order-statistics-notes - 10/12/2006 06:21 PM Order...

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10/12/2006 06:21 PM Order statistic - Wikipedia, the free encyclopedia Page 1 of 5 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.
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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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