One- or two-dimensional profile (penalized) log-likelihood function of a
selm fit and corresponding confidence interval or regions
# S3 method for selm
profile(fitted, param.type, param.name, param.values, npt,
opt.control = list(), plot.it = TRUE, log = TRUE, levels,
trace = FALSE, ...)An invisible list whose components, described below, are partly different in the one- and the two-parameter cases.
the calling statement
values of the first parameter
values of the second parameter (in the two-parameter case)
numeric vector or matrix of the profile log-likelihood values
in the one-parameter case, the confidence interval
in the one-parameter case, the confidence level
in the two-parameter case, a list of lists whose elements identify each curve of the contour plot
an object of class selm as produced by a call to
function selm with univariate response.
a character string with the required parameterization; it must be either
"CP" or "DP", or possibly their equivalent lowercase.
either a single character string or a vector of two such terms with the
name(s) of the parameter(s) for which the profile log-likelihood is
required; these names must match those appearing in
summary.selm(object, param.type).
in the one-parameter case, a numeric vector with
the values where the log-likelihood must be evaluated; in the two-parameter
case, a list of two such vectors used to build a grid of coordinates
of points. Their range must identify an interval or a rectangle which
includes the MLE or MPLE obtained by selm.
See ‘Details’ for more information.
in case the vector or any of the vectors of argument param.values
has length 2, an equally spaced grid of values is build with length equal
to the corresponding component of npt.
If the above condition is met but this argument is missing,
a default choice is made, namely 51 or (26,26) in the one- or two-parameter
case, respectively.
an optional list passed as argument control to optim to
optimize the log-likelihood; see ‘Details’ for more information.
a logical value; if TRUE (default value), a plot
is produced representing the deviance, which is described in
‘Details’ below.
In the one-parameter case, a confidence interval of prescribed level
is marked on the plot; in the two-parameter case, the contour curves are
labelled with approximate confidence levels. See however
for more information.
a logical value (default: TRUE) indicating whether the
scale and tail-weight parameter (the latter only for the ST
family) must be log-transformed, if case any of them occurs in
param.name. This applies to omega and
nu in the DP parameter set and to s.d. and
gamma2 in the CP parameter set.
a single probability value (in the one-parameter case) or a vector of such values (in the two-parameter case) for which the confidence interval or region is requited; see ‘Details’ for more information.
a logical value (default: FALSE) to activate printing
of intermediate outcome of the log-likelihood optimization process
optional graphical parameters passed to the plotting functions.
Adelchi Azzalini
This function is experimental and changes in future versions of the package may occur. Users should not rely on the persistence of the same user interface or the same name(s).
It is a known fact that, in some critical situations, peculiar outcomes are produced.
For each chosen point of the parameter(s) to be profiled, the
log-likelihood is maximized with respect to the remaining parameters.
The optimization process is accomplished using the optim
optimization function, with method="BFGS". This step can be regulated by
the user via opt.control which is passed to optim
as control argument, apart from element fnscale whose use is
reserved.
If the original fitted object included a fixed parameter value,
this is kept fixed here. If the estimation method was "MPLE",
that choice carries on here; in case the penalty function was user-defined,
it must still be accessible.
For plotting purposes and also in the numerical output, the deviance function \(D\) is used, namely $$D = 2\left[\max(\log L) - \log L\right]$$ where \(L\) denotes the likelihood.
The range of param.values must enclose the maximum (penalized)
likelihood estimates (MLE or MPLE) by an adequate extent
such that suitable confidence intervals or regions can be established from
standard asymptotic theory. If this condition does not hold, the function
still proceeds, but no confidence interval or region is delivered.
For the SN family and DP parameterization, the asymptotic
theory is actually non-standard near the important point \(\alpha=0\),
but the correspondence with the regular case of the CP
parameterization, still allows to derive confidence regions using standard
procedures; for additional information, see Section 3.1.6 of
Azzalini and Capitanio (2014).
When the MLE occurs on the frontier of the parameter space,
a message is issued and no confidence interval is produced, while in the
two-parameter case the plot is not labelled with probability values, but only
with deviance levels.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
data(ais, package="sn")
m1 <- selm(log(Fe) ~ BMI + LBM, family = "sn", data = ais)
pll <- profile(m1, "dp", param.name="alpha", param.val=c(-3,2))
profile(m1, "cp", param.name="gamma1", param.val=seq(-0.7, 0.4, by=0.1))
# in the next example, we reduce the grid points to save execution time
pll <- profile(m1, "cp", param.name=c("(Intercept.CP)", "gamma1"),
param.val = list(c(1.5, 4), c(-0.8, 0.5)), npt=c(11,11) )
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