attach(faithful) #Make data available to the R session
#Define the negative log-likelihood
nllhood=function(theta,y) {
  p=theta[1];mu=theta[2];sigma=theta[3];nu=theta[4];tau=theta[5]
  lhood=p*dnorm(y,mu,sigma)+(1-p)*dnorm(y,nu,tau)
  return(-sum(log(lhood))) }

#Use start values inferred from histogram
Faithful.fit=optim(c(0.4,52,5,80,5),nllhood,y=waiting,hessian=T)
MLE=Faithful.fit$par
ObsInfo=Faithful.fit$hess #Observed Fisher information matrix
Vhat=solve(ObsInfo) #Inverse of observed Fisher information 
Std.Errors=sqrt(diag(Vhat))
#Obtain the MLEs,estimated std errors, and approx Wald 95% CIs
Wald.table=cbind(MLE,
              Std.Errors,LowerBound=MLE-qnorm(0.975)*Std.Errors,
              UpperBound=MLE+qnorm(0.975)*Std.Errors)

parnames=c("p","mu","sigma","nu","tau")
rownames(Wald.table)=parnames
round(Wald.table,4) #Print to 4 decimal places.