Statistic

A statistic is some measurement taken directly from the sample. Formally, a statistic is a Measurable function $T:\mathcal{X}\to {\mathbb{R}}^k$. A statistic transforms the sample into some measurement of the sample. Notably, a statistic acts on the sample, but not the underlying distribution. And usually we want to use the statistic to infer something about the underlying distribution.

Example

The sample average $T(X_1,\dots ,X_n)=\frac{1}{n}{\sum }_{i=1}^n{X}_i$, is a statistic, which approximates the Mean of the underlying distribution.

First generalization. A statistic can be a measurable functional: $T:\mathcal{X}\to \mathcal{T}$, where $\mathcal{T}$ is some function space.

Example

The empirical CDF $T(X_1,\dots ,X_n)=\frac{1}{n}{\sum }_{i=1}^{n}1\{x\le X_i\}$, is a statistic, which is a function on $\mathbb{R}$ and approximates the CDF of the underlying distribution.

Second generalization. A statistic can be regarded as a Statistical Functional evaluated at the empirical distribution. Similarly, a statistical functional is some measurement taken from the underlying distribution. See Statistical Functional for more details.

Third generalization. This is a generalization of the interpretation. Due to its general definition, any output of a statistical procedure is also a statistic (if measurable), such as an estimator, test, predictor, etc.

Statistic is Random

Being a function of random variables, a statistic is also a Random Variable. Thus, if a statistic is an estimator of some function on the underlying distribution, we can discuss its convergence properties.