Variational series

From Encyclopedia of Mathematics
Jump to: navigation, search

series of order statistics

An arrangement of the values of a random sample with distribution function in ascending sequence . The series is used to construct the empirical distribution function , where is the number of terms of the series which are smaller than . Important characteristics of series of order statistics are its extremal terms (, ) and the range . The densities of the distributions of the minimum and maximum terms of a series of order statistics in the case

are defined by the expressions


Considered as a stochastic process with time index , , the series of order statistics forms a non-homogeneous Markov chain.


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


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


[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Variational series. A.I. Shalyt (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098