dp2cp {sn} | R Documentation |
Convert direct parameters (DP) to centred parameters (CP) of a skew-elliptical distribution and vice versa.
dp2cp(dp, family, obj = NULL, cp.type = "proper", upto = NULL) cp2dp(cp, family)
dp |
a vector (in the univariate case) or a list (in the multivariate
case) as described in |
cp |
a vector or a list, in agreement with |
family |
a characther string, as described in |
obj |
optionally, an S4 object of class |
cp.type |
character string, which has effect only if |
upto |
numeric value (in |
for dp2cp
, a matching vector (in the univariate case) or a list
(in the multivariate case) of cp
parameters; for cp2dp
,
a similar object of dp
parameters.
For a description of the DP
parameters, see Section ‘Details’ of makeSECdistr
. The
CP form of parameterization is cumulant-based. For a univariate
distribution, the CP components are the mean value (first cumulant),
the standard deviation (square root of the 2nd cumulant), the coefficient of
skewness (3rd standardized cumulant) and, for the ST,
the coefficient of excess kurtosis (4th standardized cumulant).
For a multivariate distribution, there exists an extension based on the \
same logic; its components represent the
vector mean value, the variance matrix, the vector of marginal coefficients of
skewness and, only for the ST, the Mardia's coefficient of excess
kurtosis. The pseudo-CP variant provides an ‘approximate form’ of
CP when not all required cumulants exist; however, this parameter set
is not uniquely invertible to DP. The names of pseudo-CP
components printed in summary output are composed by adding a ~
after the usual component name; for example, the first one is denoted
mean~
.
Background information is provided by Azzalini and Capitanio (2014). Specifically, their Section 3.1.4 presents CP in the univariate SN case, Section 4.3.4 CP for the ST case and the 'pseudo-CP' version. Section 5.2.3 presents the multivariate extension for the SN distribution, Section 6.2.5 for the multivariate ST case. For a more detailed discussion, see Arellano-Valle and Azzalini (2013).
It is possible to call the functions with dp
or cp
having more
components than those expected for a given family as described above and in
makeSECdistr
. In the univariate case, this means that dp
or cp
can be vectors of longer length than indicated earlier. This
occurrence is interpreted in the sense that the additional components after
the first one are regarded as regression coefficients of a selm
model,
and they are transferred unchanged to the matching components of the
transformed parameter set; the motivation is given in Section 3.1.4 of
Azzalini and Capitanio (2014). In the multivariate case, dp[[1]]
and
cp[[1]]
can be matrices instead of vectors; the rows beyond the first
one are transferred unchanged to cp[[1]]
and dp[[1]]
,
respectively.
Arellano-Valle, R. B. and Azzalini, A. (2013, available on-line 12 June 2011). The centred parameterization and related quantities of the skew-t distribution. J. Multiv. Analysis 113, 73-90.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
makeSECdistr
, summary.SECdistr
,
sn.cumulants
, the ‘Note’ at summary.selm
# univariate case cp <- dp2cp(c(1, 2222, 3333, 2, 3), "SN") dp <- cp2dp(cp, "SN") # notice that 2nd and 3rd component remain unchanged # # multivariate case dp3 <- list(xi=1:3, Omega=toeplitz(1/(1:3)), alpha=c(-3, 8, 5), nu=6) cp3 <- dp2cp(dp3, "ST") dp3.back <- cp2dp(cp3, "ST")