Variational series

In statistics, a variational series is a non-decreasing sequence formula_1composed from an initial series of independent and identically distributed random variables formula_2. The members of the variational series form order statistics, which form the basis for nonparametric statistical methods.
formula_3 is called the "k"th order statistic, while the values formula_4 and formula_5 (the 1st and formula_6th order statistics, respectively) are referred to as the extremal terms. The sample range is given by formula_7, and the sample median by formula_8 when formula_9 is odd and formula_10 when formula_11 is even.
The variational series serves to construct the empirical distribution function formula_12 , where formula_13 is the number of members of the series which are less than formula_14. The empirical distribution formula_15 serves as an estimate of the true distribution formula_16 of the random variablesformula_2, and according to the Glivenko–Cantelli theorem converges almost surely to formula_16.