# Order Statistics and Applications

DOI: 10.4018/978-1-5225-2255-3.ch162
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## Abstract

Order statistics refer to the collection of sample observations sorted in ascending order and are among the most fundamental tools in non-parametric statistics and inference. Statistical inference established based on order statistics assumes nothing stronger than continuity of the cumulative distribution function of the population and is simple and broadly applicable. We discuss how order statistics are applied in statistical analysis, e.g., tests of independence, tests of goodness of fit, hypothesis tests of equivalence of means, ranking and selection, and quantile estimation. These order-statistics techniques are key components of many studies.
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## Background

Let denote a sequence of mutually independent random samples from a common distribution of the continuous type having a probability density function (pdf) and a cumulative distribution function (cdf) F. Let be the th smallest of these such that . Then is called the th order statistic of the random sample Note that even though the samples are independently and identically distributed (iid), the order statistics are not independent because of the order restriction. The difference is called the sample range. It is a measure of the dispersion in the sample and should reflect the dispersion in the population.

Suppose , where “” denotes “is distributed as” and “” denotes a uniform distribution with range [a,b]. We are interested in the distribution of , which can be viewed as the second order statistic. The cdf of is

## Key Terms in this Chapter

Indifference Zone: We don’t distinguish values that are deviated less than a significant amount (the indifference amount), i.e., they are within in the indifference zone.

Tests of Independence: Statistical null hypothesis tests to determine whether sequences of observations appear to be independent.

Range Statistics: The statistics of the difference between the maximum and minimum observations. Range statistics give an estimate of the spread of the data.

Q-Q Plot: A probability plot for comparing two probability distributions by plotting their quantiles against each other.

Critical Constant: In ranking and selection, the critical constant is a required parameter for computing the required sample sizes and is the quantile of the difference of two specific random values.

Order Statistics: The collection of sample observations sorted in ascending order.

Empirical Distribution Function: The cumulative distribution function (cdf) associated with the empirical observations of the sample. The empirical distribution function estimates the true underlying cdf of the sample.

Histogram: A graphical representation showing the distribution of data.

Quantile Function (Inverse Distribution Function): The quantile function (given a probability p ) returns the value at or below which 100 p percent of the population lies.

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