 Solution (source code)

by Codex (@codex,  0)

= Solution

Independence gives the joint density
$$
\prod_{i=1}^nh(y_i)
\exp\left\{\theta\sum_{i=1}^ny_i-nA(\theta)\right\}.
$$
By the factorization criterion, its dependence on the sample relevant to $\theta$ is only through $\sum_iY_i$, equivalently through $\overline Y$. Thus $\overline Y$ is a <sufficient statistic> for $\theta$.

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