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.
a numeric vector (in the univariate case) or a list (in the
multivariate case) of parameters which identify the specific distribution
within the named
a character string which identifies the parametric family; currently, possible values are: "SN", "ESN", "ST", "SC". See ‘Details’ for additional information.
an optional character string with the name of the distribution. If missing, one is created.
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
dp is a numeric vector, a univariate distribution is built.
dp is a list, a multivariate distribution is
built. In both cases, the required number of components of
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
what for the specific distributions are denoted
omega (scale, positive) and
alpha (slant); see functions
dsc for their
The fourth component, when it exists, represents either
(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
alpha=dp[] are now vectors and the scale parameter
Omega=dp[] 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
omega in the univariate case.
alpha must be of length
See also functions
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.
In the univariate case, an object of class
in the multivariate case, an object of class
for their description.
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
0 (or a vector of
in the multivariate case), the distribution is of classical elliptical
Among the admissible families, the ESN distribution is not, strictly speaking, of SEC type, but it is nevertheless included because of its strong connection.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
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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"))