Energy Test of Univariate Normality

Performs the energy test of univariate normality for the composite hypothesis Case 4, estimated parameters.

normal.test(x, method=c("mc","limit"), R)
normal.e(x)

Arguments

x

univariate data vector

method

method for p-value

R

number of replications if Monte Carlo method

Details

If method="mc" this test function applies the parametric bootstrap method implemented in mvnorm.test.

If method="limit", the p-value of the test is computed from the asymptotic distribution of the test statistic under the null hypothesis. The asymptotic distribution is a quadratic form of centered Gaussian random variables, which has the form $$\sum_{k=1}^\infty \lambda_k Z_k^2,$$ where \(\lambda_k\) are positive constants (eigenvalues) and \(Z_k\) are iid standard normal variables. Eigenvalues are pre-computed and stored internally. A p-value is computed using Imhof's method as implemented in the CompQuadForm package.

Note that the "limit" method is intended for moderately large samples because it applies the asymptotic distribution.

The energy test of normality was proposed and implemented by Szekely and Rizzo (2005). See mvnorm.test for more details.

Value

normal.e returns the energy goodness-of-fit statistic for a univariate sample.

normal.test returns a list with class htest containing

statistic

observed value of the test statistic

p.value

p-value of the test

estimate

sample estimates: mean, sd

data.name

description of data

See also

mvnorm.test and mvnorm.e for the energy test of multivariate normality and the test statistic for multivariate samples.

References

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002 .

Mori, T. F., Szekely, G. J. and Rizzo, M. L. "On energy tests of normality." Journal of Statistical Planning and Inference 213 (2021): 1-15.

Rizzo, M. L. (2002). A New Rotation Invariant Goodness-of-Fit Test, Ph.D. dissertation, Bowling Green State University.

J. P. Imhof (1961). Computing the Distribution of Quadratic Forms in Normal Variables, Biometrika, Volume 48, Issue 3/4, 419-426.

Author

Maria L. Rizzo mrizzo@bgsu.edu and Gabor J. Szekely

Examples

  x <- iris[1:50, 1]
  normal.e(x)
#> [1] 0.4650295
  normal.test(x, R=199)
#> 
#> 	Energy test of normality: estimated parameters
#> 
#> data:  x, sample size 50, dimension 1, replicates 199
#> E-statistic = 0.46503, p-value = 0.2915
#> sample estimates:
#>      mean        sd 
#> 5.0060000 0.3524897 
#> 
  normal.test(x, method="limit")
#> 
#> 	Energy test of normality: limit distribution
#> 
#> data:  Case 4: composite hypothesis, estimated parameters
#> statistic = 0.46503, p-value = 0.2869
#> sample estimates:
#>      mean        sd 
#> 5.0060000 0.3524897 
#>