Problem Set 3

Question 1

Suppose that \(X_1, \ldots, X_n \stackrel{i.i.d.}{\sim} \text{Poisson}(\lambda)\). That is \[\begin{align*} p_X(x) & = \frac{\lambda^x e^{-\lambda}}{x!}, \end{align*}\]

for \(x \in \{0,1,2,\ldots\}\).

a. (10 points)

What is the Fisher information and observed Fisher information?

Solution:

b. (10 points)

For the remainder of the problem, let \(\lambda=4\). Simulate \(n=100,000\) observations from this distribution. Find the asymptotic 95% confidence interval for \(\lambda\).

Solution:

c. (10 points)

Draw \(B=1000\) bootstrap samples from your simulated data. What is the bootstrap standard deviation of \(\hat{\lambda}\)?

Solution:

d. (10 points)

Find a 95% confidence interval using the bootstrapped standard deviation.

Solution:

e. (10 points)

Find a 95% confidence interval using the bootstrapped empirical quantile method.

Solution:

f. (10 points)

Discuss the differences between the two bootstraps and the asymptotic confidence intervals.

Solution:

Question 2

a. (30 points)

Now drop all but the first 50 observations of your simulated data from question 1. Repeat parts (b) - (e).

Solution:

b. (10 points)

Discuss the differences with only 50 observations between the bootstrap and asymptotic intervals. What do you notice with a smaller sample size?

Solution: