Convert direct parameters (DP) to centred parameters
(CP) of a skew-elliptical distribution and *vice versa*.

1 2 3 4 |

`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 |

`op` |
a vector or a list, in agreement with |

`family` |
a characther string with the family acronym,
as described in |

`object` |
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`

and `op2dp`

, a similar object of `dp`

parameters,
provided the set of input parameters is in the admissible region.
For `dp2op`

, a similar set of `op`

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~`

.

Additional 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).

The OP parameterization is very similar to DP, from which
it differs only for the components which regulate dispersion (or scatter)
and slant. Its relevance lies essentially in the multivariate case, where
the components of the slant parameter can be interpreted component-wise and
remain unaffected if marginalization with respect to some other components
is performed.
In the multivariate SN case, the components of OP, denoted
*ξ, Ψ, λ*, are associated to the expression of the density
function (5.30) of Azzalini \& Capitanio (2014); see pp.128–131 for more
information. In the univariate case, the slant component of DP
and the one of OP coincide, that is, *α=λ*,
Parameter *ξ* and other parameters which may exist with other families
remain the same of the DP set. The term OP stands for
‘original parameterization’ since this is, up to a negligible difference,
the parameterization adopted by Azzalini & Dalla Valle (1996).

While any choice of the components of DP or OP is
admissible, this is not true for CP. An implication is that a
call to `cp2dp`

may fail with an error message `"non-admissible CP"`

for certain input values. The most extreme case is represented by the
SC family, for which CP never exists; hence it makes
to sense to call `cp2dp`

with `family="SC"`

.

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.

Azzalini, A. and Dalla Valle, A. (1996).
The multivariate skew-normal distribution.
*Biometrika* **83**, 715–726.

`makeSECdistr`

, `summary.SECdistr`

,
`sn.cumulants`

,

the ‘Note’ at `summary.selm`

for the reason why
CP is the default parameterization in that function and in
related ones,

the ‘Examples’ at `rmsn`

for use of the CP
parameterization

1 2 3 4 5 6 7 8 9 10 11 12 | ```
# 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")
#
op3 <- dp2op(dp3, "ST")
dp3back <- op2dp(op3,"ST")
``` |

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