Definition:Statistic
Definition
A statistic is broadly defined as a quantity which is calculated from sample data.
Sample Statistic
A sample statistic is a numerical description of a sample.
Test Statistic
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 $\delta$.
Sufficient Statistic
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population indexed 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 $D$ is independent of the value of $\theta$ for all $t \in I$.
Also see
- Results about statistics can be found here.
Sources
- 1974: H.T. Hayslett, MS: Statistics Made Simple (3rd ed.) ... (previous) ... (next): Chapter $\text {I}$: What is Statistics?
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): statistic