sn (version 2.0.0)

Qpenalty: Penalty function for log-likelihood of selm models

Description

Penalty function for the log-likelihood of selm models when method="MPLE". Qpenalty is the default function; MPpenalty is an example of a user-defined function effectively corresponding to a prior distributio on alpha.

Usage

Qpenalty(alpha_etc, nu = NULL, der = 0)
MPpenalty(alpha, der = 0)

Arguments

alpha_etc, alpha

in the univariate case, a single value alpha; in the multivariate case, a two-component list whose first component is the vector alpha, the second one is matrix cov2cor(Omega).

nu

degrees of freedom, only required if selm is called with family="ST".

der

a numeric value in the set 0,1,2 which indicates the required numer of derivatives of the function. In the multivariate case the function will only be called with der equal to 0 or 1.

Value

A positive number Q representing the penalty, possibly with attributes attr(Q, "der1") and attr(Q, "der2"), depending onthe input value der.

Details

The penalty is a function of alpha, but its expression may depend on other ingredients, specifically nu and cov2cor(Omega). See ‘Details’ of selm for additional information.

The penalty mechanism allows to introduce a prior distribution \(\pi\) for \(\alpha\) by setting \(Q=-\log\pi\), leading to a maximum a posteriori estimate in the stated sense.

As a simple illustration of this mechanism, function MPpenalty implements the `matching prior' distribution for the univariate SN distribution studied by Cabras et al. (2012); a brief summary of the proposal is provided in Section 3.2 of Azzalini and Capitanio (2014). Note that, besides alpha=+/-Inf, this choice also penalizes alpha=0 with Q=Inf, effectively removing alpha=0 from the parameter space.

Starting from the code of function MPpenalty, a user should be able to introduce an alternative prior distribution if so desired.

References

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.

Cabras, S., Racugno, W., Castellanos, M. E., and Ventura, L. (2012). A matching prior for the shape parameter of the skew-normal distribution. Scand. J. Statist. 39, 236--247.

See Also

selm function

Examples

# NOT RUN {
data(frontier)
m2 <- selm(frontier ~ 1)  # no penalty
m2a <- selm(frontier ~ 1, method="MPLE") # penalty="Qpenalty" is implied here
m2b <- selm(frontier ~ 1, method="MPLE", penalty="MPpenalty")    
# }