NormalityTest Class

NormalityTest Class

Performs a test for normality.

Inheritance Hierarchy

SystemObject
Imsl.StatNormalityTest

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class NormalityTest

<SerializableAttribute>
Public Class NormalityTest

[SerializableAttribute]
public ref class NormalityTest

[<SerializableAttribute>]
type NormalityTest =  class end

The NormalityTest type exposes the following members.

Constructors

	Name	Description
	NormalityTest	Constructor for NormalityTest.

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Methods

	Name	Description
	ChiSquaredTest	Performs the chi-squared goodness-of-fit test.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	LillieforsTest	Performs the Lilliefors test.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ShapiroWilkTest	Performs the Shapiro-Wilk W test using Royston's calculation.
	ShapiroWilkWTest	Performs the Shapiro-Wilk W test.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	ChiSquared	Returns the chi-square statistic for the chi-squared goodness-of-fit test.
	DegreesOfFreedom	Returns the degrees of freedom for the chi-squared goodness-of-fit test.
	MaxDifference	Returns the maximum absolute difference between the empirical and the theoretical distributions for the Lilliefors test.
	ShapiroWilkW	Returns the Shapiro-Wilk W statistic for the Shapiro-Wilk W test.

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Remarks

Three methods are provided for testing normality: the Shapiro-Wilk W test, the Lilliefors test, and the chi-squared test.

Shapiro-Wilk W Test

The Shapiro-Wilk W test is thought by D'Agostino and Stevens (1986, p. 406) to be one of the best omnibus tests of normality. The function is based on the approximations and code given by Royston (1982a, b, c). It can be used in samples as large as 2,000 or as small as 3. In the Shapiro and Wilk test, W is given by

$W = \left( {\sum {a_i x_{\left( i \right)} } } \right)^2 /\left( {\sum {\left( {x_i - \bar x} \right)^2 } } \right)$

where $x_{(i)}$ is the i-th largest order statistic and x is the sample mean. Royston (1982) gives approximations and tabled values that can be used to compute the coefficients $a_i, i = 1, \ldots, n$ , and obtains the significance level of the W statistic.

Lilliefors Test

This function computes Lilliefors test and its p-values for a normal distribution in which both the mean and variance are estimated. The one-sample, two-sided Kolmogorov-Smirnov statistic D is first computed. The p-values are then computed using an analytic approximation given by Dallal and Wilkinson (1986). Because Dallal and Wilkinson give approximations in the range (0.01, 0.10) if the computed probability of a greater D is less than 0.01, the p-value is set to 0.50. Note that because parameters are estimated, p-values in Lilliefors test are not the same as in the Kolmogorov-Smirnov Test.

Observations should not be tied. If tied observations are found, an informational message is printed. A general reference for the Lilliefors test is Conover (1980). The original reference for the test for normality is Lilliefors (1967).

Chi-Squared Test

This function computes the chi-squared statistic, its p-value, and the degrees of freedom of the test. Argument n finds the number of intervals into which the observations are to be divided. The intervals are equiprobable except for the first and last interval, which are infinite in length.

If more flexibility is desired for the specification of intervals, the same test can be performed with class ChiSquaredTest.

Reference

Imsl.Stat Namespace

Other Resources

Example