Homework 7

Problem 1

Assume a Poisson($\mu$) model for the number of home runs hit (in total by both teams) in a MLB game. Let $X_1,\dots ,X_n$ be a random sample of home run counts for $n$ games.

Suppose we want to estimate $\theta =\mu e^{-\mu }$, the probability that any single game has exactly 1 HR (for Poisson($\mu$), $P(X=1)=e^{-\mu }{u}^1/1!=\mu e^{-\mu }$). Consider two estimators of $\theta$:

• $\hat{\theta }=\overline{X}{e}^{-\overline{X}}$
• $\hat{p}=\text{sample proportion of 1s}=\frac{\text{number of games in the sample with 1 HR}}{\text{sample size}}$

1. Compute the value of $\hat{\theta }$ based on the sample (3, 0, 1, 4, 0). Write a clearly worded sentence reporting in context this estimate of $\theta$.
2. Compute the value of $\hat{p}$ based on the sample (3, 0, 1, 4, 0). Write a clearly worded sentence reporting in context your estimate of $\theta$.
3. Which of these two estimators is the MLE of $\theta$ in this situation? Explain, without doing any calculations.
4. It can be shown that $\hat{p}$ is an unbiased estimator of $\theta$. Explain in words what this means.
5. Is $\hat{\theta }$ an unbiased estimator of $\theta$? Explain. (You don’t have to derive anything; just apply a general principle.)
6. Suppose $\mu =2.3$ and $n=5$. Explain in full detail how you would use simulation to approximate the bias of $\hat{\theta }$ in this case.
7. Coding required. Conduct the simulation from the previous part and approximate bias of $\hat{\theta }$ when $\mu =2.3$ and $n=5$.
8. Explain in full detail how you would use simulation to approximate the bias function of $\hat{\theta }$ when $n=5$.
9. Coding required. Conduct the simulation from the previous part and plot the approximate bias function when $n=5$. For what values of $\mu$ does $\hat{\theta }$ tend to overestimate $\mu$? Underestimate? For what values of $\mu$ is the bias the worst?

Problem 2

Continuing Problem 1.

1. It can be shown that $\text{Var}(\hat{p})=\frac{\theta (1-\theta )}{n}$. Compute $\text{Var}(\hat{p})$ when $\mu =2.3$ and $n=5$. Then write a clearly worded sentence interpreting this value.
2. Suppose $\mu =2.3$ and $n=5$. Explain in full detail how you would use simulation to approximate the variance of $\hat{\theta }$.
3. Coding required. Conduct the simulation from the previous part and approximate the variance of $\hat{\theta }$ when $\mu =2.3$ and $n=5$. Then write a clearly worded sentence interpreting this value.
4. Which estimator has smaller variance when $\mu =2.3$ (and $n=5$)? Answer, but then explain why this information alone is not really useful.
5. Explain in full detail how you would use simulation to approximate the variance function of $\hat{\theta }$ (if $n=5$).
6. Coding required. Conduct the simulation from the previous part and plot the approximate variance function. Compare to the variance function of $\hat{p}$. Based on variability alone, which estimator is preferred?

Problem 3

Continuing Problems 1 and 2

1. Compute $\text{MSE}(\hat{p})$ when $\mu =2.3$ and $n=5$. (You can do the next part first if you want, but it helps to work with specific numbers first.)
2. Derive the MSE function of $\hat{p}$. (Hint: use facts from previous parts.)
3. Suppose $\mu =2.3$ (and $n=5$). Explain in full detail how you would use simulation to approximate the MSE of $\hat{\theta }$.
4. Coding required. Conduct the simulation from the previous part and approximate the MSE of $\hat{\theta }$ when $\mu =2.3$ (and $n=5$).
5. Which estimator has smaller MSE when $\mu =2.3$ (and $n=5$)? Answer, but then explain why this information alone is not really useful.
6. Explain in full detail how you would use simulation to approximate the MSE function of $\hat{\theta }$ (if $n=5$).
7. Coding required. Conduct the simulation from the previous part and plot the approximate MSE function. Compare to the MSE function of $\hat{p}$.
8. Compare the MSEs of the two estimators for $n=5$ and a few other values of $n$. Is there a clear preference between these two estimators? Discuss.