Sufficient statistic

A statistic $T$ is sufficient for a parameter when the conditional distribution of the full data given $T$ is independent of that parameter.

A sufficient statistic is complete when every integrable function of it with expectation zero for every parameter is almost surely zero for every parameter. Full regular one-parameter exponential families with natural parameter ranging over an open interval provide a standard completeness result. Completeness makes an unbiased function of a sufficient statistic unique, giving the Lehmann–Scheffé theorem.

For independent observations $X_i\sim N(\mu ,{\sigma }^2)$ with known variance, ${\sum }_i{X}_i$ is minimal sufficient for $\mu$. Factorization proves sufficiency, while the likelihood ratio for two samples is independent of $\mu$ exactly when their sums agree.

A sufficient statistic is minimal sufficient when it is a function of every other sufficient statistic, up to null sets. Its level sets form the coarsest sufficient partition of the sample space.

For independent samples uniform on $[-\theta ,\theta ]$ with $\theta >0$, the absolute maximum $M$ is a minimal sufficient statistic, since the likelihood is $(2\theta {)}^{-n}{1}_{\{\theta \ge M\}}$. The pair of sample minimum and maximum is a sufficient statistic but is not minimal: simultaneous sign reversal preserves $M$ and generally changes that pair.

For a dominated family with positive densities $p_{\theta }$, a statistic $T$ is minimal sufficient if

A statistic is sufficient exactly when the likelihood factors into a parameter-dependent function of the statistic times a parameter-free function of the sample.