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 $$F(x)$$.