# Statistics

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

Let a random variable take values in the sample space . Any -measurable mapping from onto a measurable space is then called a statistic, and the probability distribution of the statistic is defined by the formula

### Examples.

1) Let be independent identically-distributed random variables which have a variance. The statistics

are then unbiased estimators for the mathematical expectation and the variance , respectively.

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

constructed from the observations , are statistics.

3) Let the random variables form a stationary stochastic process with spectral density . In this case the statistic

called the periodogram, is an asymptotically-unbiased estimator for , given certain specific conditions of regularity on , i.e.

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=14854