Chapter 9.6, Problem 78E

Let Y₁, Y₂,…, Y_n denote independent and identically distributed random variables from a power family distribution with parameters α and θ = 3. Then, as in Exercise 9.43, if α > 0,

$f(y|\alpha )=\{\begin{array}{ll}\alpha y^{a-1}/3^{\alpha },\hfill & 0\le y\le 3,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

Show that E(Y₁) = 3α/(α + 1) and derive the method-of-moments estimator for α.

9.43 Let Y₁, Y₂,…, Y_n denote independent and identically distributed random variables from a power family distribution with parameters α and θ. Then, by the result in Exercise 6.17, if α, θ > 0,

$f(y|\alpha ,\theta )=\{\begin{array}{ll}\alpha y^{a-1}/{\theta }^{\alpha },\hfill & 0\le y\le \theta ,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

If θ is known, show that ${\displaystyle {\prod }_{i=1}^n{Y}_i}$ is sufficient for α.