Distributions of Continuous Order Statistics

Let , ..., be a random sample from a continuous distribution. Reorder the sample in increasing order; denote the corresponding variables by , ..., . Thus, for example, is the smallest of the variables, the second smallest, and the largest. The variable is called the order statistic. The Demonstration shows the probabilities of the order statistics (the red curves) when the sample is from a uniform, beta, exponential, gamma, normal, or extreme value distribution (the probability density function of is shown in blue).

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Snapshot 1: The data has the uniform distribution in the interval ; see the blue curve. The smallest, middle, and largest result are, with high probability, in the intervals, say, , , and, respectively. The expectations of the smallest, middle, and largest result are 0.25, 0.5, and 0.75, respectively.
Snapshot 2: The data has the exponential distribution with expectation 1; see the blue curve. The smallest, middle, and largest result are, with high probability, at most, say, 1, 2, and 4, respectively. The expectations of the smallest, middle, and largest result are 0.33, 0.83, and 1.83, respectively.
Snapshot 3: The data has the standard normal distribution with expectation 0 and standard deviation 1; see the blue curve. The smallest, middle, and largest result are, with high probability, in the intervals, say, , , and , respectively. The expectations of the smallest, middle, and largest result are -0.85, 0.0, and 0.85, respectively.
The other three distributions considered in this Demonstration are the beta distribution with parameters 3 and 2, the gamma distribution with parameters 3 and 2, and the extreme value distribution with parameters 0 and 2.
Let the probability density function and the cumulative distribution function of the data variable be and , respectively. The probability density function of the order statistics is then [1, p. 10]
.
The expectations of the order statistics are calculated in the traditional way, by integrating the product of and the density function.
The order statistics for several continuous distributions are considered in [1, Chapter 4].
Reference
[1] B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics, Philadelphia: SIAM, 2008.
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