com.imsl.stat

Class ChiSquaredTest

java.lang.Object

com.imsl.stat.ChiSquaredTest

public class ChiSquaredTest
extends Object

Chi-squared goodness-of-fit test.

ChiSquaredTest performs a chi-squared goodness-of-fit test that a random sample of observations is distributed according to a specified theoretical cumulative distribution. The theoretical distribution, which may be continuous, discrete, or a mixture of discrete and continuous distributions, is specified via a user-defined function F where F implements CdfFuntion. Because the user is allowed to specify a range for the observations in the setRange method, a test that is conditional upon the specified range is performed.

ChiSquaredTest can be constructed in two different ways. The intervals can be specified via the array cutpoints. Otherwise, the number of cutpoints can be given and equiprobable intervals computed by the constructor. The observations are divided into these intervals. Regardless of the method used to obtain them, the intervals are such that the lower endpoint is not included in the interval while the upper endpoint is always included. The user should determine the cutpoints when the cumulative distribution function has discrete elements since ChiSquaredTest cannot determine them in this case.

By default, the lower and upper endpoints of the first and last intervals are \(-\infty\) and \(+\infty\), respectively. The method setRange can be used to change the range.

A tally of counts is maintained for the observations in x as follows:

If the cutpoints are specified by the user, the tally is made in the interval to which \(x_i\) belongs, using the user-specified endpoints.

If the cutpoints are determined by the class then the cumulative probability at \(x_i\), \(F(x_i)\), is computed using cdf.

The tally for \(x_i\) is made in interval number \(\lfloor mF(x) + 1 \rfloor\), where m is the number of categories and \(\lfloor.\rfloor\) is the function that takes the greatest integer that is no larger than the argument of the function. If the cutpoints are specified by the user, the tally is made in the interval to which \(x_i\) belongs using the endpoints specified by the user. Thus, if the computer time required to calculate the cumulative distribution function is large, user-specified cutpoints may be preferred in order to reduce the total computing time.

If the expected count in any cell is less than 1, then a rule of thumb is that the chi-squared approximation may be suspect. A warning message to this effect is issued in this case, as well as when an expected value is less than 5.

See Also:

Example

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	ChiSquaredTest.DidNotConvergeException The iteration did not converge
static class	ChiSquaredTest.NoObservationsException There are no observations.
static class	ChiSquaredTest.NotCDFException The function is not a Cumulative Distribution Function (CDF).

Constructor Summary

Constructors

Constructor and Description
ChiSquaredTest(CdfFunction cdf, double[] cutpoints, int nParameters) Constructor for the Chi-squared goodness-of-fit test.
ChiSquaredTest(CdfFunction cdf, int nCutpoints, int nParameters) Constructor for the Chi-squared goodness-of-fit test

Method Summary

Modifier and Type	Method and Description
double[]	getCellCounts() Returns the cell counts.
double	getChiSquared() Returns the chi-squared statistic.
double[]	getCutpoints() Returns the cutpoints.
double	getDegreesOfFreedom() Returns the degrees of freedom in chi-squared.
double[]	getExpectedCounts() Returns the expected counts.
double	getP() Returns the p-value for the chi-squared statistic.
void	setCutpoints(double[] cutpoints) Sets the cutpoints.
void	setRange(double lower, double upper) Sets endpoints of the range of the distribution.
void	update(double x) Adds a new observation to the test.
void	update(double[] x) Adds new observations to the test.
void	update(double[] x, double[] freq) Adds new observations to the test.
void	update(double x, double freq) Adds a new observation to the test.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ChiSquaredTest

public ChiSquaredTest(CdfFunction cdf,
                      double[] cutpoints,
                      int nParameters)
               throws ChiSquaredTest.NotCDFException

Constructor for the Chi-squared goodness-of-fit test.

Parameters:

cdf - a CdfFunction object that implements the CdfFunction interface

cutpoints - a double array containing the cutpoints

nParameters - an int which specifies the number of parameters estimated in computing the Cdf. For example, with a binomial distribution nParameters=1 if p is estimated from the data and nParameters=0 if p is given in advance. The degrees of freedom in \(\chi ^2\) is: $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters.

Throws:

ChiSquaredTest.NotCDFException

ChiSquaredTest

public ChiSquaredTest(CdfFunction cdf,
                      int nCutpoints,
                      int nParameters)
               throws ChiSquaredTest.NotCDFException,
                      InverseCdf.DidNotConvergeException

Constructor for the Chi-squared goodness-of-fit test

Parameters:

cdf - a CdfFunction object that implements the CdfFunction interface

nCutpoints - an int, the number of cutpoints

nParameters - an int which specifies the number of parameters estimated in computing the Cdf. For example, with a binomial distribution nParameters=1 if p is estimated from the data and nParameters=0 if p is given in advance. The degrees of freedom in \(\chi ^2\) is: $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters.

Throws:

ChiSquaredTest.NotCDFException

InverseCdf.DidNotConvergeException

Method Detail

setRange

public void setRange(double lower,
                     double upper)
              throws ChiSquaredTest.NotCDFException

Sets endpoints of the range of the distribution. Points outside of the range are ignored so that distributions conditional on the range can be used. In this case, the point lower is excluded from the first interval, but the point upper is included in the last interval. By default, a range on the whole real line is used.

Parameters:

lower - a double, the lower range limit

upper - a double, the upper range limit

Throws:

ChiSquaredTest.NotCDFException

update

public void update(double[] x)
            throws ChiSquaredTest.NotCDFException

Adds new observations to the test.

Parameters:

x - a double array which contains the new observations to be added to the test. The frequencies of these observations are assumed to be 1.0.

Throws:

ChiSquaredTest.NotCDFException

update

public void update(double x)
            throws ChiSquaredTest.NotCDFException

Adds a new observation to the test.

Parameters:

x - a double, the new observation to be added to the test. The frequency of this observation is assumed to be 1.0.

Throws:

ChiSquaredTest.NotCDFException

update

public void update(double[] x,
                   double[] freq)
            throws ChiSquaredTest.NotCDFException

Adds new observations to the test.

Parameters:

x - a double array which contains the new observations to be added to the test

freq - a double array which contains the frequencies of the corresponding new observations in x

Throws:

ChiSquaredTest.NotCDFException

update

public void update(double x,
                   double freq)
            throws ChiSquaredTest.NotCDFException

Adds a new observation to the test.

Parameters:

x - a double, the new observation to be added to the test

freq - a double, the frequency of the new observation, x

Throws:

ChiSquaredTest.NotCDFException

getChiSquared

public double getChiSquared()
                     throws ChiSquaredTest.NotCDFException

Returns the chi-squared statistic.

Returns:

a double, the chi-squared statistic

Throws:

ChiSquaredTest.NotCDFException

getP

public double getP()
            throws ChiSquaredTest.NotCDFException

Returns the p-value for the chi-squared statistic.

Returns:

a double, the p-value for the chi-squared statistic

Throws:

ChiSquaredTest.NotCDFException

getDegreesOfFreedom

public double getDegreesOfFreedom()
                           throws ChiSquaredTest.NotCDFException

Returns the degrees of freedom in chi-squared. The degrees of freedom (df) in chi-squared is $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters, the number of estimated parameters.

Returns:

a double, the degrees of freedom in the chi-squared statistic

Throws:

ChiSquaredTest.NotCDFException

setCutpoints

public void setCutpoints(double[] cutpoints)

Sets the cutpoints. The intervals defined by the cutpoints are such that the lower endpoint is not included while the upper endpoint is included in the interval.

Parameters:

cutpoints - a double array which contains the cutpoints

getCutpoints

public double[] getCutpoints()

Returns the cutpoints.

Returns:

a double array which contains the cutpoints

getCellCounts

public double[] getCellCounts()

Returns the cell counts.

Returns:

a double array which contains the number of actual observations in each cell.

getExpectedCounts

public double[] getExpectedCounts()

Returns the expected counts.

Returns:

a double array which contains the number of expected observations in each cell.