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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
thsmallest value. Together with
rank statistics,
order statistics
are among the most fundamental tools in nonparametric 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 smallsamplesize 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.
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Order statistic  Wikipedia, the free encyclopedia
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where, following a common convention, we use uppercase letters to refer to random variables, and lowercase letters (as above) to refer to
their actual observed values.
Similarly, for a sample of size
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 Spring '08
 Staff
 Statistics

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