E-statistic (Energy) Test of Multivariate Normality

Description

Performs the E-statistic (energy) test of multivariate or univariate normality.

Usage

 mvnorm.etest(x, R = 999)
 mvnorm.e(x)
 normal.e(x)

Arguments

Details

If x is a matrix, each row is a multivariate observation. The data will be standardized to zero mean and identity covariance matrix using the sample mean vector and sample covariance matrix. If x is a vector, the univariate statistic normal.e(x) is returned. If the data contains missing values or the sample covariance matrix is singular, NA is returned.

The E-test of multivariate normality was proposed and implemented by Szekely and Rizzo (2005). The test statistic for d-variate normality is given by

E = n((2/n) sum[1:n] E||y_i-Z|| - E||Z-Z'|| - (1/n^2) sum[1:n,1:n] ||y_i-y_j||),

where y_1,…,y_n is the standardized sample, Z, Z' are iid standard d-variate normal, and || || denotes Euclidean norm.

The E-test of multivariate (univariate) normality is implemented by parametric bootstrap with R replicates.

Value

The value of the E-statistic for univariate normality is returned by normal.e. The value of the E-statistic for multivariate normality is returned by mvnorm.e.

mvnorm.etest returns a list with class htest containing

Author(s)

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

References

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

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

Szekely, G. J. (1989) Potential and Kinetic Energy in Statistics, Lecture Notes, Budapest Institute of Technology (Technical University).

Examples

 ## compute normality test statistics for iris Setosa data
 data(iris)
 mvnorm.e(iris[1:50, 1:4])
 normal.e(iris[1:50, 1])

 ## test if the iris Setosa data has multivariate normal distribution
 mvnorm.etest(iris[1:50,1:4], R = 199)
 
 ## test a univariate sample for normality
 x <- runif(50, 0, 10)
 mvnorm.etest(x, R = 199)
 

Results