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In my statistics class we've been going over exponential families and sufficiency, which deviates from what's in the textbook. As such, now that I need to solve problems about exponential families I don't have examples and explanations in the textbook to fall back on, and unfortunately the examples we go over in class are simple and not as complex as the problems I want to solve now.

My question is: is a power family distribution with parameters $\alpha$ and $\theta$ where $f(y|\alpha, \theta)=\frac{\alpha}{\theta^{\alpha}}y^{\alpha-1}$ for $0\leq y\leq\theta$, an exponential family?

I had thought that because the support $0\leq y\leq \theta$ depended on $\theta$ it couldn't be an exponential family, but a classmate told me this was incorrect. Could somebody tell me how I can get this pdf into the form $f(y, \theta)=h(y)c(\theta)e^{\sum w_i(\theta)t_i(y)}$ with the support containing $\theta$?

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You're correct that this family of distributions with varying $\theta$ is not an exponential family; you can't factor the condition $y > \theta$ into something dependent only on $y$ and something only on $\theta$.

If you hold $\theta$ fixed, the resulting family with the single parameter $\alpha$ then is an exponential family (for any single $\theta$). This is analogous to the case for, say, a binomial distribution.

The Wikipedia article discusses this issue a bit.

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