Variational series

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

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

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