R: Summary of a SEC distribution object

summary.SECdistr {sn} | R Documentation

## 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 signature 'SECdistrUv'
    summary(object, cp.type = "auto", probs)

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

### 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.

### 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.

### 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).

### 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.

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)

[Package sn version 1.5-3 Index]