Description Usage Arguments Details Value Author(s) References See Also Examples
Produce a summary of an object of class either
"SECdistrUv"
or "SECdistrMv"
, which refer to a univariate or a
multivariate SEC distribution, respectively. Both types of
objects can be produced by makeSECditr
.
1 2 3 4 5 
object 
an object of class 
cp.type 
a character string to select the required variance of
CP parameterization; possible values are 
probs 
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
pseudoCP parameter sets included in the returned object,
see dp2cp
.
The aux
slot of the returned object includes other summary quantities,
as described next.
In the univariate case, the reported quantilebased 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
slant parameter 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
components. The alpha*
and delta*
coefficients are univariate
comprehensive summary quantities of slant; see pp.1323 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,
via alpha*
.
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 pseudoDP 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 aux
and other slots of the returned object.
A list with the following components:
family 
name of the family within the SEC class, character 
dp 
DP parameters, a list or a vector 
name 
the name of the distribution, character string 
compNames 
in the multivariate case the names of the components, a character vector 
cp 
CP parameters, a list or a vector 
cp.type 
the name of the selected variant of the CP set 
aux 
a list with auxiliary ingredients (mode, coefficients of skewness and kurtosis, in the parametric and nonparametric variants, and more); see Section ‘Details’ for more information. 
The list items dp
and cp
are vectors if class(object)
is
SECdistrUv
(univariate distribution); they are lists if
class(object)
is SECdistrMv
(multivariate distribution).
Adelchi Azzalini
Azzalini, A. with the collaboration of Capitanio, A. (2014). The SkewNormal and Related Families. Cambridge University Press, IMS Monographs series.
Moors, I. J. A. (1988). A quantile alternative for kurtosis. The Statistician 37, 2532.
Groeneveld, R. A. (2006). Skewness, Bowley's measures of. In volume 12, 77713, of Encyclopedia of Statistical Sciences, 2nd edition, edited by Kotz et al. Wiley, New York.
makeSECdistr
for building a SEC distribution
extractSECdistr
for extracting a SEC
distribution from a selm
fit
methods mean
and sd
for computing the mean and the standard deviation of
SECdistrUvclass
objects,
methods mean
and vcov
for computing the mean vector and the variance matrix of
SECdistrMvclass
objects
modeSECdistr
for computing the mode directly
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  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 # quantilebased 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="ESN2d")
summary(esn2)

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