= Solution
A <sufficient statistic> has a conditional distribution of the full data given its value that is independent of the unknown parameter. A <minimal sufficient statistic> is sufficient and is a function of every other <sufficient statistic>, up to null sets. For a dominated family with common positive support, the <likelihood-ratio criterion for minimal sufficiency> states that $T$ is minimal sufficient if $T(x)=T(y)$ holds exactly when $p_\theta(x)/p_\theta(y)$ is independent of $\theta$.
For $S=\sum_iX_i$, the joint <probability mass function> is
$$
p_\theta(x)=\theta^{-n}(1-1/\theta)^{S-n},\qquad x_i\ge1.
$$
The <Fisher-Neyman factorization theorem> proves sufficiency of $S$. For two samples, the likelihood ratio is $(1-1/\theta)^{S(x)-S(y)}$, independent of $\theta>1$ exactly when their sums agree. Thus $S$ is minimal sufficient. Any one-to-one transformation of it is also minimal sufficient, so choose
$$
\boxed{T=\overline X=\frac Sn,\qquad\mathbb E_\theta T=\theta}.
$$
This chosen <estimator> is an <unbiased estimator>, since each observation has mean $\theta$. If instead the statistic is reported as $S$, its expectation is $n\theta$, and as an estimator of $\theta$ it has <bias> $(n-1)\theta$ (zero only when $n=1$). Being a <minimal sufficient statistic> is unchanged by this rescaling; being an <unbiased estimator> is not.
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