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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 2.7 in How to Use the NAG Library and its Documentation).

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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

nag_prob_chi_sq (g01ecc) is not threaded in any implementation.

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)