skewMLRM (version 1.7)

FIM: Observed Fisher information matrix for distributions in the MSMN, MSMSN, MSSMN and MSMSNC classes.

Description

FI.xxx computes the observed Fisher information (FI) matrix 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

FI.MN(P, y, X)
FI.MT(P, y, X)
FI.MSL(P, y, X)
FI.MCN(P, y, X)
FI.MSN(P, y, X)
FI.MSTN(P, y, X)
FI.MSSL(P, y, X)
FI.MSCN(P, y, X)
FI.MSTT(P, y, X, nu)
FI.MSSL2(P, y, X, nu)
FI.MSCN2(P, y, X, nu, gamma)
FI.MSNC(P, y, X)
FI.MSTEC(P, y, X, nu)
FI.MSSLEC(P, y, X, nu)
FI.MSCEC(P, y, X, nu, gamma)

Value

A matrix with the observed FI matrix for the specified model.

Arguments

P

the estimated parameters returned by a function of the form estimate.xxx, where xxx is a supported distribution.

y

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

X

The regressor matrix.

nu

nu parameter. Only for MSTT, MSSL2, MSTEC, MSSLEC and MSCEC distributions.

gamma

gamma parameter. Only for MSCN2 and MSCEC distributions.

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

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))
    mi=matrix(0,q0,m)
    for (j in 1:m) mi[((j-1)*p+1):(j*p),j]=aux
    X[,,i]<-mi
} #Simulated matrix covariates
betas<-matrix(rnorm(q0),ncol=1) ## True betas
Sigmas <- clusterGeneration::genPositiveDefMat(m,rangeVar=c(1,3), 
lambdaLow=1, ratioLambda=3)$Sigma ##True Sigma
lambda<-rnorm(m) ##True lambda
y=matrix(0,n,m)
for(i in 1:n) {
     mui<-t(X[,,i])%*%betas
     y[i,]<-rMSN(n=1,c(mui),Sigmas,lambda)}
# \donttest{
fit.MSN=estimate.MSN(y,X) ##Estimate parameters for MSN model
fit.MSN  ## Output of estimate.MSN
summary(fit.MSN)
fit.MSN$se ##Estimated standard errors by the estimate.MSN function 
##Estimated standard errors by minus the square root of 
##the diagonal from the observed FI matrix of the MSN model 
sqrt(diag(solve(-FI.MSN(fit.MSN$coefficients, y, X))))# }