# NMST 434: Exercise session 1 # February 21, 2020 # Example 6 # Higher order delta theorem B = 10^4 # no of replications n = 10^4 # no of observations p = .25 # true p, compare cases when p!=1/2 and p=1/2 X = rbinom(B,n,p)/n # B independent realisations of Bi(n,p)/n = bar(X) X = X*(1-X) # B independent realisations of hat(theta) op = par(mfrow=c(1,2)) Y<-sqrt(n)*(X-p*(1-p)) avar = p*(1-p)*(1-2*p)^2 range = c(min(min(Y),-sqrt(avar)*3),max(max(Y),sqrt(avar)*3)) hist(Y,prob=TRUE,main="Usual Delta-Theorem",xlim=range) abline(v=0,lty=2,lwd=2,col=2) abline(v=c(1,-1)*sqrt(avar)*qnorm(.975),lty=3,lwd=2) curve(dnorm(x,sd=sqrt(avar)),range[1],range[2],add=TRUE,n=1001,lwd=2,col=1) legend("topleft",c("as. mean", "as. density", "conf. int."),lty=c(2,1,3),col=c(2,1,1),lwd=2) hist(Y<--n/(p*(1-p))*(X-p*(1-p)),prob=TRUE,main="Second Order Delta-Theorem") abline(v=1,lty=2,lwd=2,col=2) curve(dchisq(x,df=1),min(Y),max(Y),add=TRUE,n=1001,lwd=2) legend("topright",c("as. mean", "as. density"),lty=c(2,1),col=c(2,1),lwd=2) par(op)