Statistic T(X) is a function defined on the sample X = (X_1, \cdots, X_n) with values in \mathbf{R}^k with k \le n
Statistic T(X) is a sufficient statistic for \theta if the conditional distribution function of X given T(X) does not depend on \theta
For some joint p.d. X of i.i.d. we wish to consider P_\theta (X = x | T(X) = T(x))
Since \{X = x\} is a subset of all events \{T(X) = T(x)\} then P_\theta (X = x | T(X) = T(x)) = \frac{P_\theta (X = x)}{P_\theta (T(X) = T(x))}
Let p(x|\theta) be a joint pdf (or pmf) of X. A statistic T(x) is sufficient if we can find
p(x|\theta) = h(x)g(T(x) | \theta)
A function of any other sufficient statistic. Has the smallest dimensionality.
If
(\frac{p(x|\theta)}{p(y|\theta)} \text{ does not depend on } \theta) \iff (T(x) = T(y))
Then T(X) is the minimal sufficient statistic.
Distribution belongs to the exponential family if its pdf can be expressed as:
p(x|\theta) = h(x)c(\theta)\exp(\sum_{j=1}^k \omega_j(\theta)t_j(x))
If i.i.d. joint distributions belong to the exponential family then
T(X) = (\sum_{i = 1}^n t_1(X_i), \cdots, \sum_{i = 1}^n t_k(X_i))^T
is a sufficient statistic for \theta