Exercise Solution 4.2

1. The first estimator is based upon the fact that, out of 50 realizations drawn from a U(0,θ) distribution, we would expect the largest of those realizations to fall near the upper bound θ. The second estimator is based upon the fact that the mean of a U(0,θ) distribution is θ/2. Accordingly, multiplying the sample mean by 2 provides an estimate of θ.
2. H₁ estimates θ as 14.66. H₂ estimates θ as 13.84.
3. No. Estimator H₁ produced an estimate that is clearly impossible. Since realization 14.66 was drawn from the interval (0,θ), it is impossible that θ equals 13.84.
4. H₁ is biased. It will always underestimate θ. H₂ is unbiased.

This exercise illustrates the tradeoffs we often face in selecting a statistical estimator. In this case, estimator H₁ is biased. It will always underestimate θ. On the other hand, estimator H₂ is unbiased, but it can produce estimates that are clearly incorrect.