Question

Perform Monte-Carlo estimation where theta = E(e^U) where U is a continuous uniform random variable between zero and one a. Estimate !using 1000 data points and obtain a 95% confidence interval for this case. b. Perform part a) 100 times. Check how often the true !actually does fall within the 100 resulting confidence intervals. Please solve in R.

Answer 1

(a) We use 1000 data points from U[0,1] and estimated theta. The R-codes are given below:

U=runif(1000,0,1)
mean(exp(U))

and the output is:

[1] 1.724483.

To calculate the 95% confidence interval we use the function CI() and we need the package Rmisc.

R-codes are given below:

install.packages("Rmisc")
library(Rmisc)
CI(x, ci = 0.95)

and the output is:

upper mean lower
1.720706 1.717510 1.714314

(b) We now perform the process(a) to get 100 estimates for theta as:

x=replicate(100,mean(exp(runif(1000,0,1))));x

The histogram of x is:

Now to calculate the percentage of estimated values of theta falls in the 95%confidence interval, we use the codes:

length(which(x>1.714314&x<1.720706))

and the output is : 18.