In the lecture, Keith said inadmissible estimators \(\approx\) useless. Here I will discuss some reasons why we may still use an inadmissible estimator.
Automatic nature/wide applicability
Example 3.18 shows that the MLE of \(\sigma^2\) is inadmissible. However, the principle of likelihood is widely applicable and we can use it easily when there is a new problem; see Appendix D. Admissibility is a sound concept but it does not automatically construct an estimator for us.
Computational efficiency
Admissibility concerns (only) the statistical efficiency of an estimator or estimation procedure. Nevertheless, the time to compute/update an estimate matters more in some applications, e.g., convergence diagnostics.
Other properties
Admissible estimators may not have some desirable theoretical properties. For example, we may want a variance estimator guaranteed to be non-negative.
If I omit a question, that usually means the hints are sufficient so you can just refer to the Remark in the assignment. I do not discuss bonus question in general.
Apart from using Example 3.13, you can also use Example 2.14.
When does \(\hat{\theta}_1 = \bar{x}\)?
Derive \(R_1(\theta, c+d\bar{x})\) stated in Remark 3.1.5 first. You can express \(\hat{\theta}_1\) in form of \(c+d\bar{x}\). This result is also useful in Ex3.1.8.
If you are going to use data related to other entities in the given dataset, please be careful of any systematic difference such as population and epidemiological situation. For example,
data = read.csv("covid.csv")
colnames(data) = c("entity","date","case","death")
data[which(data$entity=="Taiwan"), "death"]
# 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
data[which(data$entity=="United States"), "death"]
# 1995 3750 3890 3950 4037 3251 1824 2006 4466 3964
# 3930 3805 3352 1751 1405 2774 4377 4203 3760 3329
# 1775 1917 4084 3943 4001 3603 2731 1794 2031 3389