distribution, log-normal Poisson mean (Panel 4.5)

`panel4pt5.fn` <-
function(y=catbird.y,J=11,Nmax=120){
# Computes MLE of Royle-Nichols model with over-dispersed abundance
# distribution. In this case, N is Poisson with mean lambda and
# lambda has a log-normal distribution.
 
 lik<-function(parms){
  r<-expit(parms[1])
  mu<-parms[2]
  sigma<-exp(parms[3])
  uy<-unique(y)
  il<-rep(NA,Nmax+1)
  for(k in 0:Nmax){
  il[k+1]<-integrate( 
  function(x){
    dpois(k,exp(x))*dnorm(x,mu,sigma) 
   },
    lower=-Inf,upper=Inf)$value
   }

 il2<-rep(NA,J+1)
 for(k in uy){
    il2[k+1]<- sum(dbinom(k,J,1-(1-r)^(0:Nmax))*il)
 }

 -1*sum(log(il2[y+1] ))
}

nsites<-length(y)
nlm(lik,c(0,0,0),hessian=TRUE)

}