SkewOptimal {ellipticalsymmetry}

Tests for elliptical symmetry by Babic et al.

Description

Tests for elliptical symmetry: specified and unspecified location.

Usage

SkewOptimal(X, location = NA, f = "t", param = NA)

Arguments

X

A numeric matrix.

location

A vector of location parameters.

f

A string that specifies the type of the radial density upon which the test is based. Currently supported options are "t", "logistic" and "powerExp". The default is set to "t".

param

A parameter that is used when f = "t" and f = "powerExp". The default value of param represents the degrees of freedom of the multivariate t distribution and it is set to 4.

Details

X and location are the only input arguments for the specified location test. The default value for location is set to NA which implies that the unspecified location test will be performed unless the user specifies location.

For the unspecified location test, besides the data matrix X, the input arguments are f and param. The f argument is a string that specifies the type of the radial density upon which the test is based. Currently supported options are: "t" for the radial density of the multivariate t distribution, "logistic" for the multivariate logistic and "powerExp" for the radial density of the multivariate power-exponential distribution. Note that the default is set to "t". The role of the param argument is as follows. If f = "t" then param denotes the degrees of freedom of the multivariate t distribution. Given that the default radial density is "t", it follows that the default value of param represents the degrees of freedom of the multivariate t distribution and it is set to 4. Note also that the degrees of freedom have to be greater than 2. If f = "powerExp" then param denotes the kurtosis parameter. In that case the value of param has to be different from 1, because for the multivariate power exponential distribution, a kurtosis parameter equal to 1 corresponds to the multivariate Gaussian distribution (the Gaussian f is excluded due to a singular Fisher information matrix). The default value is set to 0.5.

Value

An object of class "htest" containing the following components:

statistic

The value of the test statistic.

pvalue

The p-value of the test.

alternative

A character string describing the alternative hypothesis.

method

A character string indicating what type of test was performed.

Background

Tests for elliptical symmetry both for specified and unspecified location. These tests are based on Le Cam’s theory of statistical experiments and they are optimal against generalized skew-elliptical alternatives, but they remain quite powerful under a much broader class of non-elliptical distributions. They have a simple asymptotic chi-squared distribution under the null hypothesis of ellipticity, they are affine-invariant, computationally fast, have a simple and intuitive form, only require finite moments of order 2.

References

Babic, S., Gelbgras, L., Hallin, M., & Ley, C. (2021). Optimal tests for elliptical symmetry: specified and unspecified location. Bernoulli (in press).

Examples

## sepal width and length of the versicolor subset of the Iris data
X = datasets::iris[51:100,1:2]


## location unspecified test based on the radial density of the multivariate t distribution
SkewOptimal(X)

## location unspecified test based on the radial density of the logistic distribution
SkewOptimal(X, f="logistic")

## location unspecified test based the radial density of the power exponential distribution
SkewOptimal(X, f="powerExp")