Description Usage Arguments Value Details Background References See Also Examples
Computes Fisher information for parameters of simple sample having skewnormal (SN) or skewt (ST) distribution or for a regression model with errors term having such distributions, in the DP and CP parametrizations.
1 2 3 4 5 6 7 8 9  sn.infoUv(dp=NULL, cp=NULL, x=NULL, y, w, penalty=NULL, norm2.tol=1e06)
sn.infoMv(dp, x=NULL, y, w, penalty=NULL, norm2.tol=1e06)
st.infoUv(dp = NULL, cp = NULL, x = NULL, y, w, fixed.nu = NULL,
symmetr = FALSE, penalty = NULL, norm2.tol = 1e06)
st.infoMv(dp, x = NULL, y, w, fixed.nu = NULL, symmetr = FALSE,
penalty = NULL, norm2.tol = 1e06)

dp, cp 
direct or centred parameters, respectively; one of them
can be a non 
x 
an optional matrix which represents the design matrix of a regression model 
y 
a numeric vector (for 
w 
an optional vector of weights (only meaningful for the observed
information, hence if 
fixed.nu 
an optional numeric value which declares a fixed value of the
degrees of freedom, 
symmetr 
a logical flag which indicates whether a symmetry condition of
the distribution is being imposed; default is 
penalty 
a optional character string with the name of the penalty
function used in the call to 
norm2.tol 
for the observed information case, the Mahalanobis squared
distance of the score function from 0 is evaluated; if it exceeds

a list containing the following components:
dp, cp 
one of the two arguments is the one supplied on input; the other one matches the previous one in the alternative parametrization. 
type 
the type of information matrix: "observed" or "expected". 
info.dp, info.cp 
matrices of Fisher (observed or expected) information in the two parametrizations. 
asyvar.dp, asyvar.cp 
inverse matrices of Fisher information in the two parametrizations, when available; See ‘Details’ for additional information. 
aux 
a list containing auxiliary elements, depending of the selected function and the type of computation. 
In the univariate SN case, when x
is not set, then a simple
random sample is assumed and a matrix x
with a single column of all
1's is constructed; in this case, the supplied vector dp
or cp
must have length 3. If x
is set, then the supplied vector of parameters,
dp
or cp
, must have length ncol(x)+2
.
In the multivariate case, a direct extension of this scheme applies.
If the observed information matrix is required, dp
or cp
should
represent the maximum likelihood estimates (MLE) for the given y
,
otherwise the information matrix may fail to be positivedefinite and it
would be meaningless anyway.
Therefore, the squared Mahalobis norm of the score vector is evaluated and compared with norm2.tol
.
If it exceeds this threshold, this is taken as an indication that the supplied
parameter list is not at the MLE and a warning message is issued.
The returned list still includes info.dp
and info.cp
, but in
this case these represent merely the matrices of second derivatives;
asyvar.dp
and asyvar.cp
are set to NULL
.
The information matrix for the the univariate SN distribution in the two stated parameterizations in discussed in Sections 3.1.3–4 of Azzalini and Capitanio (2014). For the multivariate distribution, Section 5.2.2 of this monograph summarizes briefly the findings of ArellanoValle and Azzalini (2008).
For ST distributions, only the observed information matrix is provided, at the moment. Computation for the univariate case is based on DiCiccio and Monti (2011). For the multivariate case, the score function is computed using an expression of ArellanoValle (2010) followed by numerical differentiation.
ArellanoValle, R. B. (2010). The information matrix of the multivariate skewt distribution. Metron, LXVIII, 371–386.
ArellanoValle, R. B., and Azzalini, A. (2008). The centred parametrization for the multivariate skewnormal distribution. J. Multiv. Anal. 99, 1362–1382. Corrigendum: 100 (2009), 816.
Azzalini, A. with the collaboration of Capitanio, A. (2014). The SkewNormal and Related Families. Cambridge University Press, IMS Monographs series.
DiCiccio, T. J. and Monti, A. C. (2011). Inferential aspects of the skew tdistribution. Quaderni di Statistica 13, 1–21.
1 2 3 4 5 6 7 8 9 10  infoE < sn.infoUv(dp=c(0,1,5)) # expected information
set.seed(1); rnd < rsn(100, dp=c(0, 1, 3))
fit < selm(rnd~1, family="SN")
infoO < sn.infoUv(cp=coef(fit), y=rnd) # observed information
#
data(wines)
X < model.matrix(~ pH + wine, data=wines)
fit < sn.mple(x=X, y=wines$alcohol)
infoE < sn.infoUv(cp=fit$cp, x=X)
infoO < sn.infoUv(cp=fit$cp, x=X, y=wines$alcohol)

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