Penalty function for log-likelihood of selm models

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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

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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 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.

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

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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")    

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