RBE3 (version 1.1)

ML.BE3: Perform the parameter estimation for the Generalized beta distribution

Description

ML.BE3 computes the maximum likelihood estimates based on the maximum likelihood method.

Usage

ML.BE3(data, tau = 0.5, link.mu = "logit")

Value

a list containing the following elements

estimate

A matrix with the estimates

logLik

The maximum likelihood values attached by the estimates parameters

Arguments

data

a list containing the response vector (\(y\)), and the matrices to model \(\mu\), the \(\tau\)-quantile of distribution, and the shape parameters \(\alpha\) and \(\beta\), labeled as \(Z_1, Z_2\) and \(Z_3\), respectively.

tau

the quantile of the distribution to be modelled (\(0<\tau<1\)).

link.mu

link function to be used for \(\mu\): logit (default), probit, loglog or cloglog.

Author

Diego Gallardo and Marcelo Bourguignon.

Details

Covariates are included as \(g_1(\mu_i(\tau))=\mathbf{Z}_{1i}^\top {\bm \theta}(\tau)\), \(g_2(\alpha_i(\tau))=\mathbf{Z}_{2i}^\top {\bm \nu}(\tau)\) and \(g_3(\beta_i(\tau))=\mathbf{Z}_{3i}^\top {\bm \eta}(\tau)\), where \({\bm \theta}(\tau)=(\theta_1(\tau),\ldots,\theta_{r_1}(\tau))\), \({\bm \nu}(\tau)=(\nu_1(\tau),\ldots,\nu_{r_2}(\tau))\) and \({\bm \eta}(\tau)=(\eta_1(\tau),\ldots,\eta_{r_3}(\tau))\), where \(r_1, r_2\) and \(r_3\) are the dimensions of \(Z_1, Z_2\) and \(Z_3\), respectively. Initial values for \({\bm \theta}(\tau)\) are used as the coefficients for the linear regresion in \(\mbox{logit}(y_i)\) using the elements of \(\mathbf{Z}_{1i}^\top\) as regressors. Initial values for the other coefficients are considered as zeros.

References

Bourguignon, M., Gallardo, D.I., Saulo, H. (2023) A parametric quantile beta regression for modeling case fatality rates of COVID-19. Submitted.

Examples

# \donttest{
##Simulating two covariates
set.seed(2100)
x1<-rnorm(200); x2<-rbinom(200, size=1, prob=0.5)
##Desing matrices: Z1 includes x1 and x2, 
##Z2 includes only x1 and Z3 includes only x2
Z1=model.matrix(~x1+x2);Z2=model.matrix(~x1);Z3=model.matrix(~x2)
##Fixing parameters
theta=c(1, 0.2, -0.5); nu=c(0.5,-0.2); eta=c(-0.5, 0.3); tau=0.4
mu=plogis(Z1%*%theta); alpha=exp(Z2%*%nu); beta=exp(Z3%*%eta)
y=rBE3(200, mu, alpha, beta, tau=tau)
data=list(y=y, Z1=Z1, Z2=Z2, Z3=Z3)
ML.BE3(data, tau=tau)
# }