# Build a skew-elliptically contoured distribution

### Description

Build an object which identifies a skew-elliptically contoured distribution (SEC), in the univariate and in the multivariate case. The term ‘skew-elliptical distribution’ is a synonym of SEC distribution.

### Usage

1 | ```
makeSECdistr(dp, family, name, compNames)
``` |

### Arguments

`dp` |
a numeric vector (in the univariate case) or a list (in the
multivariate case) of parameters which identify the specific distribution
within the named |

`family` |
a character string which identifies the parametric
family; currently, possible values are: |

`name` |
an optional character string with the name of the distribution. If missing, one is created. |

`compNames` |
in the multivariate case, an optional vector of character
strings with the names of the component variables; its length must be
equal to the dimensionality of the distribution being generated.
If missing and the first component of |

### Details

If `dp`

is a numeric vector, a univariate distribution is built.
Alternatively, if `dp`

is a list, a multivariate distribution is
built. In both cases, the required number of components of `dp`

depends on `family`

: it must be `3`

for `"SN"` and
`"SC"`; it must be `4`

for `"ESN"` and `"ST"`.

In the univariate case, the first three components of `dp`

represent
what for the specific distributions are denoted `xi`

(location),
`omega`

(scale, positive) and `alpha`

(slant); see functions
`dsn`

, `dst`

, `dsc`

for their
description.
The fourth component, when it exists, represents either `tau`

(hidden variable mean) for `"ESN"` or `nu`

(degrees of freedom)
for `"ST"`. The names of the individual parameters are attached
to the components of `dp`

in the returned object.

In the multivariate case, `dp`

is a list with components having
similar role as in the univariate case, but `xi=dp[[1]]`

and
`alpha=dp[[3]]`

are now vectors and the scale parameter
`Omega=dp[[2]]`

is a symmetric positive-definite matrix.
For a multivariate distribution of dimension 1 (which can be created,
although a warning message is issued), `Omega`

corresponds to the
square of `omega`

in the univariate case.
Vectors `xi`

and `alpha`

must be of length `ncol(Omega)`

.
See also functions `dmsn`

, `dmst`

and
`dmsc`

.
The fourth component, when it exists, is a scalar with the same role as
in the univariate case.

In the univariate case `alpha=Inf`

is allowed, but in the multivariate
case all components of the vector `alpha`

must be finite.

### Value

In the univariate case, an object of class `SECdistrUv`

;
in the multivariate case, an object of class `SECdistrMv`

.
See `SECdistrUv-class`

and `SECdistrMv-class`

for their description.

### Background

For background information, see Azzalini and Capitanio (2014), specifically Chapters 2 and 4 for univariate cases, Chapters 5 and 6 for multivariate cases; Section 6.1 provides a general formulation of SEC distributions.

If the slant parameter `alpha`

is `0`

(or a vector of `0`

's,
in the multivariate case), the distribution is of classical elliptical
type.

Among the admissible families, the ESN distribution is not, strictly speaking, of SEC type, but it is nevertheless included because of its strong connection.

### Author(s)

Adelchi Azzalini

### References

Azzalini, A. with the collaboration of Capitanio, A. (2014).
*The Skew-Normal and Related Families*.
Cambridge University Press, IMS Monographs series.

### See Also

The description of classes `SECdistrUv-class`

and
`SECdistrMv-class`

`plot.SECdistr`

for plotting and
`summary.SECdistr`

for summaries

Related functions `dsn`

, `dst`

, `dsc`

,
`dmsn`

, `dmst`

, `dp2cp`

Functions `affineTransSECdistr`

and
`conditionalSECdistr`

to manipulate objects of class
`SECdistrMv-class`

Function `extractSECdistr`

to extract objects of class
`SECdistrMv-class`

and `SECdistrUv-class`

representing the SEC distribution of a `selm`

fit

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
f1 <- makeSECdistr(dp=c(3,2,5), family="SN", name="First-SN")
show(f1)
summary(f1)
plot(f1)
plot(f1, probs=c(0.1, 0.9))
#
f2 <- makeSECdistr(dp=c(3, 5, -4, 8), family="ST", name="First-ST")
f9 <- makeSECdistr(dp=c(5, 1, Inf, 0.5), family="ESN", name="ESN,alpha=Inf")
#
dp0 <- list(xi=1:2, Omega=diag(3:4), alpha=c(3, -5))
f10 <- makeSECdistr(dp=dp0, family="SN", name="SN-2d", compNames=c("u1", "u2"))
#
dp1 <- list(xi=1:2, Omega=diag(1:2)+outer(c(3,3),c(2,2)), alpha=c(-3, 5), nu=6)
f11 <- makeSECdistr(dp=dp1, family="ST", name="ST-2d", compNames=c("t1", "t2"))
``` |