mbackcrit: Multivariate backward based on the AIC or BIC criteria
In skewMLRM: Estimation for Scale-Shape Mixtures of Skew-Normal Distributions

Description Usage Arguments Details Value Author(s) References Examples

mbackcrit implements the covariates selection based on backward and the Akaike's information criteria (AIC) or Schwartz's information criteria (BIC) in a specified multivariate model in the multivariate scale mixtures of normal (MSMN), multivariate scale mixtures of skew-normal (MSMSN), multivariate skew scale mixtures of normal (MSSMN) or multivariate scale mixtures of skew-normal-Cauchy (MSMSNC) classes. See details for avaliable distributions.

1 2	mbackcrit(y, X = NULL, max.iter = 1000, prec = 1e-04, dist = "MN", criteria = "AIC", est.var=TRUE, cluster = FALSE, ...)

`y`	The multivariate vector of responses. The univariate case also is supported.
`X`	The regressor matrix. It should include intercept term for all the variates.
`max.iter`	The maximum number of iterations.
`prec`	The convergence tolerance for parameters.
`dist`	the multivariate distribution in which the covariates selection will be implemented.
`criteria`	criteria used to perform the covariates selection. AIC (default) and BIC avaliable.
`est.var`	Logical. If TRUE the standard errors are estimated.
`cluster`	logical. If TRUE, parallel computing is used. FALSE is the default value.
`...`	Possible aditional arguments. For instance, for MSTT, MSSL2, MSTEC and MSSLEC distributions should be added nu.min and nu.fixed related to specifications for the nu parameter.

Supported models are:

In MSMN class: multivariate normal (MN), multivariate Student t (MT), multivariate slash (MSL), multivariate contaminated normal (MCN). See Lange and Sinsheimer (1993) for details.

In MSMSN class: multivariate skew-normal (MSN), multivariate skew-T (MSTT), multivariate skew-slash (MSSL2), multivariate skew-contaminated normal (MSCN2). See Zeller, Lachos and Vilca-Labra (2011) for details.

In MSSMN class: MSN, multivariate skew-t-normal (MSTN), multivariate skew-slash normal (MSSL), multivariate skew-contaminated normal (MSCN). See Louredo, Zeller and Ferreira (2021) for details.

In MSMSNC class: multivariate skew-normal-Cauchy (MSNC), multivariate skew-t-Expected-Cauchy (MSTEC), multivariate skew-slash-Expected-Cauchy (MSSLEC), multivariate skew-contaminated-Expected-Cauchy (MSCEC). See Kahrari et al. (2020) for details.

Note: the MSN distribution belongs to both, MSMSN and MSSMN classes.

an object of class "skewMLRM" is returned. The object returned for this functions is a list containing the following components:

`coefficients`	A named vector of coefficients
`se`	A named vector of the standard errors for the estimated coefficients. Valid if est.var is TRUE and the hessian matrix is invertible.
`logLik`	The log-likelihood function evaluated in the estimated parameters for the selected model
`AIC`	Akaike's Information Criterion for the selected model
`BIC`	Bayesian's Information Criterion for the selected model
`iterations`	the number of iterations until convergence (if attached)
`conv`	An integer code for the selected model. 0 indicates successful completion. 1 otherwise.
`dist`	The distribution for which was performed the estimation.
`class`	The class for which was performed the estimation.
`choose.crit`	the specified criteria to choose the distribution.
`comment`	A comment indicating how many coefficients were eliminated
`y`	The multivariate vector of responses. The univariate case also is supported.
`X`	The regressor matrix (in a list form).
`function`	a string with the name of the used function.

Clecio Ferreira, Diego Gallardo and Camila Zeller.

Kahrari, F., Arellano-Valle, R.B., Ferreira, C.S., Gallardo, D.I. (2020) Some Simulation/computation in multivariate linear models of scale mixtures of skew-normal-Cauchy distributions. Communications in Statistics - Simulation and Computation. In press. DOI: 10.1080/03610918.2020.1804582

Lange, K., Sinsheimer, J.S. (1993). Normal/independent distributions and their applications in robust regression. Journal of Computational and Graphical Statistics 2, 175-198.

Louredo, G.M.S., Zeller, C.B., Ferreira, C.S. (2021). Estimation and influence diagnostics for the multivariate linear regression models with skew scale mixtures of normal distributions. Sankhya B. In press. DOI: 10.1007/s13571-021-00257-y

Zeller, C.B., Lachos, V.H., Vilca-Labra, F.E. (2011). Local influence analysis for regression models with scale mixtures of skew-normal distributions. Journal of Applied Statistics 38, 343-368.

data(ais, package="sn") ##Australian Institute of Sport data set
attach(ais)
##It is considered a bivariate regression model
##with Hg and SSF as response variables and
##Hc, Fe, Bfat and LBM as covariates
y<-cbind(Hg,SSF)
n<-nrow(y); m<-ncol(y)
X.aux=model.matrix(~Hc+Fe+Bfat+LBM)
p<-ncol(X.aux)
X<-array(0,dim=c(2*p,m,n))
for(i in 1:n) {
    X[1:p,1,i]=X.aux[i,,drop=FALSE]
    X[p+1:p,2,i]=X.aux[i,,drop=FALSE]
}
##See the regressor matrix X
##X

##Perform covariates selection in the MN distribution
##based on the AIC criteria
##may take some time on some systems
fit.MN=mbackcrit(y, X, dist="MN")
summary(fit.MN)
##Identical process for MT distribution
fit.MT=mbackcrit(y, X, dist="MT")
summary(fit.MT)
##and for MSN distribution
fit.MSN=mbackcrit(y, X, dist="MSN")
summary(fit.MSN)