Produce a summary of an object of class either
"SECdistrMv", which refer to a univariate or a
multivariate SEC distribution, respectively. Both types of
objects can be produced by
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an object of class
a character string to select the required variance of
CP parameterization; possible values are
in the univariate case, a vector of probabilities for which
the corresponding quantiles are required. If missing, the vector
For a description of the DP, CP and
pseudo-CP parameter sets included in the returned object,
aux slot of the returned object includes other summary quantities,
as described next.
In the univariate case, the reported quantile-based measures of skewness and
kurtosis refer to the Bowley and Moors measures, respectively;
see Groeneveld (2006) and Moors (1988) for their specifications.
In the multivariate case, the Mardia's measures of skewness and kurtosis
are computed from the expressions given on p.153 and p.178 of
Azzalini and Capitanio (2014).
In the univariate case,
delta is a simple transformation of the
alpha; it takes values in (-1, 1).
In the multivariate case,
delta is a vector with components
of similar type; they correspond to the matching terms of the univariate
delta* coefficients are univariate
comprehensive summary quantities of slant; see pp.132-3 of
Azzalini and Capitanio (2014) for their expressions. These quantities
play an important role in SEC distributions; for instance,
the Mardia's measures of multivariare skewness and kurtosis depend
on the vector of slant parameters only via
delta* or, equivalently,
The mode, which is unique for all these distributions, is computed by a numerical line search between the DP location and the CP location (or the pseudo-DP location, when the latter does exists). This line search is univariate also in the multivariate case, using Propositions 5.14 and 6.2 of Azzalini and Capitanio (2014); see also Problem 5.14.
The examples below illustrate how extract various components from
and other slots of the returned object.
A list with the following components:
name of the family within the SEC class, character
DP parameters, a list or a vector
the name of the distribution, character string
in the multivariate case the names of the components, a character vector
CP parameters, a list or a vector
the name of the selected variant of the CP set
a list with auxiliary ingredients (mode, coefficients of skewness and kurtosis, in the parametric and non-parametric variants, and more); see Section ‘Details’ for more information.
The list items
cp are vectors if
SECdistrUv (univariate distribution); they are lists if
SECdistrMv (multivariate distribution).
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
Moors, I. J. A. (1988). A quantile alternative for kurtosis. The Statistician 37, 25-32.
Groeneveld, R. A. (2006). Skewness, Bowley's measures of. In volume 12, 7771-3, of Encyclopedia of Statistical Sciences, 2nd edition, edited by Kotz et al. Wiley, New York.
makeSECdistr for building a SEC distribution
for computing the mean and the standard deviation of
for computing the mean vector and the variance matrix of
modeSECdistr for computing the mode directly
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f3 <- makeSECdistr(dp=c(3,2,5), family="SC") summary(f3) s <- summary(f3, probs=(1:9)/10) print(slotNames(s)) print(names(slot(s,"aux"))) # the components of the 'aux' slot slot(s, "aux")$mode # the same of modeSECdistr(object=f3) slot(s, "aux")$q.measures # quantile-based measures of skewness and kurtosis # dp3 <- list(xi=1:3, Omega=toeplitz(1/(1:3)), alpha=c(-3, 8, 5), nu=6) st3 <- makeSECdistr(dp=dp3, family="ST", name="ST3", compNames=c("U", "V", "W")) s <- summary(st3) dp <- slot(s, "dp") # the same of slot(st3, "dp") slot(s, "cp")$var.cov # the same of vcov(st3) slot(s, "aux")$delta.star # comprehensive coefficient of shape slot(s, "aux")$mardia # Mardia's measures of skewness and kurtosis # dp2 <- list(xi=rep(0,2), Omega=matrix(c(2,2,2,4),2,2), alpha=c(3,-5), tau=-1) esn2 <- makeSECdistr(dp=dp2, family="ESN", name="ESN-2d") summary(esn2)
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