distMahal {skewMLRM}

Mahalanobis distance for fitted models in the MSMN, MSMSN, MSSMN and MSMSNC classes

Description

Compute and plot the Mahalanobis distance for 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

distMahal(object, alpha = 0.95, ...)

Arguments

object

an object of class "skewMLRM" returned by one of the following functions: estimate.xxx, choose.yyy, choose2, mbackcrit or mbacksign. See details for supported distributions.

alpha

significance level (0.05 by default).

...

aditional graphical parameters

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.

Value

distMahal provides an object of class skewMLRM related to compute the Mahalanobis distance for all the observations and a cut-off to detect possible influent observations based on the specified significance (0.05 by default).

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

Mahal

the Mahalanobis distance for all the observations

function

a string with the name of the used function.

dist

The distribution for which was performed the estimation.

class

The class for which was performed the estimation.

alpha

specified level of significance (0.05 by default).

cut

the cut-off to detect possible influent observations based on the specified significance.

y

The multivariate vector of responses. The univariate case also is supported.

X

The regressor matrix (in a list form).

Author(s)

Clecio Ferreira, Diego Gallardo and Camila Zeller

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

set.seed(2020)
n=200   # length of the sample
nv<-3   # number of explanatory variables
p<-nv+1 # nv + intercept
m<-4    # dimension of Y
q0=p*m
X<-array(0,c(q0,m,n)) 
for(i in 1:n) {
    aux=rep(1,p)
    aux[2:p]<-rMN(1,mu=rnorm(nv),Sigma=diag(nv)) ##simulating covariates
    mi=matrix(0,q0,m)
    for (j in 1:m) mi[((j-1)*p+1):(j*p),j]=aux
    X[,,i]<-mi
} ##X is the simulated regressor matrix
betas<-matrix(rnorm(q0),ncol=1) ##True betas
Sigmas <- clusterGeneration::genPositiveDefMat(m,rangeVar=c(1,3), 
lambdaLow=1, ratioLambda=3)$Sigma ##True Sigma
y=matrix(0,n,m)
for(i in 1:n) {
     mui<-t(X[,,i])%*%betas
     y[i,]<-rMN(n=1,c(mui),Sigmas) ## simulating the response vector 
}
fit.MN=estimate.MN(y,X)        #fit the MN model
mahal.MN=distMahal(fit.MN)     #compute the Mahalanobis distances for MN model
plot(mahal.MN)                 #plot the Mahalanobis distances for MN model
mahal.MN$Mahal                #presents the Malahanobis distances