Function selm fits a linear model
with skew-elliptical error term.
The term ‘skew-elliptical distribution’ is an abbreviated equivalent
of skew-elliptically contoured (SEC) distribution.
The function works for univariate and multivariate response variables.
selm(formula, family = "SN", data, weights, subset, na.action,
start = NULL, fixed.param = list(), method = "MLE", penalty=NULL,
model = TRUE, x = FALSE, y = FALSE, contrasts = NULL, offset, ...)an S4 object of class selm or mselm, depending on whether
the response variable of the fitted model is univariate or multivariate;
these objects are described in the selm class.
an object of class "formula"
(or one that can be coerced to that class): a symbolic description of the
model to be fitted, using the same syntax used for the similar parameter of
e.g. "lm", with the restriction that the constant
term must not be removed from the linear predictor.
a character string which selects the parametric family
of SEC type assumed for the error term. It must be one of
"SN" (default), "ST" or "SC", which correspond to the
skew-normal, the skew-t and the skew-Cauchy family, respectively.
See makeSECdistr for more information on these families and
the set of SEC distributions; notice that the family "ESN"
listed there is not allowed here.
an optional data frame containing the variables in
the model. If not found in data, the variables are taken from
environment(formula), typically the environment from which
selm is called.
a numeric vector of weights associated to individual
observations. Weights are supposed to represent frequencies, hence must be
non-negative integers (not all 0) and length(weights) must equal the
number of observations. If not assigned, a vector of all 1's is generated.
an optional vector specifying a subset of observations
to be used in the fitting process. It works like the same parameter
in lm.
a function which indicates what should happen
when the data contain NAs. The default is set by the
na.action setting of options.
The ‘factory-fresh’ default is na.omit.
Another possible value is NULL, no action.
a vector (in the univariate case) or a list (in the
multivariate case) of initial DP values for searching the
parameter estimates. See ‘Details’ about a choice of
start to be avoided. If start=NULL (default),
initial values are selected by the procedure.
If family="ST", an additional option exists; see ‘Details’.
a list of assignments of parameter values which must
be kept fixed in the estimation process.
Currently, there only two types of admissible constraint: one is to
set alpha=0 to impose a symmetry condition of the distribution;
the other is to set nu=<value>, to fix the degrees of freedom
at the named <value> when family="ST", for instance
list(nu=3). See ‘Details’ for additional information.
a character string which selects the estimation method to be
used for fitting. Currently, two options exist: "MLE" (default) and
"MPLE", corresponding to standard maximum likelihood and maximum
penalized likelihood estimation, respectively. See ‘Details’ for
additional information.
a character string which denotes the penalty function to be
subtracted to the log-likelihood function, when method="MPLE"; if
penalty=NULL (default), a pre-defined function is adopted. See
‘Details’ for a description of the default penalty function and for
the expected format of alternative specifications. When
method="MLE", no penalization is applied and this argument has no
effect.
logicals. If TRUE, the corresponding components
of the fit are returned.
an optional list. See the contrasts.arg of
model.matrix.default.
this can be used to specify an a priori known
component to be included in the linear predictor during fitting. This
should be NULL or a numeric vector of length equal to the number of
cases. One or more offset terms can be included in the
formula instead or as well, and if more than one are specified their sum
is used.
optional control parameters, as follows.
trace: a logical value which indicates whether intermediate
evaluations of the optimization process are printed (default:
FALSE).
info.type: a character string which indicates the type of
Fisher information matrix; possible values are "observed"
(default) and "expected". Currently, "expected" is
implemented only for the SN family.
opt.method: a character string which selects the numerical
optimization method, among the possible values
"nlminb", "Nelder-Mead", "BFGS", "CG", "SANN".
If opt.method="nlminb" (default),
function nlminb is called,
otherwise function optim is called with
method equal to opt.method.
opt.control: a list of control parameters which is passed on
either to nlminb or to optim, depending on the chosen
opt.method.
Adelchi Azzalini
The first of these notes applies to the stage preceding the use of selm and related fitting procedures. Before fitting a model of this sort, consider whether you have enough data for this task. In this respect, the passage below taken from p.63 of Azzalini and Capitanio (2014) is relevant.
“Before entering technical aspects, it is advisable to underline a qualitative effect of working with a parametric family which effectively is regulated by moments up to the third order. The implication is that the traditional rule of thumb by which a sample size is small up to ‘about \(n = 30\)’, and then starts to become ‘large’, while sensible for a normal population or other two-parameter distribution, is not really appropriate here. To give an indication of a new threshold is especially difficult, because the value of \(\alpha\) also has a role here. Under this caveat, numerical experience suggests that ‘about \(n = 50\)’ may be a more appropriate guideline in this context.”
The above passage referred to the univariate SN context. In the multivariate case, increase the sample size appropriately, especially so with the ST family. This is not to say that one cannot attempt fitting these models with small or moderate sample size. However, one must be aware of the implications and not be surprised if problems appear.
The second cautionary note refers instead to the outcome of a call to
selm and related function, or the lack of it.
The estimates are obtained by numerical optimization methods and, as
usual in similar cases, there is no guarantee that the maximum of the
objective function is achieved. Consideration of model simplicity
and of numerical experience indicate that models with SN error
terms generally produce more reliable results compared to those with
the ST family. Take into account that models involving a
traditional Student's \(t\) distribution with unknown degrees of freedom
can already be problematic; the presence of the (multivariate) slant parameter
\(\alpha\) in the ST family cannot make things any simpler.
Consequently, care must be exercised, especially so if one works with
the (multivariate) ST family.
Consider re-fitting a model with different starting values and,
in the ST case, building the profile log-likelihood for a range
of \(\nu\) values; function profile.selm can be useful here.
Details on the numerical optimization which has produced object obj
can be extracted with slot(obj, "opt.method"); inspection of this
component can be useful in problematic cases.
# Be aware that occasionally optim and nlminb declare successful
# completion of a regular minimization problem at a point where the Hessian
# matrix is not positive-definite.
By default, selm fits the selected model by maximum
likelihood estimation (MLE), making use of some numerical
optimization method. Maximization is performed in one
parameterization, usually DP, and then the estimates are mapped to
other parameter sets, CP and pseudo-CP;
see dp2cp for more information on parameterizations.
These parameter transformations are carried out trasparently to the user.
The observed information matrix is used to obtain the estimated variance
matrix of the MLE's and from this the standard errors.
Background information on MLE in the context of SEC
distributions is provided by Azzalini and Capitanio (2014);
see specifically Chapter 3, Sections 4.3, 5.2, 6.2.5--6. For additional
information, see the original research work referenced therein as well as
the sources quoted below.
Although the density functionof SEC distributions are expressed using
DP parameter sets, the methods associated to the objects created
by this function communicate, by default, their outcomes in the CP
parameter set, or its variant form pseudo-CP when CP
does not exist; the ‘Note’ at summary.selm explains why.
A more detailed discussion is provided by Azzalini and Capitanio
(1999, Section 5.2) and Arellano-Valle and Azzalini (2008, Section 4),
for the univariate and the multivariate SN case, respectively;
an abriged account is available in Sections 3.1.4--6 and 5.2.3 of
Azzalini and Capitanio (2014). For the ST case, see Arellano-Valle
and Azzalini (2013).
There is a known open issue which affects computation of the information
matrix of the multivariate skew-normal distribution when the slant
parameter \(\alpha\) approaches the null vector; see p.149 of
Azzalini and Capitanio (2014). Consequently, if a model with
multivariate response is fitted with family="SN" and the estimate
alpha of \(\alpha\) is at the origin or neary so, the
information matrix and the standard errors are not computed and a
warning message is issued. In this unusual circumstance, a simple
work-around is to re-fit the model with family="ST", which will
work except in remote cases when (i) the estimated degrees of freedom
nu diverge and (ii) still alpha remains at the origin.
The optional argument fixed.param=list(alpha=0) imposes the
constraint \(\alpha=0\) in the estimation process; in the multivariate
case, the expression is interpreted in the sense that all the components
of vector \(\alpha\) are zero, which implies symmetry of the
error distribution, irrespectively of the parameterization
subsequently adopted for summaries and diagnostics.
When this restriction is selected, the estimation method cannot be
set to "MPLE". Under the constraint \(\alpha=0\),
if family="SN", the model is fitted similarly to lm, except
that here MLE is used for estimation of the covariance matrix.
If family="ST" or family="SC", a symmetric Student's \(t\)
or Cauchy distribution is adopted.
Under the constraint \(\alpha=0\), the location parameter \(\xi\) coincides with the mode and the mean of the distribution, when the latter exists. In addition, when the covariance matrix of a ST distribution exists, it differs from \(\Omega\) only by a multiplicative factor. Consequently, the summaries of a model of this sort automatically adopt the DP parametrization.
The other possible form of constraint allows to fix the degrees of
freedom when family="ST". The two constraints can be combined
writing, for instance, fixed.param=list(alpha=0, nu=6).
The constraint nu=1 is equivalent to select family="SC".
In practice, an expression of type fixed.param=list(..) can be
abbreviated to fixed=list(..).
Argument start allows to set the initial values, with respect to the DP parameterization, of the numerical optimization. However, there is a specific choice of start to be avoided. When family="SN", do not set the shape parameter alpha exactly at 0, as this would blow-up computation of the log-likelihood gradient and the Hessian matrix. This is not due to a software bug, but to a known peculiar behaviour of the log-likelihood function at that specific point. Therefore, in the univariate case for instance, do not set e.g. start=c(12, 21, 0), but set instead something like start=c(12, 21, 0.01). Recall that, if one needs to fit a model forcing 0 asymmetry, typically to compare two log-likelihood functions with/without asymmetry, then the option to use is fixed.param=list(alpha=0).
Since version 1.6.0, a new initialization procedure has been introduced for the case family="ST", which adopts the method proposed by Azzalini & Salehi (2020), implemented in functions st.prelimFit and mst.prelimFit. Correspondingly, the start argument can now be of different type, namely a character with possible values "M0", "M2" (detault in the univariate case) and "M3" (detault in the multivariate case). The choice "M0" selects the older method, in use prior to version 1.6.0. For more information, see Azzalini & Salehi (2020).
In some cases, especially for small sample size, the MLE occurs on
the frontier of the parameter space, leading to DP estimates with
abs(alpha)=Inf or to a similar situation in the multivariate case
or in an alternative parameterization. Such outcome is regared by many as
unsatisfactory; surely it prevents using the observed information matrix to
compute standard errors. This problem motivates the use of maximum penalized
likelihood estimation (MPLE), where the regular log-likelihood
function \(\log~L\) is penalized by subtracting an amount
\(Q\), say, increasingly large as \(|\alpha|\) increases.
Hence the function which is maximized at the optimization stage is now
\(\log\,L~-~Q\). If method="MPLE" and
penalty=NULL, the default function Qpenalty is used,
which implements the penalization:
$$Q(\alpha) = c_1 \log(1 + c_2 \alpha_*^2)$$
where \(c_1\) and \(c_2\) are positive constants, which
depend on the degrees of freedom nu in the ST case,
$$\alpha_*^2 = \alpha^\top \bar\Omega \alpha$$
and \(\bar\Omega\) denotes the correlation matrix
associated to the scale matrix Omega described in connection with
makeSECdistr. In the univariate case
\(\bar\Omega=1\),
so that \(\alpha_*^2=\alpha^2\). Further information
on MPLE and this choice of the penalty function is given in
Section 3.1.8 and p.111 of Azzalini and Capitanio (2014); for a more
detailed account, see Azzalini and Arellano-Valle (2013) and references
therein.
It is possible to change the penalty function, to be declared via the
argument penalty. For instance, if the calling statement includes
penalty="anotherQ", the user must have defined
anotherQ <- function(alpha_etc, nu = NULL, der = 0)
with the following arguments.
alpha_etc: 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 equal to
cov2cor(Omega).
nu: degrees of freedom, only relevant if family="ST".
der: a numeric value which indicates the required order of
derivation; if der=0 (default value), only the penalty Q
needs to be retuned by the function;
if der=1, attr(Q, "der1") must represent the
first order derivative of Q with respect to alpha; if
der=2, also attr(Q, "der2") must be assigned, containing
the second derivative (only required in the univariate case).
This function must return a single numeric value, possibly with required
attributes when is called with der>1.
Since sn imports functions grad and
hessian from package numDeriv, one can rely
on them for numerical evaluation of the derivatives, if they are not
available in an explicit form.
This penalization scheme allows to introduce a prior distribution
\(\pi\) for \(\alpha\) by setting \(Q=-\log\pi\),
leading to a maximum a posteriori estimate in the stated sense.
See Qpenalty for more information and an illustration.
The actual computations are not performed within selm which only
sets-up ingredients for work of selm.fit and other functions
further below this one. See selm.fit for more information.
Arellano-Valle, R. B., and Azzalini, A. (2008). The centred parametrization for the multivariate skew-normal distribution. J. Multiv. Anal. 99, 1362--1382. Corrigendum: 100 (2009), 816.
Arellano-Valle, R. B., and Azzalini, A. (2013, available online 12 June 2011). The centred parametrization and related quantities for the skew-t distribution. J. Multiv. Anal. 113, 73--90.
Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J.Roy.Statist.Soc. B 61, 579--602. Full-length version available at https://arXiv.org/abs/0911.2093
Azzalini, A. and Arellano-Valle, R. B. (2013, available online 30 June 2012). Maximum penalized likelihood estimation for skew-normal and skew-t distributions. J. Stat. Planning & Inference 143, 419--433.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.
Azzalini, A. and Salehi, M. (2020). Some computational aspects of maximum likelihood estimation of the skew-t distribution. In Computational and Methodological Statistics and Biostatistics, edited by A. Bekker, Ding-Geng Chen and Johannes T. Ferreira, pp.3-28. Springer Nature Switzerland.
selm-class for classes "selm" and "mselm",
summary.selm for summaries, plot.selm for plots,
residuals.selm for residuals and fitted values
the generic functions coef, logLik,
vcov, profile, confint,
predict
the underlying function selm.fit and those further down
the selection of a penalty function of the log-likelihood,
such as Qpenalty
the function extractSECdistr to extract the SEC
error distribution from an object returned by selm
the broad underlying logic and a number of ingredients are like in
function lm
data(ais)
m1 <- selm(log(Fe) ~ BMI + LBM, family="SN", data=ais)
print(m1)
summary(m1)
s <- summary(m1, "DP", cov=TRUE, cor=TRUE)
plot(m1)
plot(m1, param.type="DP")
logLik(m1)
coef(m1)
coef(m1, "DP")
var <- vcov(m1)
#
m1a <- selm(log(Fe) ~ BMI + LBM, family="SN", method="MPLE", data=ais)
m1b <- selm(log(Fe) ~ BMI + LBM, family="ST", fixed.param=list(nu=8), data=ais)
#
data(barolo)
attach(barolo)
A75 <- (reseller=="A" & volume==75)
logPrice <- log(price[A75],10)
m <- selm(logPrice ~ 1, family="ST", opt.method="Nelder-Mead")
summary(m)
summary(m, "DP")
plot(m, which=2, col=4, main="Barolo log10(price)")
# cfr Figure 4.7 of Azzalini & Capitanio (2014), p.107
detach(barolo)
#-----
# examples with multivariate response
#
m3 <- selm(cbind(BMI, LBM) ~ WCC + RCC, family="SN", data=ais)
plot(m3, col=2, which=2)
summary(m3, "dp")
coef(m3)
coef(m3, vector=FALSE)
#
data(wines)
m28 <- selm(cbind(chloride, glycerol, magnesium) ~ 1, family="ST", data=wines)
dp28 <- coef(m28, "DP", vector=FALSE)
pcp28 <- coef(m28, "pseudo-CP", vector=FALSE)
# the next statement takes a little more time than others
plot(m28)
#
m4 <- selm(cbind(alcohol,sugar)~1, family="ST", data=wines)
m5 <- selm(cbind(alcohol,sugar)~1, family="ST", data=wines, fixed=list(alpha=0))
print(1 - pchisq(2*as.numeric(logLik(m4)-logLik(m5)), 2)) # test for symmetry
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