sn (version 2.1.1)

summary.SECdistr: Summary of a SEC distribution object

Description

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.

Usage

# S4 method for SECdistrUv
summary(object, cp.type = "auto", probs)
# S4 method for SECdistrMv
summary(object, cp.type = "auto")

Value

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 non-parametric 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).

Arguments

object

an object of class "SECdistrUv" or "SECdistrMv".

cp.type

a character string to select the required variance of CP parameterization; possible values are "proper", "pseudo", "auto" (default). For a description of these codes, see dp2cp.

probs

in the univariate case, a vector of probabilities for which the corresponding quantiles are required. If missing, the vector c(0.05, 0.25, 0.50, 0.75, 0.95) is used.

Author

Adelchi Azzalini

Details

For a description of the DP, CP and pseudo-CP 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 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 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.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, 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 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 aux and other slots of the returned object.

References

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.

See Also

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 SECdistrUv-class objects, methods mean and vcov for computing the mean vector and the variance matrix of SECdistrMv-class objects

modeSECdistr for computing the mode directly

Examples

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)