Variational Series

In statistics, a variational series is a non-decreasing sequence $X_ \leqslant X_ \leqslant \cdots \leqslant X_ \leqslant X_$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.

$X_$ is called the ''k''th order statistic, while the values $X_=\min_$ and $X_=\max_$ (the 1st and $n$th order statistics, respectively) are referred to as the extremal terms. The sample range is given by $R_n = X_-X_$, and the sample median by $X_$ when $n=2m+1$ is odd and $(X_ + X_)/2$ when $n=2m$ is even.

The variational series serves to construct the empirical distribution function $\hat(x) = \mu(x)/n$ , where $\mu(x)$ is the number of members of the series which are less than $x$. The empirical distribution $\hat(x)$ 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 $F(x)$.

References

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Nonparametric statistics