# Variational series

*series of order statistics*

An arrangement of the values of a random sample $ ( x _ {1} \dots x _ {n} ) $ with distribution function $ F( x) $ in ascending sequence $ x _ {(} 1) \leq \dots \leq x _ {(} n) $. The series is used to construct the empirical distribution function $ {F _ {n} } ( x) = {m _ {x} } /n $, where $ m _ {x} $ is the number of terms of the series which are smaller than $ x $. Important characteristics of series of order statistics are its extremal terms ( $ x _ {(} 1) = \min _ {1 \leq i \leq n } x _ {i} $, $ x _ {(} n) = \max _ {1 \leq i \leq n } x _ {i} $) and the range $ R _ {n} = {x _ {(} n) } - {x _ {(} 1) } $. The densities of the distributions of the minimum and maximum terms of a series of order statistics in the case

$$ F ( x) = \int\limits _ {- \infty } ^ { x } p ( y) dy $$

are defined by the expressions

$$ p _ {(} 1) ( x) = n [ 1 - F ( x)] ^ {n - 1 } p ( x) $$

and

$$ p _ {(} n) ( x) = nF ^ { n - 1 } ( x) p( x). $$

Considered as a stochastic process with time index $ i $, $ i = 1 \dots n $, the series of order statistics forms a non-homogeneous Markov chain.

#### Comments

The phrase "variational series" is almost never used in the West. Cf. also Order statistic.

#### References

[1] | S.S. Wilks, "Mathematical statistics" , Wiley (1962) |

[a1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |

**How to Cite This Entry:**

Variational series.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Variational_series&oldid=53729