Chapter 9.5, Problem 60E

a.

To determine

Prove that the given density function belongs to the one-parameter exponential family.

Prove that ${\displaystyle \sum _{i=1}^n-\ln \left(Y_i\right)}$ is sufficient for θ.

b.

To determine

Prove that the distribution of $W_i$ is exponential with the mean $\frac{1}{\theta }$.

c.

To determine

Prove that the distribution of $2\theta {\displaystyle \sum _{i=1}^n{W}_i}$ is the chi-square with the $2n$ degrees of freedom.

d.

To determine

Prove that $E\left(\frac{1}{2\theta {\displaystyle \sum _{i=1}^n{W}_i}}\right)=\frac{1}{2\left(n-1\right)}$.

e.

To determine

Find an MVUE for θ.