Problem 1
Let \(X_1\), \(X_2\), \(X_3\), … , \(X_n\) be i.i.d exponential random variables
with mean \(\theta\).
- One possible estimator for \(\theta\) is \(\overline X\). Recall that the variance of
\(X_i\) is \(\theta^2\), so we can estimate \(Var(X_i)\) as \((\overline X) ^2\). By using these
estimators for the mean and variance, write down the asymptotic (i.e.,
large sample) confidence interval for \(\theta\) with the confidence level \(1 - \alpha\). (Note that you should use
\(\overline X ^2\) as the variance
estimator, not \(S^2\).)
Solution: \[
\Big( \ldots \Big),
\]
- Set \(n = 4\), \(\theta = 2\), \(\alpha = 0.1\). Generate a vector \(x\) that contains the random sample \(X_1\), … , \(X_n\) and calculate the bounds of the
asymptotic confidence interval which you wrote in 1. (Be careful with
the generation of exponential r.v.s in R, since R functions for these
random variables use parameter “rate” = \(1/\theta\).) Does this interval contain
\(\theta\)?
Solution:
set.seed(123)
# n =
# theta =
# alpha =
# x = rexp(...)
#now calculate the lower and upper bounds of the confidence interval (lb, ub):
# lb = ...
# lb
# ub = ...
# ub
Comment: \(\ldots\)
- Repeat the calculation that you programmed for 2. for \(N = 10,000\) times and record whether the
CI contained \(\theta\) in a vector of
length \(N\). What is the proportion of
the cases when \(\theta\) lies inside
the interval? Comment on your results: What can you say about the
performance of the asymptotic CI? Why does this happen?
Solution:
N = 10000
y = rep(0, N) #pre-fill y with zeros.
#for (i in 1:N){
# ... #generate a sample and calculate the lower and upper bounds of the CI.
# if (... && ...){ #if theta is between bounds, set y[i] to 1.
# y[i] = ...
# }
#}
# finally, calculate the mean of y
# ...
Comment: \(\ldots\)
- Change \(n\) to \(n = 40\) and repeat 3. Comment on your
results.
Solution:
#n = ...
#y = rep(...)
#for (...){
# ....
# if (...){
# y[i] = ...
# }
#}
# calculate the mean of y
# ...
Comment: \(\ldots\)