R: Penalty function for log-likelihood of 'selm' models
Qpenalty
R Documentation
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.
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 π
for α by setting Q=-log(π),
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.
Value
A positive number Q representing the penalty, possibly
with attributes attr(Q, "der1") and attr(Q, "der2"),
depending onthe input value der.
Author(s)
Adelchi Azzalini
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) # no penalty
m2a <- selm(frontier ~ 1, method="MPLE") # penalty="Qpenalty" is implied here
m2b <- selm(frontier ~ 1, method="MPLE", penalty="MPpenalty")