Definition:Sufficient Statistic

Definition

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 $\theta$.

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

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 $t \in I$.

Let $S$ be a sample from $N$.

Then the sample mean $\overline x$ is a sufficient statistic for estimation of $\mu$.

This is because knowledge of the individual sample values provides no further information about $\mu$.

This in turn is because the distribution of $S$, conditional upon $\overline x$, is independent of the population mean.

The concept of a sufficient statistic was introduced by Ronald Aylmer Fisher in $1921$.