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 <nag.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)

The routine may be called by the names g08cgf or nagf_nonpar_test_chisq.

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: $nclass$ – Integer Input

On entry:

k

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

Constraint:

nclass \geq 2

2: $ifreq (nclass)$ – Integer array Input

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) array Input

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) array Input

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: $npest$ – Integer Input

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

Constraint:

0 \leq npest < nclass - 1

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

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.

Constraints:

dist ='A'

$prob (i) > 0.0$ , for $i = 1, 2, \dots, k$ ;
$\sum_{i = 1}^{k} prob (i) = 1.0$ .

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: $ndf$ – Integer Output

On exit: contains

(nclass - 1 - npest)

, the degrees of freedom associated with the test.

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

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) array Output

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. 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).

Errors or warnings detected by the routine:

Note: in some cases g08cgf may return useful information.

$ifail = 1$: 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.

g08cg: FL CL CPP AD

NAG FL Interfaceg08cgf (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

NAG FL Interface
g08cgf (test_chisq)