nag_chi_sq_goodness_of_fit_test (g08cgc) : NAG Library, Mark 24

2 Specification

#include <nag.h>

#include <nagg08.h>

void	nag_chi_sq_goodness_of_fit_test (Integer nclass, const Integer ifreq[], const double cint[], Nag_Distributions dist, const double par[], Integer npest, const double prob[], double chisq, double p, Integer ndf, double eval[], double chisqi[], NagError fail)

3 Description

The

χ^{2}

goodness-of-fit test performed by nag_chi_sq_goodness_of_fit_test (g08cgc) 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 have 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 function are:

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

You must supply the frequencies and classes. Given a set of data and classes the frequencies may be calculated using nag_frequency_table (g01aec).

nag_chi_sq_goodness_of_fit_test (g08cgc) 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.

5 Arguments

1: nclass – IntegerInput

On entry: the number of classes,

k

, into which the data is divided.

Constraint:

nclass \geq 2

2: ifreq[nclass] – const IntegerInput

On entry:

ifreq [i - 1]

must specify the frequency of the

i

th class,

O_{i}

, for

i = 1, 2, \dots, k

Constraint:

ifreq [i - 1] \geq 0

, for

i = 1, 2, \dots, k

3: cint[

nclass - 1

] – const doubleInput

On entry:

cint [i - 1]

must specify the upper boundary value for the

i

th class, for

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

Constraints:

$cint [0] < cint [1] < \dots < cint [nclass - 2]$ ;
For the exponential, gamma and $χ^{2}$ distributions $cint [0] \geq 0.0$ .

4: dist – Nag_DistributionsInput

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

$dist = Nag_Normal$: The Normal distribution is used.
$dist = Nag_Uniform$: The uniform distribution is used.
$dist = Nag_Exponential$: The exponential distribution is used.
$dist = Nag_ChiSquare$: The $χ^{2}$ distribution is used.
$dist = Nag_Gamma$: The gamma distribution is used.
$dist = Nag_UserProb$: You must supply the class probabilities in the array prob.

Constraint:

dist = Nag_Normal

Nag_Uniform

Nag_Exponential

Nag_ChiSquare

Nag_Gamma

Nag_UserProb

5: par[

2

] – const doubleInput

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

dist = Nag_UserProb

) the array par is not referenced.

If a Normal distribution is used then

par [0]

and

par [1]

must contain the mean,

μ

, and the variance,

σ^{2}

, respectively.

If a uniform distribution is used then

par [0]

and

par [1]

must contain the boundaries

a

and

b

respectively.

If an exponential distribution is used then

par [0]

must contain the argument

λ

par [1]

is not used.

If a

χ^{2}

distribution is used then

par [0]

must contain the number of degrees of freedom.

par [1]

is not used.

If a gamma distribution is used

par [0]

and

par [1]

must contain the arguments

α

and

β

respectively.

Constraints:

if $dist = Nag_Normal$ , $par [1] > 0.0$ ;
if $dist = Nag_Uniform$ , $par [0] < par [1]$ and $par [0] \leq cint [0]$ ;
otherwise $par [1] \geq cint (nclass - 2)$ ;
if $dist = Nag_Exponential$ , $par [0] > 0.0$ ;
if $dist = Nag_ChiSquare$ , $par [0] > 0.0$ ;
if $dist = Nag_Gamma$ , $par [0]$ and $par [1] > 0.0$ .

6: npest – IntegerInput

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

Constraint:

0 \leq npest < nclass - 1

7: prob[nclass] – const doubleInput

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

dist = Nag_UserProb

) then

prob [i - 1]

must contain the probability that

X

lies in the

i

th class.

dist \neq Nag_UserProb

, prob is not referenced.

Constraint: if

dist = Nag_UserProb

prob [i - 1] > 0.0

and

\sum_{i = 1}^{k} prob [i - 1] = 1.0

, for

i = 1, 2, \dots, k

8: chisq – double *Output

On exit: the test statistic,

X^{2}

, for the

χ^{2}

goodness-of-fit test.

9: p – double *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] – doubleOutput

On exit:

eval [i - 1]

contains the expected frequency for the

i

th class,

E_{i}

, for

i = 1, 2, \dots, k

12: chisqi[nclass] – doubleOutput

On exit:

chisqi [i - 1]

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: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ARRAY_CONS

The contents of array prob are not valid.
Constraint: Sum of

prob [i - 1] = 1

, for

i = 1, 2, \dots, nclass

, when

dist = Nag_UserProb

NE_ARRAY_INPUT

On entry, the values provided in par are invalid.

NE_BAD_PARAM

On entry, argument dist had an illegal value.

NE_G08CG_CLASS_VAL

This is a warning that expected values for certain classes are less than 1.0. This implies that one cannot be confident that the

χ^{2}

distribution is a good approximation to the distribution of the test statistic.

NE_G08CG_CONV

The solution obtained when calculating the probability for a certain class for the gamma or

χ^{2}

distribution did not converge in 600 iterations. The solution may be an adequate approximation.

NE_G08CG_FREQ

An expected frequency is equal to zero when the observed frequency is not.

NE_INT_2

On entry,

npest = ⟨value⟩

nclass = ⟨value⟩

.
Constraint:

0 \leq npest < nclass - 1

NE_INT_ARG_LT

On entry,

nclass = ⟨value⟩

.
Constraint:

nclass \geq 2

NE_INT_ARRAY_CONS

On entry,

ifreq [⟨value⟩] = ⟨value⟩

.
Constraint:

ifreq [i - 1] \geq 0

, for

i = 1, 2, \dots, nclass

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_NOT_STRICTLY_INCREASING

The sequence cint is not strictly increasing

cint [⟨value⟩] = ⟨value⟩

cint [⟨value⟩ - 1] = ⟨value⟩

NE_REAL_ARRAY_CONS

On entry,

prob [⟨value⟩] = ⟨value⟩

.
Constraint:

prob [i - 1] > 0

, for

i = 1, 2, \dots, nclass

, when

dist = Nag_UserProb

NE_REAL_ARRAY_ELEM_CONS

On entry,

cint [0] = ⟨value⟩

.
Constraint:

cint [0] \geq 0.0

, if

dist = Nag_Exponential ‖Nag_ChiSquare‖ Nag_Gamma

NAG Library Function Document

nag_chi_sq_goodness_of_fit_test (g08cgc)

+− 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 Documentnag_chi_sq_goodness_of_fit_test (g08cgc)

+− 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_chi_sq_goodness_of_fit_test (g08cgc)