Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

1 2 3 | ```
mvnorm.test(x, R)
mvnorm.etest(x, R)
mvnorm.e(x)
``` |

`x` |
data matrix of multivariate sample, or univariate data vector |

`R` |
number of bootstrap replicates |

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, `mvnorm.e`

returns the univariate statistic
`normal.e(x)`

.
If the data contains missing values or the sample covariance matrix is
singular, `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 `R`

replicates.

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

.

`mvnorm.test`

returns a list with class `htest`

containing

`method` |
description of test |

`statistic` |
observed value of the test statistic |

`p.value` |
approximate p-value of the test |

`data.name` |
description of data |

`mvnorm.etest`

is replaced by `mvnorm.test`

.

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.e`

.

`normal.test`

also provides an optional method for the
test based on the asymptotic sampling distribution of the test
statistic.

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).

`normal.test`

for the energy test of univariate
normality and `normal.e`

for the statistic.

1 2 3 4 5 6 |

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