Order Statistics

Example 1: Continuous     Logistic distribution

Let random variable X have a Logistic distribution with pdf :

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Let denote a sample of size n drawn on X, and let denote the ordered sample, so that . The pdf of the r ^th order statistic, , is given by the mathStatica function:

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The following diagram plots the pdf of the r ^th order statistic, as r increases from 1 to 10, given a sample of size n = 10:

The joint pdf of and , for , is given by:

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Example 2: Discrete    Negative Binomial distribution

A new feature in mathStatica 1.2 is that the OrderStat function now also supports discrete random variables. To illustrate, let random variable with pmf :

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Let denote a random sample of size n drawn on X, and let denote the order statistics.

Then, the pmf of the r ^th order statistic, , denoted , is given immediately by:

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with domain of support:

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The following diagram plots the pmf of the r ^th order statistic that we have just derived, when , and r = 1 and the sample size is n = 10:

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