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Is there any example of curved exponential family with $T$ being a complete statistic? Here $T$ is the sufficient statistic.

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  • $\begingroup$ "Curved exponential families may arise when the parameters of an exponential family satisfy constraints. For these families the minimal sufficient statistic may not be complete, and UMVU estimation may not be possible." R. Keener $\endgroup$
    – Xi'an
    Jan 18 '21 at 20:05
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    $\begingroup$ Keener gives the example of two Normal samples with both variances being equal, in which case there exists a complete sufficient statistic. $\endgroup$
    – Xi'an
    Jan 18 '21 at 20:07
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Besides the bivariate Normal example, Keener gives an exercise in relation with the question:

2. When $X\sim \mathcal B(p)$ and $Y\sim\mathcal B(h(p))$, the joint distribution of $(X,Y)$ is curved except for a specific function $h_0(p)$. Give two functions $h(p)$ for which $(X,Y)$ is minimal sufficient but not complete and for which $(X,Y)$ is complete.

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    $\begingroup$ One solution to this problem is $h(p)=cp$, then $(X,Y)$ is not complete. When $h(p)=p^c$, $(X,Y)$ is complete. Here $c$ is a non-zero constant. Other solutions are also possible. $\endgroup$
    – Tan
    Jan 18 '21 at 21:25

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