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`

.

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`alpha_etc, alpha` |
in the univariate case, a single value |

`nu` |
degrees of freedom, only required if |

`der` |
a numeric value in the set |

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.

A positive number `Q`

representing the penalty, possibly
with attributes `attr(Q, "der1")`

and `attr(Q, "der2")`

,
depending onthe input value `der`

.

Adelchi Azzalini

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.

`selm`

function

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