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# syn.nbc.r
# Table 10.9: Hilbe, Negative Binomial Regression, 2 ed, Cambridge Univ Press
# with assistance of: Andrew Robinson, University of Melbourne, Australia
# Synthetic NB-C estimated using optimiation
#
library(MASS)
nobs <- 50000
x2 <- runif(nobs)
x1 <- runif(nobs)
a <- 1.15 # value of alpha: 1.15
xb <- 1.25*x1 + .1*x2 - 1.5
mu <- 1/((exp(-xb)-1)*a)
p <- 1/(1+a*mu)
r <- 1/a
nbcy <- rnbinom(50000, size=r, prob = p)
nbc.reg.ml <- function(b.hat, X, y) {
a.hat <- b.hat[1]
xb.hat <- X %*% b.hat[-1]
mu.hat <- 1 / ((exp(-xb.hat)-1)*a.hat)
p.hat <- 1 / (1 + a.hat*mu.hat)
r.hat <- 1 / a.hat
sum(dnbinom(y,
size = r.hat,
prob = p.hat,
log = TRUE))
}
nbcX <- cbind(1, x1, x2)
p.0 <- c(a.hat = 1,
b.0 = -2,
b.1 = 1,
b.2 = 1)
fit <- optim(p.0, ## Maximize the JCLL
nbc.reg.ml,
X = nbcX,
y = nbcy,
control = list(fnscale = -1),
hessian = TRUE
)
fit$par ## ML estimates
sqrt(diag(solve(-fit$hessian))) ## SEs
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