# 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

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

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

### Examples.

1) Let $X _ {1} \dots X _ {n}$ be independent identically-distributed random variables which have a variance. The statistics

$$\overline{X}\; = \frac{1}{n} \sum _ {i=1} ^ { n } X _ {i} \ \textrm{ and } \ \ s ^ {2} = \frac{1}{(} n- 1) \sum _{i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2}$$

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)

$$X _ {(} 1) \leq \dots \leq X _ {(} n) ,$$

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.

$$\lim\limits _ {n \rightarrow \infty } {\mathsf E} I _ {n} ( \lambda ) = \ f( \lambda ),\ \ \lambda \in [- \pi , \pi ].$$

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

 [1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1988) [2] V.G. Voinov, M.S. Nikulin, "Unbiased estimates and their applications" , Moscow (1989) (In Russian)
How to Cite This Entry:
Statistics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistics&oldid=54901
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article