Sufficient Statistics

A statistic $t$ is called a sufficient statistic for $θ$ for a given $y$ if:

$p (y | t, θ) = p (y | t)$

Let $Y_{i} \sim Bernoulli (θ)$ for $i = 1, \dots, n$ , and $T = \sum_{i = 1}^{n} Y_{i}$ . Then it can be shown that $t = \sum_{i = 1}^{n} y_{i}$ is a sufficient statistic for $θ$ given $y = (y_{1}, \dots, y_{n})$ .

Fisher-Neyman Theorem

The Fisher-Neyman theorem, or the factorization theorem, helps us find sufficient statistics more readily. It states that:

A statistic $t$ is sufficient for $θ$ if and only if there are functions $f$ and $g$ such that:

$p (y | θ) = f (t, θ) g (y)$

where $t = t (y)$ .