NAG Library Function Document

nag_prob_chi_sq (g01ecc)

+− 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_chi_sq (g01ecc) returns the lower or upper tail probability for the

χ^{2}

-distribution with real degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_chi_sq (Nag_TailProbability tail, double x, double df, NagError *fail)

3 Description

The lower tail probability for the

χ^{2}

-distribution with

ν

degrees of freedom,

P (X \leq x : ν)

is defined by:

P (X \leq x : ν) = \frac{1}{2^{ν / 2} Γ (ν / 2)} \int_{0.0}^{x} X^{ν / 2 - 1} e^{- X / 2} d X, x \geq 0, ν > 0 .

To calculate

P (X \leq x : ν)

a transformation of a gamma distribution is employed, i.e., a

χ^{2}

-distribution with

ν

degrees of freedom is equal to a gamma distribution with scale parameter

2

and shape parameter

ν / 2

4 References

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

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: tail – Nag_TailProbabilityInput

On entry: indicates whether the upper or lower tail probability is required.

$tail = Nag_LowerTail$: The lower tail probability is returned, i.e., $P (X \leq x : ν)$ .
$tail = Nag_UpperTail$: The upper tail probability is returned, i.e., $P (X \geq x : ν)$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

2: x – doubleInput

On entry:

x

, the value of the

χ^{2}

variate with

ν

degrees of freedom.

Constraint:

x \geq 0.0

3: df – doubleInput

On entry:

ν

, the degrees of freedom of the

χ^{2}

-distribution.

Constraint:

df > 0.0

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALG_NOT_CONV: The series used to calculate the gamma probabilities has failed to converge. The result returned should represent an approximation to the solution.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
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_ARG_LE: On entry, $df = ⟨value⟩$ .
Constraint: $df > 0.0$ .
NE_REAL_ARG_LT: On entry, $x = ⟨value⟩$ .
Constraint: $x \geq 0.0$ .

7 Accuracy

A relative accuracy of five significant figures is obtained in most cases.

8 Parallelism and Performance

Not applicable.

9 Further Comments

For higher accuracy the transformation described in Section 3 may be used with a direct call to nag_incomplete_gamma (s14bac).

10 Example

Values from various

χ^{2}

-distributions are read, the lower tail probabilities calculated, and all these values printed out, until the end of data is reached.

NAG Library Function Documentnag_prob_chi_sq (g01ecc)

+− 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

NAG Library Function Document

nag_prob_chi_sq (g01ecc)