NAG Library Routine Document

Integer, Intent (In)	::	nclass, ifreq(nclass), npest
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ndf
Real (Kind=nag_wp), Intent (In)	::	cb(nclass-1), par(2), prob(nclass)
Real (Kind=nag_wp), Intent (Out)	::	chisq, p, eval(nclass), chisqi(nclass)
Character (1), Intent (In)	::	dist

C Header Interface

#include <nagmk26.h>

void	g08cgf_ (const Integer nclass, const Integer ifreq[], const double cb[], const char dist, const double par[], const Integer npest, const double prob[], double chisq, double p, Integer ndf, double eval[], double chisqi[], Integer *ifail, const Charlen length_dist)

3

Description

The

χ^{2}

goodness-of-fit test performed by g08cgf is used to test the null hypothesis that a random sample arises from a specified distribution against the alternative hypothesis that the sample does not arise from the specified distribution.

Given a sample of size

n

, denoted by

x_{1}, x_{2}, \dots, x_{n}

, drawn from a random variable

X

, and that the data has been grouped into

k

classes,

\begin{matrix} x \leq c_{1}, \\ c_{i - 1} < x \leq c_{i}, & i = 2, 3, \dots, k - 1, \\ x > c_{k - 1}, \end{matrix}

then the

χ^{2}

goodness-of-fit test statistic is defined by

X^{2} = \sum_{i = 1}^{k} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}},

where

O_{i}

is the observed frequency of the

i

th class, and

E_{i}

is the expected frequency of the

i

th class.

The expected frequencies are computed as

E_{i} = p_{i} \times n,

where

p_{i}

is the probability that

X

lies in the

i

th class, that is

\begin{matrix} p_{1} = P (X \leq c_{1}), \\ p_{i} = P (c_{i - 1} < X \leq c_{i}), & i = 2, 3, \dots, k - 1, \\ p_{k} = P (X > c_{k - 1}) . \end{matrix}

These probabilities are either taken from a common probability distribution or are supplied by you. The available probability distributions within this routine are:

Normal distribution with mean $μ$ , variance $σ^{2}$ ;
uniform distribution on the interval $[a, b]$ ;
exponential distribution with probability density function $(pdf) = λ e^{- λ x}$ ;
$χ^{2}$ -distribution with $f$ degrees of freedom; and
gamma distribution with $pdf = \frac{x^{α - 1} e^{- x / β}}{Γ (α) β^{α}}$ .

You must supply the frequencies and classes. Given a set of data and classes the frequencies may be calculated using g01aef.

g08cgf returns the

χ^{2}

test statistic,

X^{2}

, together with its degrees of freedom and the upper tail probability from the

χ^{2}

-distribution associated with the test statistic. Note that the use of the

χ^{2}

-distribution as an approximation to the distribution of the test statistic improves as the expected values in each class increase.

4

References

Conover W J (1980) Practical Nonparametric Statistics Wiley

Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

5

Arguments

1: $nclass$ – IntegerInput

On entry:

k

, the number of classes into which the data is divided.

Constraint:

nclass \geq 2

2: $ifreq (nclass)$ – Integer arrayInput

On entry:

ifreq (i)

must specify the frequency of the

i

th class,

O_{i}

, for

i = 1, 2, \dots, k

Constraint:

ifreq (i) \geq 0

, for

i = 1, 2, \dots, k

3: $cb (nclass - 1)$ – Real (Kind=nag_wp) arrayInput

On entry:

cb (i)

must specify the upper boundary value for the

i

th class, for

i = 1, 2, \dots, k - 1

Constraint:

cb (1) < cb (2) < \dots < cb (nclass - 1)

. For the exponential, gamma and

χ^{2}

-distributions

cb (1) \geq 0.0

4: $dist$ – Character(1)Input

On entry: indicates for which distribution the test is to be carried out.

$dist ='N'$: The Normal distribution is used.
$dist ='U'$: The uniform distribution is used.
$dist ='E'$: The exponential distribution is used.
$dist ='C'$: The $χ^{2}$ -distribution is used.
$dist ='G'$: The gamma distribution is used.
$dist ='A'$: You must supply the class probabilities in the array prob.

Constraint:

dist ='N'

'U'

'E'

'C'

'G'

'A'

5: $par (2)$ – Real (Kind=nag_wp) arrayInput

On entry: must contain the parameters of the distribution which is being tested. If you supply the probabilities (i.e.,

dist ='A'

) the array par is not referenced.

If a Normal distribution is used then

par (1)

and

par (2)

must contain the mean,

μ

, and the variance,

σ^{2}

, respectively.

If a uniform distribution is used then

par (1)

and

par (2)

must contain the boundaries

a

and

b

respectively.

If an exponential distribution is used then

par (1)

must contain the parameter

λ

par (2)

is not used.

If a

χ^{2}

-distribution is used then

par (1)

must contain the number of degrees of freedom.

par (2)

is not used.

If a gamma distribution is used

par (1)

and

par (2)

must contain the parameters

α

and

β

respectively.

Constraints:

if $dist ='N'$ , $par (2) > 0.0$ ;
if $dist ='U'$ , $par (1) < par (2)$ and $par (1) \leq cb (1)$ and $par (2) \geq cb (nclass - 1)$ ;
if $dist ='E'$ , $par (1) > 0.0$ ;
if $dist ='C'$ , $par (1) > 0.0$ ;
if $dist ='G'$ , $par (1) > 0.0$ and $par (2) > 0.0$ .

6: $npest$ – IntegerInput

On entry: the number of estimated parameters of the distribution.

Constraint:

0 \leq npest < nclass - 1

7: $prob (nclass)$ – Real (Kind=nag_wp) arrayInput

On entry: if you are supplying the probability distribution (i.e.,

dist ='A'

) then

prob (i)

must contain the probability that

X

lies in the

i

th class.

dist \neq'A'

, prob is not referenced.

Constraint: if

dist ='A'

\sum_{i = 1}^{k} prob (i) = 1.0

prob (i) > 0.0

, for

i = 1, 2, \dots, k

8: $chisq$ – Real (Kind=nag_wp)Output

On exit: the test statistic,

X^{2}

, for the

χ^{2}

goodness-of-fit test.

9: $p$ – Real (Kind=nag_wp)Output

On exit: the upper tail probability from the

χ^{2}

-distribution associated with the test statistic,

X^{2}

, and the number of degrees of freedom.

10: $ndf$ – IntegerOutput

On exit: contains

(nclass - 1 - npest)

, the degrees of freedom associated with the test.

11: $eval (nclass)$ – Real (Kind=nag_wp) arrayOutput

On exit:

eval (i)

contains the expected frequency for the

i

th class,

E_{i}

, for

i = 1, 2, \dots, k

12: $chisqi (nclass)$ – Real (Kind=nag_wp) arrayOutput

On exit:

chisqi (i)

contains the contribution from the

i

th class to the test statistic, that is,

{(O_{i} - E_{i})}^{2} / E_{i}

, for

i = 1, 2, \dots, k

13: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if

ifail \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Note: g08cgf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $nclass = 〈value〉$ .
Constraint: $nclass \geq 2$ .

$ifail = 2$: On entry, $dist = 〈value〉$ .
Constraint: $dist ='N'$ , $'U'$ , $'E'$ , $'C'$ , $'G'$ or $'A'$ .

$ifail = 3$: On entry, $npest = 〈value〉$ .
Constraint: $0 \leq npest < nclass - 1$ .

$ifail = 4$: On entry, $i = 〈value〉$ and $ifreq (i) = 〈value〉$ .
Constraint: $ifreq (i) \geq 0$ .

$ifail = 5$: On entry, $i = 〈value〉$ , $cb (i - 1) = 〈value〉$ and $cb (i) = 〈value〉$ .
Constraint: $cb (i - 1) < cb (i)$ .

$ifail = 6$: On entry, $cb (1) = 〈value〉$ .
Constraint: $cb (1) \geq 0.0$ .

$ifail = 7$: On entry, $par (1) = 〈value〉$ .
Constraint: for the exponential distribution, $par (1) > 0.0$ .

On entry, $par (1) = 〈value〉$ .
Constraint: for the $χ^{2}$ distribution, $par (1) > 0.0$ .

On entry, $par (1) = 〈value〉$ and $par (2) = 〈value〉$ .
Constraint: for the gamma distribution, $par (1) > 0.0$ and $par (2) > 0.0$ .

On entry, $par (1) = 〈value〉$ and $par (2) = 〈value〉$ .
Constraint: for the uniform distribution, $par (1) < par (2)$ , $par (1) \leq cb (1)$ and $par (2) \geq cb (nclass - 1)$ .

On entry, $par (2) = 〈value〉$ .
Constraint: for the Normal distribution, $par (2) > 0.0$ .

$ifail = 8$: On entry, $i = 〈value〉$ and $prob (i) = 〈value〉$ .
Constraint: $prob > 0.0$

On entry, $\sum_{i} prob (i) = 〈value〉$ .
Constraint: $\sum_{i} prob (i) = 1.0$ .

$ifail = 9$: An expected frequency equals zero, when the observed frequency was not.

$ifail = 10$: At least one class has an expected frequency less than $1$ . The $χ^{2}$ distribution may not be a good approximation to the distribution of the test statistic.

$ifail = 11$: The solution has failed to converge whilst computing the expected values. The returned solution may be an adequate approximation.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

g08cgf is not threaded in any implementation.

9

Further Comments

The time taken by g08cgf is dependent both on the distribution chosen and on the number of classes,

k

10

Example

This example applies the

χ^{2}

goodness-of-fit test to test whether there is evidence to suggest that a sample of

100

randomly generated observations do not arise from a uniform distribution

U (0, 1)

. The class intervals are calculated such that the interval

(0, 1)

is divided into five equal classes. The frequencies for each class are calculated using g01aef.

NAG Library Routine Document

g08cgf (test_chisq)

▸▿ 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

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