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.
References
[1] | S.S. Wilks, "Mathematical statistics" , Wiley (1962) |
Comments
The phrase "variational series" is almost never used in the West. Cf. also Order statistic.
References
[a1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
Variational series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variational_series&oldid=53729