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I think this is not an exponential family but does it mean that we can't find a sufficient statistic for $\theta$ if $X_1, X_2,..., X_n$ are a random sample from this density? $$ f_{\theta} (x) = \frac{6x(\theta - x)}{\theta^3} $$$$ 0<x<\theta$$

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  • $\begingroup$ If random variable $X$ has density $f(x) = 6x(1-x),$ for $0 < x < 1,$ then $X\sim\mathsf{Beta}(2,2).$ So your question is about a sample from a beta distribution modified ('generalized") to have support $(0, \theta).$ See Wikipedia on beta distributions or your text. $\endgroup$
    – BruceET
    Jun 29, 2021 at 15:32

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There are a few exceptions to the rule that only exponential families have sufficient statistics of fixed dimension (not depending on sample size $n$), like the uniform distribution family. In that case the parameter is the upper interval limit, and that is the case for your density also.

So it is better to look carefully at your case! I will only give a hint: Use the factorization theorem.

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