Category:Definitions/Sufficient Statistics

This category contains definitions related to Sufficient Statistics.
Related results can be found in Category:Sufficient Statistics.

Definition $1$

Let $T$ be a sample statistic such that $T$ contains all the information in a random sample that is relevant to the point estimation of a particular parameter $\theta$.

Then $T$ is known as a sufficient statistic for $\theta$.

Definition $2$

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population whose probability distribution is determined by a parameter $\theta$.

Let $T$ be a sample statistic.

Let $I = \Img {\map T {X_1, X_2, \ldots, X_n} }$.

Let $D$ be the conditional joint distribution of $X_1, X_2, \ldots, X_n$ given $T = t$ and $\theta$.

We call $T$ a sufficient statistic for $\theta$ if and only if $D$ is independent of the value of $\theta$ for all $t \in I$.