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NAG Library Function Document

nag_deviates_chi_sq (g01fcc)

▸▿ 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_deviates_chi_sq (g01fcc) returns the deviate associated with the given lower tail probability of the

χ^{2}

-distribution with real degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_chi_sq (double p, double df, NagError *fail)

3 Description

The deviate,

x_{p}

, associated with the lower tail probability

p

of the

χ^{2}

-distribution with

ν

degrees of freedom is defined as the solution to

P (X \leq x_{p} : ν) = p = \frac{1}{2^{ν / 2} Γ (ν / 2)} \int_{0}^{x_{p}} e^{- X / 2} X^{v / 2 - 1} d X, 0 \leq x_{p} < \infty; ν > 0 .

The required

x_{p}

is found by using the relationship between a

χ^{2}

-distribution and a gamma distribution, 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

For very large values of

ν

, greater than

10^{5}

, Wilson and Hilferty's normal approximation to the

χ^{2}

is used; see Kendall and Stuart (1969).

4 References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

distribution Appl. Statist. 24 385–388

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

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

5 Arguments

1: $p$ – doubleInput: On entry: $p$ , the lower tail probability from the required $χ^{2}$ -distribution.

Constraint: $0.0 \leq p < 1.0$ .
2: $df$ – doubleInput: On entry: $ν$ , the degrees of freedom of the $χ^{2}$ -distribution.

Constraint: $df > 0.0$ .
3: $fail$ – 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

On any of the error conditions listed below except $fail . code =$ NE_ALG_NOT_CONV nag_deviates_chi_sq (g01fcc) returns $0.0$ .
NE_ALG_NOT_CONV: The algorithm has failed to converge in $〈value〉$ iterations. The result should be a reasonable approximation.
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_GAM_NOT_CONV: The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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_PROBAB_CLOSE_TO_TAIL: The probability is too close to $0.0$ or $1.0$ .
NE_REAL_ARG_GE: On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .
NE_REAL_ARG_LE: On entry, $df = 〈value〉$ .
Constraint: $df > 0.0$ .
NE_REAL_ARG_LT: On entry, $p = 〈value〉$ .
Constraint: $p \geq 0.0$ .

7 Accuracy

The results should be accurate to five significant digits for most argument values. Some accuracy is lost for

p

close to

0.0

8 Parallelism and Performance

nag_deviates_chi_sq (g01fcc) is not threaded in any implementation.

9 Further Comments

For higher accuracy the relationship described in Section 3 may be used and a direct call to nag_deviates_gamma_dist (g01ffc) made.

10 Example

This example reads lower tail probabilities for several

χ^{2}

-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

NAG Library Function Documentnag_deviates_chi_sq (g01fcc)