Sufficient statistic

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A statistic is sufficient (exhaustive) means that we can make a function $T$ that is not dependent on $\theta$. If we can write $f$ as

@ f(x, \theta) = g(T(x), \theta) * h(x) @

then $T$ is a sufficient statistic. Remember that $x$ is the vector of our distribution's values. In a lot of cases, we have

@ \mathbb{E}_\theta[T(x)] = \theta @

$T$ is not unique. If we have

@ \forall{T}\quad S = H(T) @

Then $S$ is a minimal sufficient statistic (statistique exhaustive minimale).