Chapter 9.5, Problem 61E

Refer to Exercise 9.49. Use Y_(n) to find an MVUE of θ. (See Example 9.1.)

9.49 Let $Y_1,Y_2,\dots ,Y_n$ denote a random sample from the uniform distribution over the interval (0, θ). Show that $Y_{\left(n\right)}=\max \left(Y_1,Y_2,\dots ,Y_n\right)$ is sufficient for θ.

EXAMPLE 9.1

Let $Y_1,Y_2,\dots ,Y_n$ denote a random sample from the uniform distribution over the interval (0, θ). Two unbiased estimators for θ are

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

where $Y_{\left(n\right)}=\max \left(Y_1,Y_2,\dots ,Y_n\right)$. Find the efficiency of ${\hat{\theta }}_1$, relative to ${\hat{\theta }}_2$.