R: Energy Test of Univariate Normality

normal.test {energy} | R Documentation

## Energy Test of Univariate Normality

### Description

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

### Usage

    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

∑_{k=1}^∞ λ_k Z_k^2,

where λ_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

### 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.

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

### See Also

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

### Examples

      x <- iris[1:50, 1]
      normal.e(x)
      normal.test(x, R=199)

[Package energy version 1.7-6 Index]