Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf

$p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\ 0 & \text{otherwise} % \end{cases}$

I would like to show that the statistic $T(x)=\max(x)$ is sufficient and complete. I've only previously shown sufficiency and completeness for distributions such as normal where one can reformulate the pmf and shown completeness through exponential family, so this one confuses me.

  • 2
    $\begingroup$ You might begin by working out the distribution function for $\max(x).$ $\endgroup$ – whuber Jun 5 '18 at 21:17

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