sn (version 1.2-0)

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.

concept

• penalized likelihood
• prior distribution

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 an 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.

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

data(frontier)
m2 <- selm(frontier ~ 1)
m2a <- selm(frontier ~ 1, method="MPLE")
m2b <- selm(frontier ~ 1, method="MPLE", penalty="MPpenalty")