SkewOptimal {ellipticalsymmetry} | R Documentation |

Tests for elliptical symmetry: specified and unspecified location.

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

`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 |

`param` |
A parameter that is used when |

`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.

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. |

Tests for elliptical symmetry both for specified and unspecified location. These tests are based on Le Camâ€™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.

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

```
## 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")
```

[Package *ellipticalsymmetry* version 0.1.2 Index]