Chapter 9.4, Problem 48E

Refer to Exercise 9.44. If β is known, show that the Pareto distribution is in the exponential family. What is a sufficient statistic for α? Argue that there is no contradiction between your answer to this exercise and the answer you found in Exercise 9.44.

9.44 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,

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

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