13 Properties of Estimators and Tests

13.1 Comparing estimators

A normal distribution is symmetry about its mean, so that its mean and median coincide. The question therefore arises: Should be use the sample mean or the sample median to estimate that parameter? Both seem equally natural, but the mean corresponds to the MLE. We can run some simulations to compare these estimators. And indeed, the sample mean is somewhat better than the sample median (in terms of mean squared error).

If instead of considering a normal distribution we consider a double-exponential distribution, the situation is exactly reversed.

13.2 Uniformly most powerful tests

Consider testing \(Q_0\) versus \(Q_1\), two distributions on the same discrete space. The Neyman–Pearson lemma implies that any likelihood ratio test is most powerful at level equal to its size. In turn, computing a likelihood ratio test involves solving an integer program.

 [1] 1 1 0 0 0 0 0 0 0 0