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Tests for normality

test-normalityTests.Rd

A collection of functions of one sample tests for testing normality of financial return series.

The functions for testing normality are:

ksnormTestKolmogorov-Smirnov normality test,
shapiroTestShapiro-Wilk's test for normality,
jarqueberaTestJarque-Bera test for normality,
dagoTestD'Agostino normality test.

Functions for high precision Jarque Bera LM and ALM tests:

jbTestPerforms finite sample adjusted JB, LM and ALM test.

Additional functions for testing normality from the 'nortest' package:

adTestAnderson–Darling normality test,
cvmTestCramer–von Mises normality test,
lillieTestLilliefors (Kolmogorov-Smirnov) normality test,
pchiTestPearson chi-square normality test,
sfTestShapiro-Francia normality test.

For SPlus/Finmetrics Compatibility:

normalTesttest suite for some normality tests.

Usage

ksnormTest(x, title = NULL, description = NULL)

jbTest(x, title = NULL, description = NULL)
shapiroTest(x, title = NULL, description = NULL)
normalTest(x, method = c("sw", "jb"), na.rm = FALSE)

jarqueberaTest(x, title = NULL, description = NULL)
dagoTest(x, title = NULL, description = NULL)

adTest(x, title = NULL, description = NULL)
cvmTest(x, title = NULL, description = NULL)
lillieTest(x, title = NULL, description = NULL)
pchiTest(x, title = NULL, description = NULL)
sfTest(x, title = NULL, description = NULL)

Arguments

x

a numeric vector of data values or an S4 object of class "timeSeries".

title

an optional character string, if not specified the inputs data name is deparsed.

description

optional description string, or a vector of character strings.

method

for normalTest only, indicates one of four different methods for the normality test, one of "ks" (Kolmogorov-Smirnov one-sample test, the the default), "sw" (Shapiro-Wilk test), "jb" (Jarque-Bera Test), and "da" (D'Agostino Test).

na.rm

for normalTest only, a logical value. Should missing values removed before computing the tests? The default value is FALSE.

Details

The hypothesis tests may be of interest for many financial and economic applications, especially for the investigation of univariate time series returns.

Several tests for testing if the records from a data set are normally distributed are available. The input to all these functions may be just a vector x or a univariate time series object x of class timeSeries.

First, there exists a wrapper function which allows to call one from two normal tests either the Shapiro–Wilks test or the Jarque–Bera test. This wrapper was introduced for compatibility with S-Plus' FinMetrics package.

Also available are the Kolmogorov–Smirnov one sample test and the D'Agostino normality test.

The remaining five normal tests are the Anderson–Darling test, the Cramer–von Mises test, the Lilliefors (Kolmogorov–Smirnov) test, the Pearson chi–square test, and the Shapiro–Francia test. They are calling functions from R's contributed package nortest. The difference to the original test functions implemented in R and from contributed R packages is that the Rmetrics functions accept time series objects as input and give a more detailed output report.

The Anderson-Darling test is used to test if a sample of data came from a population with a specific distribution, here the normal distribution. The adTest goodness-of-fit test can be considered as a modification of the Kolmogorov–Smirnov test which gives more weight to the tails than does the ksnormTest.

Note that jarqueBeraTest computes the asymptotic statistic and p-value, while jbTesT gives final sample approximations.

Value

an object from class fHTEST.

Slot test is a list containing the following (optionally empty) elements (in addition to those described in fHTEST):

ksnormTest

the 'D' statistic and p-values for the three alternatives 'two-sided, 'less' and 'greater'.

shapiroTest

the 'W' statistic and the p-value.

jarqueberaTest

no additional elements.

jbTest

the 'Chi-squared' statistic with 2 degrees of freedom and the asymptotic p-value. jbTest is the finite sample version of the Jarque Bera Lagrange multiplier, LM, and adjusted Lagrange multiplier test, ALM.

dagoTest

the 'Chi-squared', the 'Z3' (Skewness) and 'Z4' (Kurtosis) statistic together with the corresponding p values.

adTest

the 'A' statistic and the p-value.

cvmTest

the 'W' statistic and the p-value.

lillieTest

the 'D' statistic and the p-value.

pchiTest

the value for the 'P' statistic and the p-values for the adjusted and not adjusted test cases. In addition the number of classes is printed, taking the default value due to Moore (1986) computed from the expression n.classes = ceiling(2 * (n^(2/5))), where n is the number of observations.

sfTest

the 'W' statistic and the p-value.

Note

Some of the test implementations are selected from R's ctest and nortest packages.

References

Anderson T.W., Darling D.A. (1954); A Test of Goodness of Fit, JASA 49:765–69.

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

D'Agostino R.B., Pearson E.S. (1973); Tests for Departure from Normality, Biometrika 60, 613–22.

D'Agostino R.B., Rosman B. (1974); The Power of Geary's Test of Normality, Biometrika 61, 181–84.

Durbin J. (1961); Some Methods of Constructing Exact Tests, Biometrika 48, 41–55.

Durbin,J. (1973); Distribution Theory Based on the Sample Distribution Function, SIAM, Philadelphia.

Geary R.C. (1947); Testing for Normality; Biometrika 36, 68–97.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Linnet K. (1988); Testing Normality of Transformed Data, Applied Statistics 32, 180–186.

Moore, D.S. (1986); Tests of the chi-squared type, In: D'Agostino, R.B. and Stephens, M.A., eds., Goodness-of-Fit Techniques, Marcel Dekker, New York.

Shapiro S.S., Francia R.S. (1972); An Approximate Analysis of Variance Test for Normality, JASA 67, 215–216.

Shapiro S.S., Wilk M.B., Chen V. (1968); A Comparative Study of Various Tests for Normality, JASA 63, 1343–72.

Thode H.C. (2002); Testing for Normality, Marcel Dekker, New York.

Weiss M.S. (1978); Modification of the Kolmogorov-Smirnov Statistic for Use with Correlated Data, JASA 73, 872–75.

Wuertz D., Katzgraber H.G. (2005); Precise finite-sample quantiles of the Jarque-Bera adjusted Lagrange multiplier test, ETHZ Preprint.

Author

R-core team for the tests from R's ctest package,
Adrian Trapletti for the runs test from R's tseries package,
Juergen Gross for the normal tests from R's nortest package,
James Filliben for the Fortran program producing the runs report,
Diethelm Wuertz and Helmut Katzgraber for the finite sample JB tests,
Diethelm Wuertz for the Rmetrics R-port.
Earlier versions of theses functions were based on Fortran code of Paul Johnson.

Examples

x <- rnorm(100)

## Kolmogorov-Smirnov one-sampe test
ksnormTest(x)
#> 
#> Title:
#>  Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> Test Results:
#>   STATISTIC:
#>     D: 0.055
#>   P VALUE:
#>     Alternative Two-Sided: 0.9224 
#>     Alternative      Less: 0.6972 
#>     Alternative   Greater: 0.5456 
#> 

## Shapiro-Wilk test
shapiroTest(x)
#> 
#> Title:
#>  Shapiro - Wilk Normality Test
#> 
#> Test Results:
#>   STATISTIC:
#>     W: 0.9905
#>   P VALUE:
#>     0.7034 
#> 

## Jarque-Bera Test
jarqueberaTest(x)
#> 
#> Title:
#>  Jarque - Bera Normalality Test
#> 
#> Test Results:
#>   STATISTIC:
#>     X-squared: 0.7091
#>   P VALUE:
#>     Asymptotic p Value: 0.7015 
#> 
jbTest(x)
#> 
#> Title:
#>  Jarque - Bera Normality Test
#> 
#> Test Results:
#>   PARAMETER:
#>     Sample Size: 100
#>   STATISTIC:
#>     LM: 0.709
#>     ALM: 0.607
#>   P VALUE:
#>     LM p-value: 0.667 
#>     ALM p-value: 0.713 
#>     Asymptotic: 0.701 
#>