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 3.6 in the Essential Introduction).

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_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.
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

Not applicable.

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)