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)