Suppose we have data $X = X_1,\ldots,X_n$, $Y = Y_1,\ldots,Y_n$ that is i.i.d. generated by a distribution $\mathbb{P}_\theta$. Let $T$ be a test statistic such that that $T(X) = T(Y)$ if and only if $\mathbb{P}_\theta [X] = \mathbb{P}_\theta [Y]$ for all possible values of $\theta$.

Question: is $T$ a minimal sufficient statistic? It is clear that it is a sufficient statistic. But it is not clear to me that it's minimal sufficient.


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