Minimal sufficient statistic

Remarks

Example: for $X_1, \dots, X_n$ i.i.d. Bernoulli($p$), $p \in (0, 1)$, the sum $T = \sum_i X_i$ is minimal sufficient. Indeed, the likelihood ratio $P_p(X = x)/P_p(X = y) = (p/(1 - p))^{\sum_i x_i - \sum_i y_i}$ is free of $p$ only when $\sum_i x_i = \sum_i y_i$, and by the criterion below this makes $T$ minimal. Non-example: for $n \ge 2$, the whole sample $X$ is sufficient but not minimal. Clause 2 fails with $U = T$: $X$ is not a function of $T$, since $(1, 0, \dots, 0)$ and $(0, 1, 0, \dots, 0)$ have the same sum. A common criterion, for a model with densities $f_\theta$: if $T(x) = T(y)$ holds exactly when $f_\theta(x)/f_\theta(y)$ does not depend on $\theta$, then $T$ is minimal sufficient. A minimal sufficient statistic is unique up to one-to-one measurable functions.