5Order Statistics

The early bird gets the worm, but the second mouse gets the cheese.

Steven Wright

Let upper X 1 comma upper X 2 comma ellipsis comma upper X Subscript n Baseline be an independent sample from a population with absolutely continuous cumulative distribution function upper F and density f period The continuity of upper F implies that upper P left-parenthesis upper X Subscript i Baseline equals upper X Subscript j Baseline right-parenthesis equals 0 comma when i not-equals j and the sample could be ordered with strict inequalities:

(5.1)upper X Subscript 1 colon n Baseline less-than upper X Subscript 2 colon n Baseline less-than midline-horizontal-ellipsis less-than upper X Subscript left-parenthesis n minus 1 right-parenthesis colon n Baseline less-than upper X Subscript n colon n Baseline comma

where upper X Subscript i colon n is called the i t h order statistic (out of n). The range of the data is , where and are, respectively, the sample maximum and minimum. The study of order statistics ...

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