .
Constraint: $lambda \geq 0.0$ .
On entry, $x = ⟨value⟩$ .
Constraint: $x \geq 0.0$ .
NE_TERM_LARGE: The value of a term required in (2) (see Section 3) is too large to be evaluated accurately. The most likely cause of this error is both x and lambda are too large.

7 Accuracy

The summations described in Section 3 are made until an upper bound on the truncation error relative to the current summation value is less than tol.

8 Parallelism and Performance

Not applicable.

9 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 $λ$ .
(ii)	The rate at which the central $χ^{2}$ probabilities tend to zero. This will be slower for larger values of $ν$ and $x$ .

10 Example

This example reads values from various noncentral

χ^{2}

-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

NAG Library Function Documentnag_prob_non_central_chi_sq (g01gcc)