# Parametric bootstrapping for a binomial distribution
# We have data from a binomial(8,theta) distribution with theta unkwnow.
# We find a bootstrap confidence interval for theta

#Size of binomial
binomSize = 8 

#Data 
x = c(6, 5, 5, 5, 7, 4)
n = length(x)

# Estimate of theta from the data (This is the MLE)
thetahat = mean(x)/binomSize 

# 1. Draw 100 parametric samles using thetahat as the parameter
# 2. Compute thetastar and deltastar (dstar) for each one. Store dstar in a list
nboot = 100
dstar.list = rep(0,nboot)
for (j in 1:nboot)
{
xstar = rbinom(n,binomSize,thetahat)
thetastar = mean(xstar)/binomSize
dstar.list[j] = thetastar - thetahat
}

# Compute the alpha/2 and 1-alpha/2 quantiles and make the confidence interval
alpha = .2
dstar_alpha2 = quantile(dstar.list, alpha/2, names=FALSE)
dstar_1minusalpha2 = quantile(dstar.list, 1-alpha/2, names=FALSE)
CI = thetahat - c(dstar_1minusalpha2,  dstar_alpha2)

# Display the results
print(CI)