Definition:Statistic

Definition

A statistic is broadly defined as a quantity which is calculated from sample data.

A sample statistic is a numerical description of a sample.

Let $\theta$ be a population parameter of some population $P$.

Let $\Omega$ be the parameter space of $\theta$.

Let $\mathbf X$ be a random sample from $P$.

Let $T = \map f {\mathbf X}$ be a sample statistic.

Let $\delta$ be a test procedure of the form:

reject $H_0$ if $T \in C$

for some null hypothesis $H_0$ and some $C \subset \Omega$.

We refer to $T$ as the test statistic of $\delta$.

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population indexed 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 $D$ is independent of the value of $\theta$ for all $t \in I$.