Variational series

In statistics, a variational series is a non-decreasing sequence [math]\displaystyle{ X_{(1)} \leqslant X_{(2)} \leqslant \cdots \leqslant X_{(n-1)} \leqslant X_{(n)} }[/math]composed from an initial series of independent and identically distributed random variables [math]\displaystyle{ X_1,\ldots,X_n }[/math]. The members of the variational series form order statistics, which form the basis for nonparametric statistical methods.

[math]\displaystyle{ X_{(k)} }[/math] is called the kth order statistic, while the values [math]\displaystyle{ X_{(1)}=\min_{1 \leq k \leq n}{X_k} }[/math] and [math]\displaystyle{ X_{(n)}=\max_{1 \leq k \leq n}{X_k} }[/math] (the 1st and [math]\displaystyle{ n }[/math]th order statistics, respectively) are referred to as the extremal terms. The sample range is given by [math]\displaystyle{ R_n = X_{(n)}-X_{(1)} }[/math], and the sample median by [math]\displaystyle{ X_{(m+1)} }[/math] when [math]\displaystyle{ n=2m+1 }[/math] is odd and [math]\displaystyle{ (X_{(m+1)} + X_{(m)})/2 }[/math] when [math]\displaystyle{ n=2m }[/math] is even.

The variational series serves to construct the empirical distribution function [math]\displaystyle{ \hat{F}(x) = \mu(x)/n }[/math] , where [math]\displaystyle{ \mu(x) }[/math] is the number of members of the series which are less than [math]\displaystyle{ x }[/math]. The empirical distribution [math]\displaystyle{ \hat{F}(x) }[/math] serves as an estimate of the true distribution [math]\displaystyle{ F(x) }[/math] of the random variables[math]\displaystyle{ X_1,\ldots,X_n }[/math], and according to the Glivenko–Cantelli theorem converges almost surely to [math]\displaystyle{ F(x) }[/math].