Chapter 9.3, Problem 15E

Refer to Exercise 9.3. Show that both ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are consistent estimators for θ.

9.3 Let Y₁, Y₂,…, Y_n denote a random sample from the uniform distribution on the interval (θ, θ + 1). Let

${\hat{\theta }}_1=\bar{Y}-\frac{1}{2}\text{ and }{\hat{\theta }}_2=Y_{\left(n\right)}-\frac{n}{n+1}.$

1. a Show that both ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are unbiased estimators of θ.
2. b Find the efficiency of ${\hat{\theta }}_1$ relative to ${\hat{\theta }}_2$.