Order Statistic

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



References

2016

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.