Order Statistic

An Order Statistic is a nonparametric statistic that depends on statistical sample size and ordering of the data.

• Example(s):
  • Minimum value of a statistical sample.
  • Maximum value of a statistical sample.
  • …
• Counter-Example(s):
  • L-Statistic.
• See: L-Statistic, Rankit, Fisher–Tippett Distribution, Bapat–Beg Theorem, Bernstein Polynomial, L-estimator, Rank-size Distribution, Selection Algorithm, Nonparametric Statistics.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Order_statistic Retrieved 2016-08-21
  • In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

2016

• (UAH, 2016) ⇒ http://www.math.uah.edu/stat/sample/OrderStatistics.html Retrieved 2016-08-21
  • Suppose that x is a real-valued variable for a population and that [math]\displaystyle{ x=(x_1,x_2,…,x_n) }[/math] are the observed values of a sample of size n corresponding to this variable. The order statistic of rank k is the k-th smallest value in the data set, and is usually denoted x(k). To emphasize the dependence on the sample size, another common notation is [math]\displaystyle{ x_{n:k} }[/math] Thus,

[math]\displaystyle{ x_{(1)}\leq x_{(2)}\leq \cdots \leq x_{(n−1)}\leq x_{(n)} }[/math]

Naturally, the underlying variable x should be at least at the ordinal level of measurement. The order statistics have the same physical units as x. One of the first steps in exploratory data analysis is to order the data, so order statistics occur naturally. In particular, note that the extreme order statistics are

[math]\displaystyle{ x_{(1)}=min\{x_1,x_2…,x_n\},\quad x_(n)=max{x_1,x_2,…,x_n} }[/math]

The sample range is [math]\displaystyle{ r=x_{(n)}−x_{(1)} }[/math] and the sample midrange is [math]\displaystyle{ r_2=1/2[x_{(n)}−x_{(1)}] }[/math]. These statistics have the same physical units as [math]\displaystyle{ x }[/math] and are measures of the dispersion of the data set.

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