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Order statistic - Wikipedia, the free encyclopedia
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Order statistic
From Wikipedia, the free encyclopedia
In statistics, the
k
th
order statistic
of a statistical sample is equal its
k
th-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 analyse 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.
Contents
1 Notation and examples
2 Probabilistic analysis
2.1 Distribution of each order statistic of an absolutely continuous distribution
2.2 Probability distributions of order statistics
2.2.1 The order statistics of the uniform distribution
2.2.2 Joint distributions
3 Application: confidence intervals for quantiles
3.1 Estimating the median
3.1.1 A small-sample-size example
3.1.2 Large sample sizes
3.2 Estimating quantiles
4 Dealing with discrete variables
5 Computing order statistics
6 See also
7 External links
Notation and examples
For example, suppose that four numbers are observed or recorded, resulting in a sample of size
n
= 4. if the sample values are
6, 9, 3, 8,
they will usually be denoted
where the subscript
i
in
x
i
indicates simply the order in which the observations were recorded and is usually assumed not to be significant.
A case when the order is significant is when the observations are part of a time series.
The order statistics would be denoted
where the subscript (
i
) enclosed in parentheses indicates the
i
th order statistic of the sample.
The
first order statistic
(or
smallest order statistic
) is always the
minimum
of the sample, that is,
Probability distributions for the
n
= 5 order statistics of an
exponential distribution with
θ
=
3.

10/12/2006 06:21 PM
Order statistic - Wikipedia, the free encyclopedia
Page 2 of 5
where, following a common convention, we use upper-case letters to refer to random variables, and lower-case letters (as above) to refer to
their actual observed values.
Similarly, for a sample of size

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