Performs the E-statistic (energy) test of multivariate or univariate normality.
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data matrix of multivariate sample, or univariate data vector
number of bootstrap replicates
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
is a vector,
mvnorm.e returns the univariate statistic
If the data contains missing values or the sample covariance matrix is
mvnorm.e returns NA.
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
The value of the E-statistic for multivariate
normality is returned by
mvnorm.test returns a list with class
description of test
observed value of the test statistic
approximate p-value of the test
description of data
mvnorm.etest is replaced by
If the data is univariate, the test statistic is formally
the same as the multivariate case, but a more efficient computational
formula is applied in
normal.test also provides an optional method for the
test based on the asymptotic sampling distribution of the test
Maria L. Rizzo mrizzo @ bgsu.edu and Gabor J. Szekely
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.
Szekely, G. J. (1989) Potential and Kinetic Energy in Statistics, Lecture Notes, Budapest Institute of Technology (Technical University).
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