# est
nll<-function(p)-dbinom(2,10,p, log=TRUE)
est<-nlminb(0.5, nll, lower=0, upper=1)$par

# test in Chisq
p=.5
G <- -2*(dbinom(2,10,p, log=TRUE)-dbinom(2,10,est, log=TRUE))
1-pchisq(G,1)

# test by sim
P=.5
G.obs <- -2*(dbinom(2,10,p, log=TRUE)-dbinom(2,10,est, log=TRUE))
sim <- rbinom(1000000, 10,p)  
G.sim <- -2*(dbinom(sim,10,p, log=TRUE)-dbinom(sim,10,sim/10, log=TRUE))
mean(G.sim>=G.obs)

# CI by approx
pval.chi<-function(p){
  G <- -2*(dbinom(2,10,p, log=TRUE)-dbinom(2,10,est, log=TRUE))
  1-pchisq(G,1)
}
pval.chi<-Vectorize(pval.chi)
par(mfrow=c(3,1))
plot(pval.chi, 0, .6)
abline(h=0.05, col="gray", lwd=3)
nll<-function(p)-(dbinom(2,10,p,log=TRUE)-dbinom(2,10,est, log=TRUE))
plot(nll,0,.6)
abline(h=qchisq(0.95,1)/2, col="gray", lwd=3)
low<-uniroot(function(p)nll(p)-qchisq(0.95,1)/2, interval=c(0, est))$root
hig<-uniroot(function(p)nll(p)-qchisq(0.95,1)/2, interval=c(est,1))$root
abline(v=c(low,hig), col="red")

# CI by sim
pval.one <- function(p){
  G.obs <- -2*(dbinom(2,10,p, log=TRUE)-dbinom(2,10,2/10, log=TRUE))
  set.seed(12345)
  sim <- rbinom(100000, 10,p)  
  G.sim <- -2*(dbinom(sim,10,p, log=TRUE)-dbinom(sim,10,sim/10, log=TRUE))
  mean(G.sim>=G.obs)
}
pval<-Vectorize(pval.one)
plot(pval, 0,.6)
abline(h=0.05, lwd=3, col="gray")
low<-uniroot(function(p)pval(p)-0.05, interval=c(0, est))$root
hig<-uniroot(function(p)pval(p)-0.05, interval=c(est,1))$root
abline(v=c(low,hig), col="red")