Statistics

A term used in mathematical statistics as a name for functions of the results of observations.

Let a random variable $X$ take values in the sample space $(\mathfrak{X},\mathcal{B},{\mathsf{P}}^X)$. Any $\mathcal{B}$- measurable mapping $T(\cdot )$ from $\mathfrak{X}$ onto a measurable space $(\mathfrak{Y},\mathcal{A})$ is then called a statistic, and the probability distribution of the statistic $T$ is defined by the formula

$${\mathsf{P}}^T\{B\}=\mathsf{P}\{T(X)\in B\}=\mathsf{P}\{X\in T^{-1}(B)\}=$$

$$={\mathsf{P}}^X\{T^{-1}(B)\}(\mathrm{\forall }B\in \mathcal{A}).$$

Examples.[edit]

1) Let $X_1\dots X_n$ be independent identically-distributed random variables which have a variance. The statistics

$$\bar{X}=\frac{1}{n}\sum _{i=1}^n{X}_i\text{ and }{s}^2=\frac{1}{(}n-1)\sum _{i=1}^n(X_i-\bar{X}{)}^2$$

are then unbiased estimators for the mathematical expectation $\mathsf{E}{X}_1$ and the variance $\mathsf{D}{X}_1$, respectively.

2) The terms of the variational series (series of order statistics, cf. Order statistic)

$$X_{(}1)\le \cdots \le X_{(}n),$$

constructed from the observations $X_1\dots X_n$, are statistics.

3) Let the random variables $X_1\dots X_n$ form a stationary stochastic process with spectral density $f(\cdot )$. In this case the statistic

$$I_n(\lambda )=\frac{1}{2\pi n}{\left|\sum _{k=1}^n{X}_k{e}^{-ik\lambda }\right|}^2,\lambda \in [-\pi ,\pi ],$$

called the periodogram, is an asymptotically-unbiased estimator for $f(\cdot )$, given certain specific conditions of regularity on $f(\cdot )$, i.e.

$$\underset{n\to \mathrm{\infty }}{lim}\mathsf{E}{I}_n(\lambda )=f(\lambda ),\lambda \in [-\pi ,\pi ].$$

In the theory of estimation and statistical hypotheses testing, great importance is attached to the concept of a sufficient statistic, which brings about a reduction of data without any loss of information on the (parametric) family of distributions under consideration.

References[edit]