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.