Chapter 9.4, Problem 44E

Let Y₁, Y₂, … , Y_n denote independent and identically distributed random variables from a Pareto distribution with parameters α and β. Then, by the result in Exercise 6.18, if α, β > 0,

$f(y|\alpha ,\beta )=\{\begin{array}{ll}\alpha {\beta }^{\alpha }{y}^{-\left(\alpha +1\right)},\hfill & y\ge \beta ,\hfill \\ 0,\hfill & \text{elsewhere}\text{.}\hfill \end{array}$

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