Chapter 9.2, Problem 4E

Let Y₁, Y₂, … , Y_n denote a random sample of size n from a uniform distribution on the interval (0, θ). If Y₍₁₎ = min(Y₁, Y₂, … , Y_n), the result of Exercise 8.18 is that ${\hat{\theta }}_1=(n+1)Y_{(1)}$ is an unbiased estimator for θ. If Y_(n) = max(Y₁, Y₂, … , Y_n), the results of Example 9.1 imply that ${\hat{\theta }}_2=[(n+1)/n]Y_{(n)}$ is another unbiased estimator for θ. Show that the efficiency of ${\hat{\theta }}_1$ to ${\hat{\theta }}_2$ is 1/n². Notice that this implies that ${\hat{\theta }}_2$ is a markedly superior estimator.