The parameter set of a Closed Skew-Normal (CSN) distribution is converted into the parameter set of the equivalent Unified Skew-Normal (SUN) distribution.
convertCSN2SUNpar(mu, Sigma, D, nu, Delta)a list representing the dp parameter set of the
corresponding SUN distribution
a numeric vector of length p, say.
a positive definite variance matrix of size c(p,p).
an arbitrary numeric matrix of size say c(q, p), say.
a numeric vector of length q.
a positive definite variance matrix of size c(q,q).
Adelchi Azzalini
The arguments of the function match the parameters \((\mu, \Sigma, D, \nu, \Delta)\) of the CSN distribution presented by González-Farías et alii (2004a, 2004b). These parameters are converted into those of the equivalent SUN distribution, which is unique. The converse operation, that is, mapping parameters from the SUN to the CSN family, is not handled here. Its solution would be non-unique, because the CSN family is over-parameterized.
Note that, having retained the exact notation of the above-quoted papers,
there is a Delta argument which must not be confused with one of the
arguments for the SUN distribution in SUNdistr-base.
The coincidence of these names is entirely accidental.
The CSN parameters must only satisfy the requirements that
\(\Sigma\) and \(\Delta\) are symmetric positive definite matrices.
Since these conditions are somewhat simpler to check than those for the
SUN parameters, as indicated in SUNdistr-base,
this function may provide a simple option for the specification of a
CSN/SUN distribution.
The parameter list dp produced by this function can be used as an
input for the functions in SUNdistr-base or for
makeSUNdistr.
González-Farías, G., Domínguez-Molina, J. A., & Gupta, A. K. (2004a). Additive properties of skew normal random vectors. J. Statist. Plann. Inference 126, 521-534.
González-Farías, G., Domínguez-Molina, J. A., & Gupta, A. K. (2004b). The closed skew-normal distribution. In M. G. Genton (Ed.), Skew-elliptical Distributions and Their Applications: a Journey Beyond Normality, Chapter 2, (pp. 25–42). Chapman & Hall/CRC.
SUNdistr-base, makeSUNdistr
p <- 3
q <- 2
mu <- 1:p
Sigma <- toeplitz(1/(1:p))
D <- matrix(sqrt(1:(p*q)), q, p)
nu <- 1/(1:q)
Delta <- diag(q) + outer(rep(1,q), rep(1,q))
dp <- convertCSN2SUNpar(mu, Sigma, D, nu, Delta)
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