Fbetaf function

Inverse of the Fisher information matrix of the fixed and random effects in Model 1

This function calculates the inverse of the Fisher information matrix of the fixed effects (beta) and the random effects (u) and the score vectors S.beta and S.u, for the model with one independent random effect in each category of the response variable (Model 1). modelfit1 uses the output of this function to estimate the fixed and random effects by the PQL method.

Fbetaf(sigmap, X, Z, phi, y, mu, u)

Arguments

• sigmap: a list with the model variance-covariance matrices for each domain.
• X: list of matrices with the auxiliary variables obtained from data.mme. The dimension of the list is the number of categories of the response variable.
• Z: design matrix of random effects.
• phi: vector with the values of the variance components obtained from modelfit1.
• y: matrix with the response variable except the reference category. The rows are the domains and the columns are the categories of the response variable minus one.
• mu: matrix with the estimated mean of the response variable obtained from prmu.
• u: matrix with the values of random effects obtained from modelfit1.

Returns

A list containing the following components. - F.beta.beta: the first diagonal element of the inverse of the Fisher information matrix.

• F.beta.u: the element out of the diagonal of the inverse of the Fisher information matrix.

• F.u.u: the second diagonal element of the inverse of the Fisher information matrix.

• S.beta: beta scores.

• S.u: u scores.

Examples

k=3 #number of categories of the response variable
pp=c(1,1) #vector with the number of auxiliary variables in each category
data(simdata) #data
mod=1 #type of model
datar=data.mme(simdata,k,pp,mod)
initial=datar$initial
mean=prmu(datar$n,datar$Xk,initial$beta.0,initial$u.0)
sigmap=wmatrix(datar$n,mean$estimated.probabilities)

#Inverse of the Fisher information matrix
Fisher=Fbetaf(sigmap,datar$X,datar$Z,initial$phi.0,datar$y[,1:(k-1)],
       mean$mean,initial$u.0)

References

Lopez-Vizcaino, ME, Lombardia, MJ and Morales, D (2013). Multinomial-based small area estimation of labour force indicators. Statistical Modelling, 13 ,153-178.

See Also

data.mme, initial.values, wmatrix, phi.mult, prmu, phi.direct, sPhikf, ci, modelfit1, msef, mseb.