In statistics, a variational series is a non-decreasing sequence
composed from an initial series of independent and identically distributed random variables
The members of the variational series form order statistics, which form the basis for nonparametric statistical methods.
is called the kth order statistic, while the values
min
th order statistics, respectively) are referred to as the extremal terms.
[1] The sample range is given by
,[1] and the sample median by
is odd and
The variational series serves to construct the empirical distribution function
( x ) = μ ( x )
is the number of members of the series which are less than
The empirical distribution
serves as an estimate of the true distribution
of the random variables
, and according to the Glivenko–Cantelli theorem converges almost surely to
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