estimateM: Fitting a model in the MSMN, MSMSN, MSSMN and MSMSNC classes

Description

estimate.Mxxx computes the maximum likelihood estimates for the distribution xxx, where xxx is any supported 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 supported distributions.

Usage

estimate.MN(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE)
estimate.MT(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.min = 2.0001)
estimate.MSL(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.min = 2.0001)
estimate.MCN(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE)
estimate.MSN(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE)
estimate.MSTN(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.min = 2.0001)
estimate.MSSL(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.min = 2.0001)
estimate.MSCN(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE)
estimate.MSTT(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.fixed = 3, nu.min = 2.0001)
estimate.MSSL2(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
     nu.fixed = 3, nu.min = 2.0001)
estimate.MSCN2(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
      nu.fixed = 0.5, gamma.fixed = 0.5)
estimate.MSNC(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE)
estimate.MSTEC(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
      nu.fixed = 3, nu.min = 2.0001)
estimate.MSSLEC(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
      nu.fixed = 3, nu.min = 2.0001)
estimate.MSCEC(y, X, max.iter = 1000, prec = 1e-04, est.var = TRUE, 
      nu.fixed = 0.5, gamma.fixed = 0.5)

Value

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.
nu: The estimated or fixed nu (only for MSTT, MSSL2, MSCN2, MSTEC, MSSLEC and MSCEC models)
gamma: The estimated or fixed gamma (only for MSCN2 and MSCEC models)
logLik: The log-likelihood function evaluated in the estimated parameters
AIC: Akaike's Information Criterion
BIC: Bayesian's Information Criterion
iterations: the number of iterations until convergence (if attached)
time: execution time in seconds
conv: An integer code. 0 indicates successful completion. 1 otherwise.
dist: The distribution for which was performed the estimation.
class: The class for which was performed the estimation.
n: The sample size
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.

Arguments

y: The multivariate vector of responses. The univariate case also is supported.
X: The regressor matrix.
max.iter: The maximum number of iterations.
prec: The convergence tolerance for parameters.
est.var: Logical. If TRUE the standard errors are estimated.
nu.fixed: If a numerical value is provided, the estimation consider nu as fixed. To estimate nu, use nu.fixed=FALSE. Avaliable for MSTT, MSSL2, MSCN2, MSTEC, MSSLEC and MSCEC distributions. For MSTT, MSSL2, MSTEC and MSSLEC, the default value is 3 and nu should be greater than 1. For MSCN2 and MSCEC, the default value is 0.5 and nu should be in the (0,1) interval.
gamma.fixed: If a numerical value is provided, the estimation consider gamma as fixed. To estimate gamma, use gamma.fixed=FALSE. Avaliable for MSCN2 and MSCEC distributions. For MSCN2 and MSCEC, the default value is 0.5 and gamma should be in the (0,1) interval.
nu.min: Lower value to perform the maximization for nu. For MSTT, MSSL2, MSTEC and MSSLEC is used only when nu.fixed=FALSE.

Author

Clecio Ferreira, Diego Gallardo and Camila Zeller

Details

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.

References

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.

Examples

Run this code

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 covariate matrix X
##X
fit.MN=estimate.MN(y, X) ##Estimate the parameters for the MN regression model
summary(fit.MN)
fit.MT=estimate.MT(y, X) ##Estimate the parameters for the MT regression model
summary(fit.MT)
# \donttest{
##may take some time on some systems
fit.MSSL=estimate.MSSL(y, X)   ##Estimate the parameters for the MSSL regression model
summary(fit.MSSL)
fit.MSTT=estimate.MSTT(y, X)   ##Estimate the parameters for the MSTT regression model
summary(fit.MSTT)
fit.MSNC=estimate.MSNC(y, X)   ##Estimate the parameters for the MSNC regression model
summary(fit.MSNC)
fit.MSCEC=estimate.MSCEC(y, X) ##Estimate the parameters for the MSCEC regression model
summary(fit.MSCEC)
# }

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