Variational series

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

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

The variational series serves to construct the empirical distribution function ${\displaystyle {\hat {F}}(x)=\mu (x)/n}$ , where ${\displaystyle \mu (x)}$ is the number of members of the series which are less than ${\displaystyle x}$. The empirical distribution ${\displaystyle {\hat {F}}(x)}$ serves as an estimate of the true distribution ${\displaystyle F(x)}$ of the random variables${\displaystyle X_{1},\ldots ,X_{n}}$, and according to the Glivenko–Cantelli theorem converges almost surely to ${\displaystyle F(x)}$.