R: Penalty function for log-likelihood of 'selm' models

Qpenalty {sn} | 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 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

    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")

[Package sn version 1.5-0 Index]