sn (version 1.2-2)

sn-st.info: Expected and observed Fisher information for SN and ST distributions

Description

Computes Fisher information for parameters of simple sample having skew-normal (SN) or skew-$t$ (ST) distribution or for a regression model with errors term having such distributions, in the DP and CP parametrizations.

Usage

sn.infoUv(dp=NULL, cp=NULL, x=NULL, y, w, penalty=NULL,  norm2.tol=1e-06) 
    
sn.infoMv(dp, x=NULL, y, w, penalty=NULL, norm2.tol=1e-06)
st.infoUv(dp = NULL, cp = NULL, x = NULL, y, w, fixed.nu = NULL, 
    symmetr = FALSE, penalty = NULL, norm2.tol = 1e-06) 
st.infoMv(dp, x = NULL, y, w, fixed.nu = NULL, symmetr = FALSE, 
    penalty = NULL, norm2.tol = 1e-06)

Arguments

dp, cp

direct or centred parameters, respectively; one of the two vectors must be supplied, but not both. For the univariate SN distribution, sn.infoUv is to be used, and these arguments are vectors. In the multivariate case

x

an optional matrix which represents the design matrix of a regression model

y

a numeric vector (for sn.infoUv and st.infoUv) or a matrix (for sn.infoMv and st.infoMv) representing the response. In the SN case ( sn.infoUv and sn.infoM

w

an optional vector of weights; if missing, a vector of 1's is generated.

fixed.nu

an optional numeric value which declared a fixed value of the degrees of freedom, nu. If not NULL, the information matrix has a dimension reduced by 1.

symmetr

a logical flag which indicates whether a symmetry condition of the distribution is being imposed; default is symmetr=FALSE.

penalty

a optional character string with the name of the penalty function used in the call to selm; see this function for its description;

norm2.tol

for the observed information case, the Mahalanobis squared distance of the score 0 is evaluated; if it exceeds norm2.tol, a warning message is issued, since the information matrix so evaluated may be not positive-def

Value

• 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.

Details

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 dp should represent the maximum likelihood estimates (MLE) for the given y, otherwise the information matrix may fail to be positive-definite. Therefore, the squared Mahalobis norm of the score vector is evaluated and compared with norm2.tol. If it exceeds this threshold, it is taken as an indication that the parameter 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.

Background

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 Arellano-Valle and Azzalini (2008). For ST distributions, only the observed information matrix is provided currently. Computation for the univariate case is based on DiCiccio and Monti (2011). For the multivariate case, the score function is computed using expression of Arellano-Valle (2010) followed by numerical differentiation.

References

Arellano-Valle, R. B. (2010). The information matrix of the multivariate skew-$t$ distribution. Metron, LXVIII, 371--386.

Arellano-Valle, R. B., and Azzalini, A. (2008). The centred parametrization for the multivariate skew-normal distribution. J. Multiv. Anal. 99, 1362--1382. Corrigendum: vol.,100 (2009), p.,816.

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series. DiCiccio, T. J. and Monti, A. C. (2011). Inferential aspects of the skew $t$-distribution. Quaderni di Statistica 13, 1--21.

See Also

dsn, dmsn, dp2cp

Examples

infoE <- sn.infoUv(dp=c(0,1,5))
infoO <- sn.infoUv(cp=c(0,1,0.8), y=rsn(50, dp=c(0,1,5)))
#
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)